cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A000204 Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.

Original entry on oeis.org

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803, 141422324
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

See A000032 for the version beginning 2, 1, 3, 4, 7, ...
Also called Schoute's accessory series (see Jean, 1984). - N. J. A. Sloane, Jun 08 2011
L(n) is the number of matchings in a cycle on n vertices: L(4)=7 because the matchings in a square with edges a, b, c, d (labeled consecutively) are the empty set, a, b, c, d, ac and bd. - Emeric Deutsch, Jun 18 2001
This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1..m - 1, a(m) = m + 1. The generating function is (x + m*x^m)/(1 - x - x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n - 1 - (m - 1)*i, i - 1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m = 2: A000204, m = 3: A001609, m = 4: A014097, m = 5: A058368, m = 6: A058367, m = 7: A058366, m = 8: A058365, m = 9: A058364.
L(n) is the number of points of period n in the golden mean shift. The number of orbits of length n in the golden mean shift is given by the n-th term of the sequence A006206. - Thomas Ward, Mar 13 2001
Row sums of A029635 are 1, 1, 3, 4, 7, ... - Paul Barry, Jan 30 2005
a(n) counts circular n-bit strings with no repeated 1's. E.g., for a(5): 00000 00001 00010 00100 00101 01000 01001 01010 10000 10010 10100. Note #{0...} = fib(n+1), #{1...} = fib(n-1), #{000..., 001..., 100...} = a(n-1), #{010..., 101...} = a(n-2). - Len Smiley, Oct 14 2001
Row sums of the triangle in A182579. - Reinhard Zumkeller, May 07 2012
If p is prime then L(p) == 1 (mod p). L(2^k) == -1 (mod 2^(k+1)) for k = 0,1,2,... - Thomas Ordowski, Sep 25 2013
Satisfies Benford's law [Brown-Duncan, 1970; Berger-Hill, 2017]. - N. J. A. Sloane, Feb 08 2017

Examples

			G.f. = x + 3*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 18*x^6 + 29*x^7 + 47*x^8 + ...

References

  • P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 69.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
  • Leonhard Euler, Introductio in analysin infinitorum (1748), sections 216 and 229.
  • G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 148.
  • Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
  • V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
  • R. V. Jean, Mathematical Approach to Pattern and Form in Plant Growth, Wiley, 1984. See p. 5. - N. J. A. Sloane, Jun 08 2011
  • Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

Links

Crossrefs

Cf. A000032, A000045, A061084, A027960, A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A006206, A101033, A101032, A100492, A099731, A094216, A094638, A000108, A090946 (complement).

Programs

  • Haskell
    a000204 n = a000204_list !! n
a000204_list = 1 : 3 : zipWith (+) a000204_list (tail a000204_list)
-- Reinhard Zumkeller, Dec 18 2011
  • Magma
    [Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 17 2017
  • Maple
    A000204 := proc(n) option remember; if n <=2 then 2*n-1; else procname(n-1)+procname(n-2); fi; end;
with(combinat): A000204 := n->fibonacci(n+1)+fibonacci(n-1);
# alternative Maple program:
L:= n-> (<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1]:
seq(L(n), n=1..50);  # Alois P. Heinz, Jul 25 2008
# Alternative:
a := n -> `if`(n=1, 1, `if`(n=2, 3, hypergeom([(1-n)/2, -n/2], [1-n], -4))):
seq(simplify(a(n)), n=1..39); # Peter Luschny, Sep 03 2019
  • Mathematica
    c = (1 + Sqrt[5])/2; Table[Expand[c^n + (1 - c)^n], {n, 30}] (* Artur Jasinski, Oct 05 2008 *)
Table[LucasL[n, 1], {n, 36}] (* Zerinvary Lajos, Jul 09 2009 *)
LinearRecurrence[{1, 1},{1, 3}, 50] (* Sture Sjöstedt, Nov 28 2011 *)
lukeNum[n_] := If[n < 1, 0, LucasL[n]]; (* Michael Somos, May 18 2015 *)
lukeNum[n_] := SeriesCoefficient[x D[Log[1 / (1 - x - x^2)], x], {x, 0, n}]; (* Michael Somos, May 18 2015 *)
  • PARI
    A000204(n)=fibonacci(n+1)+fibonacci(n-1) \\ Michael B. Porter, Nov 05 2009
  • Python
    from functools import cache
@cache
def a(n): return [1, 3][n-1] if n < 3 else a(n-1) + a(n-2)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Nov 13 2022
  • Python
    [(i:=-1)+(j:=2)] + [(j:=i+j)+(i:=j-i) for  in range(100)] # _Jwalin Bhatt, Apr 02 2025
  • Sage
    def A000204():
    x, y = 1, 2
    while true:
       yield x
       x, y = x + y, x
a = A000204(); print([next(a) for i in range(39)])  # Peter Luschny, Dec 17 2015
  • Scala
    def lucas(n: BigInt): BigInt = {
  val zero = BigInt(0)
  def fibTail(n: BigInt, a: BigInt, b: BigInt): BigInt = n match {
    case `zero` => a
    case _ => fibTail(n - 1, b, a + b)
  }
  fibTail(n, 2, 1)
}
(1 to 50).map(lucas()) // _Alonso del Arte, Oct 20 2019

Formula

Expansion of x(1 + 2x)/(1 - x - x^2). - Simon Plouffe, dissertation 1992; multiplied by x. - R. J. Mathar, Nov 14 2007
a(n) = A000045(2n)/A000045(n). - Benoit Cloitre, Jan 05 2003
For n > 1, L(n) = F(n + 2) - F(n - 2), where F(n) is the n-th Fibonacci number (A000045). - Gerald McGarvey, Jul 10 2004
a(n+1) = 4*A054886(n+3) - A022388(n) - 2*A022120(n+1) (a conjecture; note that the above sequences have different offsets). - Creighton Dement, Nov 27 2004
a(n) = Sum_{k=0..floor((n+1)/2)} (n+1)*binomial(n - k + 1, k)/(n - k + 1). - Paul Barry, Jan 30 2005
L(n) = A000045(n+3) - 2*A000045(n). - Creighton Dement, Oct 07 2005
L(n) = A000045(n+1) + A000045(n-1). - John Blythe Dobson, Sep 29 2007
a(n) = 2*Fibonacci(n-1) + Fibonacci(n), n >= 1. - Zerinvary Lajos, Oct 05 2007
L(n) is the term (1, 1) in the 1 X 2 matrix [2, -1].[1, 1; 1, 0]^n. - Alois P. Heinz, Jul 25 2008
a(n) = phi^n + (1 - phi)^n = phi^n + (-phi)^(-n) = ((1 + sqrt(5))^n + (1 - sqrt(5))^n)/(2^n) where phi is the golden ratio (A001622). - Artur Jasinski, Oct 05 2008
a(n) = A014217(n+1) - A014217(n-1). See A153263. - Paul Curtz, Dec 22 2008
a(n) = ((1 + sqrt(5))^n - (1 - sqrt(5))^n)/(2^n*sqrt(5)) + ((1 + sqrt(5))^(n - 1) - (1 - sqrt(5))^(n - 1))/(2^(n - 2)*sqrt(5)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009, Jan 14 2009
From Hieronymus Fischer, Oct 20 2010 (Start)
Continued fraction for n odd: [L(n); L(n), L(n), ...] = L(n) + fract(Fib(n) * phi).
Continued fraction for n even: [L(n); -L(n), L(n), -L(n), L(n), ...] = L(n) - 1 + fract(Fib(n)*phi). Also: [L(n) - 2; 1, L(n) - 2, 1, L(n) - 2, ...] = L(n) - 2 + fract(Fib(n)*phi). (End)
INVERT transform of (1, 2, -1, -2, 1, 2, ...). - Gary W. Adamson, Mar 07 2012
L(2n - 1) = floor(phi^(2n - 1)); L(2n) = ceiling(phi^(2n)). - Thomas Ordowski, Jun 15 2012
a(n) = hypergeom([(1 - n)/2, -n/2], [1 - n], -4) for n >= 3. - Peter Luschny, Sep 03 2019
E.g.f.: 2*(exp(x/2)*cosh(sqrt(5)*x/2) - 1). - Stefano Spezia, Jul 26 2022

Extensions

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
Plouffe Maple line edited by N. J. A. Sloane, May 13 2008

A000930 Narayana's cows sequence: a(0) = a(1) = a(2) = 1; thereafter a(n) = a(n-1) + a(n-3).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Named after a 14th-century Indian mathematician. [The sequence first appeared in the book "Ganita Kaumudi" (1356) by the Indian mathematician Narayana Pandita (c. 1340 - c. 1400). - Amiram Eldar, Apr 15 2021]
Number of compositions of n into parts 1 and 3. - Joerg Arndt, Jun 25 2011
A Lamé sequence of higher order.
Could have begun 1,0,0,1,1,1,2,3,4,6,9,... (A078012) but that would spoil many nice properties.
Number of tilings of a 3 X n rectangle with straight trominoes.
Number of ways to arrange n-1 tatami mats in a 2 X (n-1) room such that no 4 meet at a point. For example, there are 6 ways to cover a 2 X 5 room, described by 11111, 2111, 1211, 1121, 1112, 212.
Equivalently, number of compositions (ordered partitions) of n-1 into parts 1 and 2 with no two 2's adjacent. E.g., there are 6 such ways to partition 5, namely 11111, 2111, 1211, 1121, 1112, 212, so a(6) = 6. [Minor edit by Keyang Li, Oct 10 2020]
This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 0...m-1. The generating function is 1/(1-x-x^m). Also a(n) = Sum_{i=0..floor(n/m)} binomial(n-(m-1)*i, i). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that are m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.
a(n+2) is the number of n-bit 0-1 sequences that avoid both 00 and 010. - David Callan, Mar 25 2004 [This can easily be proved by the Cluster Method - see for example the Noonan-Zeilberger article. - N. J. A. Sloane, Aug 29 2013]
a(n-4) is the number of n-bit sequences that start and end with 0 but avoid both 00 and 010. For n >= 6, such a sequence necessarily starts 011 and ends 110; deleting these 6 bits is a bijection to the preceding item. - David Callan, Mar 25 2004
Also number of compositions of n+1 into parts congruent to 1 mod m. Here m=3, A003269 for m=4, etc. - Vladeta Jovovic, Feb 09 2005
Row sums of Riordan array (1/(1-x^3), x/(1-x^3)). - Paul Barry, Feb 25 2005
Row sums of Riordan array (1,x(1+x^2)). - Paul Barry, Jan 12 2006
Starting with offset 1 = row sums of triangle A145580. - Gary W. Adamson, Oct 13 2008
Number of digits in A061582. - Dmitry Kamenetsky, Jan 17 2009
From Jon Perry, Nov 15 2010: (Start)
The family a(n) = a(n-1) + a(n-m) with a(n)=1 for n=0..m-1 can be generated by considering the sums (A102547):
1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10
1 3 6 10 15 21 28
1 4 10 20
1
------------------------------
1 1 1 2 3 4 6 9 13 19 28 41 60
with (in this case 3) leading zeros added to each row.
(End)
Number of pairs of rabbits existing at period n generated by 1 pair. All pairs become fertile after 3 periods and generate thereafter a new pair at all following periods. - Carmine Suriano, Mar 20 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=3, 2*a(n-3) equals the number of 2-colored compositions of n with all parts >= 3, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
For n>=2, row sums of Pascal's triangle (A007318) with triplicated diagonals. - Vladimir Shevelev, Apr 12 2012
Pisano period lengths of the sequence read mod m, m >= 1: 1, 7, 8, 14, 31, 56, 57, 28, 24, 217, 60, 56, 168, ... (A271953) If m=3, for example, the remainder sequence becomes 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, ... with a period of length 8. - R. J. Mathar, Oct 18 2012
Diagonal sums of triangle A011973. - John Molokach, Jul 06 2013
"In how many ways can a kangaroo jump through all points of the integer interval [1,n+1] starting at 1 and ending at n+1, while making hops that are restricted to {-1,1,2}? (The OGF is the rational function 1/(1 - z - z^3) corresponding to A000930.)" [Flajolet and Sedgewick, p. 373] - N. J. A. Sloane, Aug 29 2013
a(n) is the number of length n binary words in which the length of every maximal run of consecutive 0's is a multiple of 3. a(5) = 4 because we have: 00011, 10001, 11000, 11111. - Geoffrey Critzer, Jan 07 2014
a(n) is the top left entry of the n-th power of the 3X3 matrix [1, 0, 1; 1, 0, 0; 0, 1, 0] or of the 3 X 3 matrix [1, 1, 0; 0, 0, 1; 1, 0, 0]. - R. J. Mathar, Feb 03 2014
a(n-3) is the top left entry of the n-th power of any of the 3 X 3 matrices [0, 1, 0; 0, 1, 1; 1, 0, 0], [0, 0, 1; 1, 1, 0; 0, 1, 0], [0, 1, 0; 0, 0, 1; 1, 0, 1] or [0, 0, 1; 1, 0, 0; 0, 1, 1]. - R. J. Mathar, Feb 03 2014
Counts closed walks of length (n+3) on a unidirectional triangle, containing a loop at one of remaining vertices. - David Neil McGrath, Sep 15 2014
a(n+2) equals the number of binary words of length n, having at least two zeros between every two successive ones. - Milan Janjic, Feb 07 2015
a(n+1)/a(n) tends to x = 1.465571... (decimal expansion given in A092526) in the limit n -> infinity. This is the real solution of x^3 - x^2 -1 = 0. See also the formula by Benoit Cloitre, Nov 30 2002. - Wolfdieter Lang, Apr 24 2015
a(n+2) equals the number of subsets of {1,2,..,n} in which any two elements differ by at least 3. - Robert FERREOL, Feb 17 2016
Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x. If a positive integer N such that r = N^(1/3) is not an integer, then the number of (not necessarily distinct) integers in g(n) is A000930(n), for n >= 1. (See A274142.) - Clark Kimberling, Jun 13 2016
a(n-3) is the number of compositions of n excluding 1 and 2, n >= 3. - Gregory L. Simay, Jul 12 2016
Antidiagonal sums of array A277627. - Paul Curtz, May 16 2019
a(n+1) is the number of multus bitstrings of length n with no runs of 3 ones. - Steven Finch, Mar 25 2020
Suppose we have a(n) samples, exactly one of which is positive. Assume the cost for testing a mix of k samples is 3 if one of the samples is positive (but you will not know which sample was positive if you test more than 1) and 1 if none of the samples is positive. Then the cheapest strategy for finding the positive sample is to have a(n-3) undergo the first test and then continue with testing either a(n-4) if none were positive or with a(n-6) otherwise. The total cost of the tests will be n. - Ruediger Jehn, Dec 24 2020

Examples

			The number of compositions of 11 without any 1's and 2's is a(11-3) = a(8) = 13. The compositions are (11), (8,3), (3,8), (7,4), (4,7), (6,5), (5,6), (5,3,3), (3,5,3), (3,3,5), (4,4,3), (4,3,4), (3,4,4). - _Gregory L. Simay_, Jul 12 2016
The compositions from the above example may be mapped to the a(8) compositions of 8 into 1's and 3's using this (more generally applicable) method: replace all numbers greater than 3 with a 3 followed by 1's to make the same total, then remove the initial 3 from the composition. Maintaining the example's order, they become (1,1,1,1,1,1,1,1), (1,1,1,1,1,3), (3,1,1,1,1,1), (1,1,1,1,3,1), (1,3,1,1,1,1), (1,1,1,3,1,1), (1,1,3,1,1,1), (1,1,3,3), (3,1,1,3), (3,3,1,1), (1,3,1,3), (1,3,3,1), (3,1,3,1). - _Peter Munn_, May 31 2017

References

  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, id. 8,80.
  • R. K. Guy, "Anyone for Twopins?" in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [See p. 12, line 3]
  • H. Langman, Play Mathematics. Hafner, NY, 1962, p. 13.
  • David Sankoff and Lani Haque, Power Boosts for Cluster Tests, in Comparative Genomics, Lecture Notes in Computer Science, Volume 3678/2005, Springer-Verlag. - N. J. A. Sloane, Jul 09 2009
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

For Lamé sequences of orders 1 through 9 see A000045, this sequence, and A017898 - A017904.
Cf. A000073, A000213, A048715, A069241, A092526, A170954.
See also A000079, A003269, A003520, A005708, A005709, A005710.
Essentially the same as A068921 and A078012.
See also A001609, A145580, A179070, A214551 (same rule except divide by GCD).
A271901 and A271953 give the period of this sequence mod n.
Cf. A007318, A060576, A102547, A277627, A289207.
A120562 has the same recurrence for odd n.

Programs

  • GAP
    a:=[1,1,1];; for n in [4..50] do a[n]:=a[n-1]+a[n-3]; od; a; # Muniru A Asiru, Aug 13 2018
  • Haskell
    a000930 n = a000930_list !! n
a000930_list = 1 : 1 : 1 : zipWith (+) a000930_list (drop 2 a000930_list)
-- Reinhard Zumkeller, Sep 25 2011
  • Magma
    [1,1] cat [ n le 3 select n else Self(n-1)+Self(n-3): n in [1..50] ]; // Vincenzo Librandi, Apr 25 2015
  • Maple
    f := proc(r) local t1,i; t1 := []; for i from 1 to r do t1 := [op(t1),0]; od: for i from 1 to r+1 do t1 := [op(t1),1]; od: for i from 2*r+2 to 50 do t1 := [op(t1),t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order
with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 2)}, unlabeled]: seq(count(SeqSetU, size=j), j=3..40); # Zerinvary Lajos, Oct 10 2006
A000930 := proc(n)
    add(binomial(n-2*k,k),k=0..floor(n/3)) ;
end proc: # Zerinvary Lajos, Apr 03 2007
a:= n-> (<<1|1|0>, <0|0|1>, <1|0|0>>^n)[1,1]:
seq(a(n), n=0..50); # Alois P. Heinz, Jun 20 2008
  • Mathematica
    a[0] = 1; a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 3]; Table[ a[n], {n, 0, 40} ]
CoefficientList[Series[1/(1 - x - x^3), {x, 0, 45}], x] (* Zerinvary Lajos, Mar 22 2007 *)
LinearRecurrence[{1, 0, 1}, {1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
a[n_] := HypergeometricPFQ[{(1 - n)/3, (2 - n)/3, -n/3}, {(1 - n)/ 2, -n/2}, -27/4]; Table[a[n], {n, 0, 43}] (* Jean-François Alcover, Feb 26 2013 *)
Table[-RootSum[1 + #^2 - #^3 &, 3 #^(n + 2) - 11 #^(n + 3) + 2 #^(n + 4) &]/31, {n, 20}] (* Eric W. Weisstein, Feb 14 2025 *)
  • Maxima
    makelist(sum(binomial(n-2*k,k),k,0,n/3),n,0,18); /* Emanuele Munarini, May 24 2011 */
  • PARI
    a(n)=polcoeff(exp(sum(m=1,n,((1+sqrt(1+4*x))^m + (1-sqrt(1+4*x))^m)*(x/2)^m/m)+x*O(x^n)),n) \\ Paul D. Hanna, Oct 08 2009
  • PARI
    x='x+O('x^66); Vec(1/(1-(x+x^3))) \\ Joerg Arndt, May 24 2011
  • PARI
    a(n)=([0,1,0;0,0,1;1,0,1]^n*[1;1;1])[1,1] \\ Charles R Greathouse IV, Feb 26 2017
  • Python
    from itertools import islice
def A000930_gen(): # generator of terms
    blist = [1]*3
    while True:
        yield blist[0]
        blist = blist[1:]+[blist[0]+blist[2]]
A000930_list = list(islice(A000930_gen(),30)) # Chai Wah Wu, Feb 04 2022
  • SageMath
    @CachedFunction
def a(n): # A000930
    if (n<3): return 1
    else: return a(n-1) + a(n-3)
[a(n) for n in (0..80)] # G. C. Greubel, Jul 29 2022

Formula

G.f.: 1/(1-x-x^3). - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{i=0..floor(n/3)} binomial(n-2*i, i).
a(n) = a(n-2) + a(n-3) + a(n-4) for n>3.
a(n) = floor(d*c^n + 1/2) where c is the real root of x^3-x^2-1 and d is the real root of 31*x^3-31*x^2+9*x-1 (c = 1.465571... = A092526 and d = 0.611491991950812...). - Benoit Cloitre, Nov 30 2002
a(n) = Sum_{k=0..n} binomial(floor((n+2k-2)/3), k). - Paul Barry, Jul 06 2004
a(n) = Sum_{k=0..n} binomial(k, floor((n-k)/2))(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006
a(n) = Sum_{k=0..n} binomial((n+2k)/3,(n-k)/3)*(2*cos(2*Pi*(n-k)/3)+1)/3. - Paul Barry, Dec 15 2006
a(n) = term (1,1) in matrix [1,1,0; 0,0,1; 1,0,0]^n. - Alois P. Heinz, Jun 20 2008
G.f.: exp( Sum_{n>=1} ((1+sqrt(1+4*x))^n + (1-sqrt(1+4*x))^n)*(x/2)^n/n ).
Logarithmic derivative equals A001609. - Paul D. Hanna, Oct 08 2009
a(n) = a(n-1) + a(n-2) - a(n-5) for n>4. - Paul Weisenhorn, Oct 28 2011
For n >= 2, a(2*n-1) = a(2*n-2)+a(2*n-4); a(2*n) = a(2*n-1)+a(2*n-3). - Vladimir Shevelev, Apr 12 2012
INVERT transform of (1,0,0,1,0,0,1,0,0,1,...) = (1, 1, 1, 2, 3, 4, 6, ...); but INVERT transform of (1,0,1,0,0,0,...) = (1, 1, 2, 3, 4, 6, ...). - Gary W. Adamson, Jul 05 2012
G.f.: 1/(G(0)-x) where G(k) = 1 - x^2/(1 - x^2/(x^2 - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2012
G.f.: 1 + x/(G(0)-x) where G(k) = 1 - x^2*(2*k^2 + 3*k +2) + x^2*(k+1)^2*(1 - x^2*(k^2 + 3*k +2))/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 27 2012
a(2*n) = A002478(n), a(2*n+1) = A141015(n+1), a(3*n) = A052544(n), a(3*n+1) = A124820(n), a(3*n+2) = A052529(n+1). - Johannes W. Meijer, Jul 21 2013, corrected by Greg Dresden, Jul 06 2020
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x^2)/( x*(4*k+3 + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013
a(n) = v1*w1^n+v3*w2^n+v2*w3^n, where v1,2,3 are the roots of (-1+9*x-31*x^2+31*x^3): [v1=0.6114919920, v2=0.1942540040 - 0.1225496913*I, v3=conjugate(v2)] and w1,2,3 are the roots of (-1-x^2+x^3): [w1=1.4655712319, w2=-0.2327856159 - 0.7925519925*I, w3=conjugate(w2)]. - Gerry Martens, Jun 27 2015
a(n) = (6*A001609(n+3) + A001609(n-7))/31 for n>=7. - Areebah Mahdia, Jun 07 2020
a(n+6)^2 + a(n+1)^2 + a(n)^2 = a(n+5)^2 + a(n+4)^2 + 3*a(n+3)^2 + a(n+2)^2. - Greg Dresden, Jul 07 2021
a(n) = Sum_{i=(n-7)..(n-1)} a(i) / 2. - Jules Beauchamp, May 10 2025

Extensions

Name expanded by N. J. A. Sloane, Sep 07 2012

A003215 Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).

Original entry on oeis.org

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487, 6769
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Crystal ball sequence for A_2 lattice. - Michael Somos, Jun 03 2012
Sixth spoke of hexagonal spiral (cf. A056105-A056109).
Number of ordered integer triples (a,b,c), -n <= a,b,c <= n, such that a+b+c=0. - Benoit Cloitre, Jun 14 2003
Also the number of partitions of 6n into at most 3 parts, A001399(6n). - R. K. Guy, Oct 20 2003
Also, a(n) is the number of partitions of 6(n+1) into exactly 3 distinct parts. - William J. Keith, Jul 01 2004
Number of dots in a centered hexagonal figure with n+1 dots on each side.
Values of second Bessel polynomial y_2(n) (see A001498).
First differences of cubes (A000578). - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
Final digits of Hex numbers (hex(n) mod 10) are periodic with palindromic period of length 5 {1, 7, 9, 7, 1}. Last two digits of Hex numbers (hex(n) mod 100) are periodic with palindromic period of length 100. - Alexander Adamchuk, Aug 11 2006
All divisors of a(n) are congruent to 1, modulo 6. Proof: If p is an odd prime different from 3 then 3n^2 + 3n + 1 = 0 (mod p) implies 9(2n + 1)^2 = -3 (mod p), whence p = 1 (mod 6). - Nick Hobson, Nov 13 2006
For n>=1, a(n) is the side of Outer Napoleon Triangle whose reference triangle is a right triangle with legs (3a(n))^(1/2) and 3n(a(n))^(1/2). - Tom Schicker (tschicke(AT)email.smith.edu), Apr 25 2007
Number of triples (a,b,c) where 0<=(a,b)<=n and c=n (at least once the term n). E.g., for n = 1: (0,0,1), (0,1,0), (1,0,0), (0,1,1), (1,0,1), (1,1,0), (1,1,1), so a(1)=7. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007
Equals the triangular numbers convolved with [1, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009
From Terry Stickels, Dec 07 2009: (Start)
Also the maximum number of viewable cubes from any one static point while viewing a cube stack of identical cubes of varying magnitude.
For example, viewing a 2 X 2 X 2 stack will yield 7 maximum viewable cubes.
If the stack is 3 X 3 X 3, the maximum number of viewable cubes from any one static position is 19, and so on.
The number of cubes in the stack must always be the same number for width, length, height (at true regular cubic stack) and the maximum number of visible cubes can always be found by taking any cubic number and subtracting the number of the cube that is one less.
Examples: 125 - 64 = 61, 64 - 27 = 37, 27 - 8 = 19. (End)
The sequence of digital roots of the a(n) is period 3: repeat [1,7,1]. - Ant King, Jun 17 2012
The average of the first n (n>0) centered hexagonal numbers is the n-th square. - Philippe Deléham, Feb 04 2013
A002024 is the following array A read along antidiagonals:
1, 2, 3, 4, 5, 6, ...
2, 3, 4, 5, 6, 7, ...
3, 4, 5, 6, 7, 8, ...
4, 5, 6, 7, 8, 9, ...
5, 6, 7, 8, 9, 10, ...
6, 7, 8, 9, 10, 11, ...
and a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013
a(n) is the sum of the terms in the n+1 X n+1 matrices minus those in n X n matrices in an array formed by considering A158405 an array (the beginning terms in each row are 1,3,5,7,9,11,...). - J. M. Bergot, Jul 05 2013
The formula also equals the product of the three distinct combinations of two consecutive numbers: n^2, (n+1)^2, and n*(n+1). - J. M. Bergot, Mar 28 2014
The sides of any triangle ABC are divided into 2n + 1 equal segments by 2n points: A_1, A_2, ..., A_2n in side a, and also on the sides b and c cyclically. If A'B'C' is the triangle delimited by AA_n, BB_n and CC_n cevians, we have (ABC)/(A'B'C') = a(n) (see Java applet link). - Ignacio Larrosa Cañestro, Jan 02 2015
a(n) is the maximal number of parts into which (n+1) triangles can intersect one another. - Ivan N. Ianakiev, Feb 18 2015
((2^m-1)n)^t mod a(n) = ((2^m-1)(n+1))^t mod a(n) = ((2^m-1)(2n+1))^t mod a(n), where m any positive integer, and t = 0(mod 6). - Alzhekeyev Ascar M, Oct 07 2016
((2^m-1)n)^t mod a(n) = ((2^m-1)(n+1))^t mod a(n) = a(n) - (((2^m-1)(2n+1))^t mod a(n)), where m any positive integer, and t = 3(mod 6). - Alzhekeyev Ascar M, Oct 07 2016
(3n+1)^(a(n)-1) mod a(n) = (3n+2)^(a(n)-1) mod a(n) = 1. If a(n) not prime, then always strong pseudoprime. - Alzhekeyev Ascar M, Oct 07 2016
Every positive integer is the sum of 8 hex numbers (zero included), at most 3 of which are greater than 1. - Mauro Fiorentini, Jan 01 2018
Area enclosed by the segment of Archimedean spiral between n*Pi/2 and (n+1)*Pi/2 in Pi^3/48 units. - Carmine Suriano, Apr 10 2018
This sequence contains all numbers k such that 12*k - 3 is a square. - Klaus Purath, Oct 19 2021
The continued fraction expansion of sqrt(3*a(n)) is [3n+1; {1, 1, 2n, 1, 1, 6n+2}]. For n = 0, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 12 2022

Examples

			G.f. = 1 + 7*x + 19*x^2 + 37*x^3 + 61*x^4 + 91*x^5 + 127*x^6 + 169*x^7 + 217*x^8 + ...
From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms:
.
.                                 o o o o
.                   o o o        o o o o o
.         o o      o o o o      o o o o o o
.   o    o o o    o o o o o    o o o o o o o
.         o o      o o o o      o o o o o o
.                   o o o        o o o o o
.                                 o o o o
.
.   1      7          19             37
.
(End)
From _Klaus Purath_, Dec 03 2021: (Start)
(1) a(19) is not a prime number, because besides a(19) = a(9) + P(29), a(19) = a(15) + P(20) = a(2) + P(33) is also true.
(2) a(25) is prime, because except for a(25) = a(12) + P(38) there is no other equation of this pattern. (End)

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.
  • M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 18.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000124, A000166, A000217, A000290, A000578 (the cubes, or partial sums), A001263, A001498, A002061, A002378, A002407 (primes), A003514, A005408, A005449, A005891, A028896, A048766, A056105, A056106, A056107, A056108, A056109, A063496, A056220, A130298, A132111 (second diagonal), A158405, A215630, A239449, A243201.
Column k=3 of A080853, and column k=2 of A047969.
See also A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.
Cf. A287326(A000124(n), 1).
Cf. A008292.
Cf. A005448, A140091, A001844, A000384, A060544, A045943.
Cf. A154105.

Programs

Formula

a(n) = 3*n*(n+1) + 1, n >= 0 (see the name).
a(n) = (n+1)^3 - n^3 = a(-1-n).
G.f.: (1 + 4*x + x^2) / (1 - x)^3. - Simon Plouffe in his 1992 dissertation
a(n) = 6*A000217(n) + 1.
a(n) = a(n-1) + 6*n = 2a(n-1) - a(n-2) + 6 = 3*a(n-1) - 3*a(n-2) + a(n-3) = A056105(n) + 5n = A056106(n) + 4*n = A056107(n) + 3*n = A056108(n) + 2*n = A056108(n) + n.
n-th partial arithmetic mean is n^2. - Amarnath Murthy, May 27 2003
a(n) = 1 + Sum_{j=0..n} (6*j). E.g., a(2)=19 because 1+ 6*0 + 6*1 + 6*2 = 19. - Xavier Acloque, Oct 06 2003
The sum of the first n hexagonal numbers is n^3. That is, Sum_{n>=1} (3*n*(n-1) + 1) = n^3. - Edward Weed (eweed(AT)gdrs.com), Oct 23 2003
a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 2 1 0 / 3 3 1]. M^n * [1 1 1] = [1 2n+1 a(n)]. E.g., a(4) = 61, right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 9 61] = [1 2n+1 a(4)]. - Gary W. Adamson, Dec 22 2004
Row sums of triangle A130298. - Gary W. Adamson, Jun 07 2007
a(n) = 3*n^2 + 3*n + 1. Proof: 1) If n occurs once, it may be in 3 positions; for the two other ones, n terms are independently possible, then we have 3*n^2 different triples. 2) If the term n occurs twice, the third one may be placed in 3 positions and have n possible values, then we have 3*n more different triples. 3) The term n may occurs 3 times in one way only that gives the formula. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007
Binomial transform of [1, 6, 6, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 6, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = (n-1)*A000166(n) + (n-2)*A000166(n-1) = (n-1)floor(n!*e^(-1)+1) + (n-2)*floor((n-1)!*e^(-1)+1) (with offset 0). - Gary Detlefs, Dec 06 2009
a(n) = A028896(n) + 1. - Omar E. Pol, Oct 03 2011
a(n) = integral( (sin((n+1/2)x)/sin(x/2))^3, x=0..Pi)/Pi. - Yalcin Aktar, Dec 03 2011
Sum_{n>=0} 1/a(n) = Pi/sqrt(3)*tanh(Pi/(2*sqrt(3))) = 1.305284153013581... - Ant King, Jun 17 2012
a(n) = A000290(n) + A000217(2n+1). - Ivan N. Ianakiev, Sep 24 2013
a(n) = A002378(n+1) + A056220(n) = A005408(n) + 2*A005449(n) = 6*A000217(n) + 1. - Ivan N. Ianakiev, Sep 26 2013
a(n) = 6*A000124(n) - 5. - Ivan N. Ianakiev, Oct 13 2013
a(n) = A239426(n+1) / A239449(n+1) = A215630(2*n+1,n+1). - Reinhard Zumkeller, Mar 19 2014
a(n) = A243201(n) / A002061(n + 1). - Mathew Englander, Jun 03 2014
a(n) = A101321(6,n). - R. J. Mathar, Jul 28 2016
E.g.f.: (1 + 6*x + 3*x^2)*exp(x). - Ilya Gutkovskiy, Jul 28 2016
a(n) = (A001844(n) + A016754(n))/2. - Bruce J. Nicholson, Aug 06 2017
a(n) = A045943(2n+1). - Miquel Cerda, Jan 22 2018
a(n) = 3*Integral_{x=n..n+1} x^2 dx. - Carmine Suriano, Apr 10 2018
a(n) = A287326(A000124(n), 1). - Kolosov Petro, Oct 22 2018
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=0} a(n)/n! = 10*e.
Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 2/e. (End)
G.f.: polylog(-3, x)*(1-x)/x. See the Simon Plouffe formula above, and the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 08 2021
a(n) = T(n-1)^2 - 2*T(n)^2 + T(n+1)^2, n >= 1, T = triangular number A000217. - Klaus Purath, Oct 11 2021
a(n) = 1 + 2*Sum_{j=n..2n} j. - Klaus Purath, Oct 19 2021
a(n) = A069099(n+1) - A000217(n). - Klaus Purath, Nov 03 2021
From Leo Tavares, Dec 03 2021: (Start)
a(n) = A005448(n) + A140091(n);
a(n) = A001844(n) + A002378(n);
a(n) = A005891(n) + A000217(n);
a(n) = A000290(n) + A000384(n+1);
a(n) = A060544(n-1) + 3*A000217(n);
a(n) = A060544(n-1) + A045943(n).
a(2*n+1) = A154105(n).
(End)

Extensions

Partially edited by Joerg Arndt, Mar 11 2010

A000215 Fermat numbers: a(n) = 2^(2^n) + 1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

It is conjectured that just the first 5 numbers in this sequence are primes.
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004
For n>0, Fermat numbers F(n) have digital roots 5 or 8 depending on whether n is even or odd (Koshy). - Lekraj Beedassy, Mar 17 2005
This is the special case k=2 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058. - Seppo Mustonen, Sep 04 2005
For n>1 final two digits of a(n) are periodically repeated with period 4: {17, 57, 37, 97}. - Alexander Adamchuk, Apr 07 2007
For 1 < k <= 2^n, a(A007814(k-1)) divides a(n) + 2^k. More generally, for any number k, let r = k mod 2^n and suppose r != 1, then a(A007814(r-1)) divides a(n) + 2^k. - T. D. Noe, Jul 12 2007
From Daniel Forgues, Jun 20 2011: (Start)
The Fermat numbers F_n are F_n(a,b) = a^(2^n) + b^(2^n) with a = 2 and b = 1.
For n >= 2, all factors of F_n = 2^(2^n) + 1 are of the form k*(2^(n+2)) + 1 (k >= 1).
The products of distinct Fermat numbers (in their binary representation, see A080176) give rows of Sierpiński's triangle (A006943). (End)
Let F(n) be a Fermat number. For n > 2, F(n) is prime if and only if 5^((F(n)-1)/4) == sqrt(F(n)-1) (mod F(n)). - Arkadiusz Wesolowski, Jul 16 2011
Conjecture: let the smallest prime factor of Fermat number F(n) be P(F(n)). If F(n) is composite, then P(F(n)) < 3*2^(2^n/2 - n - 2). - Arkadiusz Wesolowski, Aug 10 2012
The Fermat primes are not Brazilian numbers, so they belong to A220627, but the Fermat composites are Brazilian numbers so they belong to A220571. For a proof, see Proposition 3 page 36 on "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
It appears that this sequence is generated by starting with a(0)=3 and following the rule "Write in binary and read in base 4". For an example of "Write in binary and read in ternary", see A014118. - John W. Layman, Jul 30 2013
Conjecture: the numbers > 5 in this sequence, i.e., 2^2^k + 1 for k>1, are exactly the numbers n such that (n-1)^4-1 divides 2^(n-1)-1. - M. F. Hasler, Jul 24 2015

Examples

			a(0) = 1*2^1 + 1 = 3 = 1*(2*1) + 1.
a(1) = 1*2^2 + 1 = 5 = 1*(2*2) + 1.
a(2) = 1*2^4 + 1 = 17 = 2*(2*4) + 1.
a(3) = 1*2^8 + 1 = 257 = 16*(2*8) + 1.
a(4) = 1*2^16 + 1 = 65537 = 2048*(2*16) + 1.
a(5) = 1*2^32 + 1 = 4294967297 = 641*6700417 = (10*(2*32) + 1)*(104694*(2*32) + 1).
a(6) = 1*2^64 + 1 = 18446744073709551617 = 274177*67280421310721 = (2142*(2*64) + 1)*(525628291490*(2*64) + 1).

References

  • M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 7.
  • P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 87.
  • James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.
  • R. K. Guy, Unsolved Problems in Number Theory, A3.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 14.
  • E. Hille, Analytic Function Theory, Vol. I, Chelsea, N.Y., see p. 64.
  • T. Koshy, "The Digital Root Of A Fermat Number", Journal of Recreational Mathematics Vol. 32 No. 2 2002-3 Baywood NY.
  • M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001.
  • C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 36.
  • Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 18, 59.
  • C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 202.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 6-7, 70-75.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 148-149.

Links

Crossrefs

a(n) = A001146(n) + 1 = A051179(n) + 2.
Cf. A019434, A050922, A051158, A051179, A063486, A073617, A085866.
See A004249 for a similar sequence.
Cf. A080176 for binary representation of Fermat numbers.
Cf. A220627, A220570, A220571, A125134, A249119.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.
Cf. A000058, A000120, A000225, A006943, A007814, A014118, A077585, A242619, A242620, A257916, A257917.

Programs

  • Haskell
    a000215 = (+ 1) . (2 ^) . (2 ^)  -- Reinhard Zumkeller, Feb 13 2015
  • Maple
    A000215 := n->2^(2^n)+1;
  • Mathematica
    Table[2^(2^n) + 1, {n, 0, 8}] (* Alonso del Arte, Jun 07 2011 *)
  • Maxima
    A000215(n):=2^(2^n)+1$ makelist(A000215(n),n,0,10); /* Martin Ettl, Dec 10 2012 */
  • PARI
    a(n)=if(n<1,3*(n==0),(a(n-1)-1)^2+1)
  • Python
    def a(n): return 2**(2**n) + 1
print([a(n) for n in range(9)]) # Michael S. Branicky, Apr 19 2021

Formula

a(0) = 3; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(n) = a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get the empty product, i.e., 1, plus 2, giving 3 = a(0). - Benoit Cloitre, Sep 15 2002 [edited by Daniel Forgues, Jun 20 2011]
The above formula implies that the Fermat numbers (being all odd) are coprime.
Conjecture: F is a Fermat prime if and only if phi(F-2) = (F-1)/2. - Benoit Cloitre, Sep 15 2002
A000120(a(n)) = 2. - Reinhard Zumkeller, Aug 07 2010
If a(n) is composite, then a(n) = A242619(n)^2 + A242620(n)^2 = A257916(n)^2 - A257917(n)^2. - Arkadiusz Wesolowski, May 13 2015
Sum_{n>=0} 1/a(n) = A051158. - Amiram Eldar, Oct 27 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = A249119.
Product_{n>=0} (1 - 1/a(n)) = 1/2. (End)
a(n) = 2*A077585(n) + 3. - César Aguilera, Jul 26 2023
a(n) = 2*2^A000225(n) + 1. - César Aguilera, Jul 11 2024

A000607 Number of partitions of n into prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 30, 35, 40, 46, 52, 60, 67, 77, 87, 98, 111, 124, 140, 157, 175, 197, 219, 244, 272, 302, 336, 372, 413, 456, 504, 557, 614, 677, 744, 819, 899, 987, 1083, 1186, 1298, 1420, 1552, 1695, 1850, 2018, 2198, 2394, 2605, 2833, 3079, 3344
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) gives the number of values of k such that A001414(k) = n. - Howard A. Landman, Sep 25 2001
Let W(n) = {prime p: There is at least one number m whose spf is p, and sopfr(m) = n}. Let V(n,p) = {m: sopfr(m) = n, p belongs to W(n)}. Then a(n) = sigma(|V(n,p)|). E.g.: W(10) = {2,3,5}, V(10,2) = {30,32,36}, V(10,3) = {21}, V(10,5) = {25}, so a(10) = 3+1+1 = 5. - David James Sycamore, Apr 14 2018
From Gus Wiseman, Jan 18 2020: (Start)
Also the number of integer partitions such that the sum of primes indexed by the parts is n. For example, the sum of primes indexed by the parts of the partition (3,2,1,1) is prime(3)+prime(2)+prime(1)+prime(1) = 12, so (3,2,1,1) is counted under a(12). The a(2) = 1 through a(14) = 10 partitions are:
1 2 11 3 22 4 32 41 33 5 43 6 44
21 111 31 221 222 42 322 331 51 52
211 1111 311 321 411 421 332 431
2111 2211 2221 2222 422 3222
11111 3111 3211 3221 3311
21111 22111 4111 4211
111111 22211 22221
31111 32111
211111 221111
1111111
(End)

Examples

			n = 10 has a(10) = 5 partitions into prime parts: 10 = 2 + 2 + 2 + 2 + 2 = 2 + 2 + 3 + 3 = 2 + 3 + 5 = 3 + 7 = 5 + 5.
n = 15 has a(15) = 12 partitions into prime parts: 15 = 2 + 2 + 2 + 2 + 2 + 2 + 3 = 2 + 2 + 2 + 3 + 3 + 3 = 2 + 2 + 2 + 2 + 2 + 5 = 2 + 2 + 2 + 2 + 7 = 2 + 2 + 3 + 3 + 5 = 2 + 3 + 5 + 5 = 2 + 3 + 3 + 7 = 2 + 2 + 11 = 2 + 13 = 3 + 3 + 3 + 3 + 3 = 3 + 5 + 7 = 5 + 5 + 5.

References

  • R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 203.
  • Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
  • B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 49-78 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.
  • D. M. Burton, Elementary Number Theory, 5th ed., McGraw-Hill, 2002.
  • L. M. Chawla and S. A. Shad, On a trio-set of partition functions and their tables, J. Natural Sciences and Mathematics, 9 (1969), 87-96.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

G.f. = 1 / g.f. for A046675. See A046113 for the ordered (compositions) version.
Cf. A046676, A048165, A004526, A051034, A000040, A001414, A000586 (distinct parts), A000041, A070214, A192541, A112021, A056768, A128515, A319265, A319266.
Row sums of array A116865 and of triangle A261013.
Column sums of A331416.
Partitions whose Heinz number is divisible by their sum of primes are A330953.
Partitions of whose sum of primes is divisible by their sum are A331379.
Cf. A000720, A014689, A331385, A331387, A331415, A331417.

Programs

  • Haskell
    a000607 = p a000040_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 05 2012
  • Magma
    [1] cat [#RestrictedPartitions(n,{p:p in PrimesUpTo(n)}): n in [1..100]]; // Marius A. Burtea, Jan 02 2019
  • Maple
    with(gfun):
t1:=mul(1/(1-q^ithprime(n)),n=1..51):
t2:=series(t1,q,50):
t3:=seriestolist(t2); # fixed by Vaclav Kotesovec, Sep 14 2014
  • Mathematica
    CoefficientList[ Series[1/Product[1 - x^Prime[i], {i, 1, 50}], {x, 0, 50}], x]
f[n_] := Length@ IntegerPartitions[n, All, Prime@ Range@ PrimePi@ n]; Array[f, 57] (* Robert G. Wilson v, Jul 23 2010 *)
Table[Length[Select[IntegerPartitions[n],And@@PrimeQ/@#&]],{n,0,60}] (* Harvey P. Dale, Apr 22 2012 *)
a[n_] := a[n] = If[PrimeQ[n], 1, 0]; c[n_] := c[n] = Plus @@ Map[# a[#] &, Divisors[n]]; b[n_] := b[n] = (c[n] + Sum[c[k] b[n - k], {k, 1, n - 1}])/n; Table[b[n], {n, 1, 20}] (* Thomas Vogler, Dec 10 2015: Uses Euler transform, caches computed values, faster than IntegerPartitions[] function. *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = -1; Do[p = Prime[k]; Do[poly[[j + 1]] -= poly[[j + 1 - p]], {j, nmax, p, -1}];, {k, 2, pmax}]; s = Sum[poly[[k + 1]]*x^k, {k, 0, Length[poly] - 1}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 11 2021 *)
  • PARI
    N=66;x='x+O('x^N); Vec(1/prod(k=1,N,1-x^prime(k))) \\ Joerg Arndt, Sep 04 2014
  • Python
    from sympy import primefactors
l = [1, 0]
for n in range(2, 101):
    l.append(sum(sum(primefactors(k)) * l[n - k] for k in range(1, n + 1)) // n)
l  # Indranil Ghosh, Jul 13 2017
  • Sage
    [Partitions(n, parts_in=prime_range(n + 1)).cardinality() for n in range(100)]  # Giuseppe Coppoletta, Jul 11 2016

Formula

Asymptotically a(n) ~ exp(2 Pi sqrt(n/log n) / sqrt(3)) (Ayoub).
a(n) = (1/n)*Sum_{k=1..n} A008472(k)*a(n-k). - Vladeta Jovovic, Aug 27 2002
G.f.: 1/Product_{k>=1} (1-x^prime(k)).
See the partition arrays A116864 and A116865.
From Vaclav Kotesovec, Sep 15 2014 [Corrected by Andrey Zabolotskiy, May 26 2017]: (Start)
It is surprising that the ratio of the formula for log(a(n)) to the approximation 2 * Pi * sqrt(n/(3*log(n))) exceeds 1. For n=20000 the ratio is 1.00953, and for n=50000 (using the value from Havermann's tables) the ratio is 1.02458, so the ratio is increasing. See graph above.
A more refined asymptotic formula is found by Vaughan in Ramanujan J. 15 (2008), pp. 109-121, and corrected by Bartel et al. (2017): log(a(n)) = 2*Pi*sqrt(n/(3*log(n))) * (1 - log(log(n))/(2*log(n)) + O(1/log(n))).
See Bartel, Bhaduri, Brack, Murthy (2017) for a more complete asymptotic expansion. (End)
G.f.: 1 + Sum_{i>=1} x^prime(i) / Product_{j=1..i} (1 - x^prime(j)). - Ilya Gutkovskiy, May 07 2017
a(n) = A184198(n) + A184199(n). - Vaclav Kotesovec, Jan 11 2021

A003114 Number of partitions of n into parts 5k+1 or 5k+4.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 61, 70, 79, 91, 102, 117, 131, 149, 167, 189, 211, 239, 266, 299, 333, 374, 415, 465, 515, 575, 637, 709, 783, 871, 961, 1065, 1174, 1299, 1429, 1579, 1735, 1913, 2100, 2311, 2533, 2785
Offset: 0

Views

Author

N. J. A. Sloane and Herman P. Robinson

Keywords

Comments

Expansion of Rogers-Ramanujan function G(x) in powers of x.
Same as number of partitions into distinct parts where the difference between successive parts is >= 2.
As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence. - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001
The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))).
Coefficients in expansion of permanent of infinite tridiagonal matrix:
1 1 0 0 0 0 0 0 ...
x 1 1 0 0 0 0 0 ...
0 x^2 1 1 0 0 0 ...
0 0 x^3 1 1 0 0 ...
0 0 0 x^4 1 1 0 ...
................... - Vladeta Jovovic, Jul 17 2004
Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Jul 17 2004
Also number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Feb 27 2006
Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3, 3, 3], [2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 16 2006
a(n) = number of NW partitions of n, for n >= 1; see A237981.
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[1](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109700 and A109697. - Vaclav Kotesovec, Jan 21 2017

Examples

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
G.f. = 1/q + q^59 + q^119 + q^179 + 2*q^239 + 2*q^299 + 3*q^359 + 3*q^419 + ...
From _Joerg Arndt_, Dec 27 2012: (Start)
The a(16)=17 partitions of 16 where all parts are 1 or 4 (mod 5) are
  [ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
  [ 2]  [ 4 1 1 1 1 1 1 1 1 1 1 1 1 ]
  [ 3]  [ 4 4 1 1 1 1 1 1 1 1 ]
  [ 4]  [ 4 4 4 1 1 1 1 ]
  [ 5]  [ 4 4 4 4 ]
  [ 6]  [ 6 1 1 1 1 1 1 1 1 1 1 ]
  [ 7]  [ 6 4 1 1 1 1 1 1 ]
  [ 8]  [ 6 4 4 1 1 ]
  [ 9]  [ 6 6 1 1 1 1 ]
  [10]  [ 6 6 4 ]
  [11]  [ 9 1 1 1 1 1 1 1 ]
  [12]  [ 9 4 1 1 1 ]
  [13]  [ 9 6 1 ]
  [14]  [ 11 1 1 1 1 1 ]
  [15]  [ 11 4 1 ]
  [16]  [ 14 1 1 ]
  [17]  [ 16 ]
The a(16)=17 partitions of 16 where successive parts differ by at least 2 are
  [ 1]  [ 7 5 3 1 ]
  [ 2]  [ 8 5 3 ]
  [ 3]  [ 8 6 2 ]
  [ 4]  [ 9 5 2 ]
  [ 5]  [ 9 6 1 ]
  [ 6]  [ 9 7 ]
  [ 7]  [ 10 4 2 ]
  [ 8]  [ 10 5 1 ]
  [ 9]  [ 10 6 ]
  [10]  [ 11 4 1 ]
  [11]  [ 11 5 ]
  [12]  [ 12 3 1 ]
  [13]  [ 12 4 ]
  [14]  [ 13 3 ]
  [15]  [ 14 2 ]
  [16]  [ 15 1 ]
  [17]  [ 16 ]
(End)

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 238.
  • G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(e), p. 591.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 107.
  • G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 90-92.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 290-291.
  • H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900.
Cf. A188216 (least part k occurs at least k times).
Cf. A047209, A203776, A237981.
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Row sums of A268187.

Programs

  • Haskell
    a003114 = p a047209_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 05 2011
  • Haskell
    a003114 = p 1 where
   p _ 0 = 1
   p k m = if k > m then 0 else p (k + 2) (m - k) + p (k + 1) m
-- Reinhard Zumkeller, Feb 19 2013
  • Maple
    g:=sum(x^(k^2)/product(1-x^j,j=1..k),k=0..10): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60); # Emeric Deutsch, Feb 27 2006
  • Mathematica
    CoefficientList[ Series[Sum[x^k^2/Product[1 - x^j, {j, 1, k}], {k, 0, 10}], {x, 0, 65}], x][[1 ;; 61]] (* Jean-François Alcover, Apr 08 2011, after Emeric Deutsch *)
Table[Count[IntegerPartitions[n], p_ /; Min[p] >= Length[p]], {n, 0, 24}] (* Clark Kimberling, Feb 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 0, 0, -1, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *)
nmax = 60; kmax = nmax/5;
s = Flatten[{Range[0, kmax]*5 + 1}~Join~{Range[0, kmax]*5 + 4}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)
  • PARI
    {a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1), n))}; /* Michael Somos, Oct 15 2008 */

Formula

G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i).
The g.f. above is the special case D=2 of sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ), the g.f. for partitions into distinct part where the difference between successive parts is >= D. - Joerg Arndt, Mar 31 2014
G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)). - Jon Perry, Jul 06 2004
G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 1, 0, 0, 1, 0, ...]. - Michael Somos, Oct 15 2008
Expansion of f(-x^5) / f(-x^1, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, May 17 2015
Expansion of f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015
a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * 5^(1/2) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(60*sqrt(15))) / sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 23 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

A000587 Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).

Original entry on oeis.org

1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, -2180, -17731, -50533, 110176, 1966797, 9938669, 8638718, -278475061, -2540956509, -9816860358, 27172288399, 725503033401, 5592543175252, 15823587507881, -168392610536153, -2848115497132448, -20819319685262839
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Alternating row sums of Stirling2 triangle A048993.
Related to the matrix-exponential of the Pascal-matrix, see A000110 and A011971. - Gottfried Helms, Apr 08 2007
Closely linked to A000110 and especially the contribution there of Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007, by offering what is a complementary finding.
Number of set partitions of 1..n with an even number of parts, minus the number of such partitions with an odd number of parts. - Franklin T. Adams-Watters, May 04 2010
After -2, the smallest prime is a(36) = -1454252568471818731501051, no others through a(100). What is the first prime >0 in the sequence? - Jonathan Vos Post, Feb 02 2011
a(723) ~ 1.9*10^1265 is almost certainly prime. - D. S. McNeil, Feb 02 2011
Stirling transform of a(n) = [1, -1, 0, 1, 1, ...] is A033999(n) = [1, -1, 1, -1, 1, ...]. - Michael Somos, Mar 28 2012
Negated coefficients in the asymptotic expansion: A005165(n)/n! ~ 1 - 1/n + 1/n^2 + 0/n^3 - 1/n^4 - 1/n^5 + 2/n^6 + 9/n^7 + 9/n^8 - 50/n^9 - 267/n^10 - 413/n^11 + O(1/n^12), starting from the O(1/n) term. - Vladimir Reshetnikov, Nov 09 2016
Named after Venkata Ramamohana Rao Uppuluri and John A. Carpenter of the Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee. They are called "Rényi numbers" by Fekete (1999), after the Hungarian mathematician Alfréd Rényi (1921-1970). - Amiram Eldar, Mar 11 2022

Examples

			G.f. = 1 - x + x^3 + x^4 - 2*x^5 - 9*x^6 - 9*x^7 + 50*x^8 + 267*x^9 + 413*x^10 - ...

References

  • N. A. Kolokolnikova, Relations between sums of certain special numbers (Russian), in Asymptotic and enumeration problems of combinatorial analysis, pp. 117-124, Krasnojarsk. Gos. Univ., Krasnoyarsk, 1976.
  • Alfréd Rényi, Új modszerek es eredmenyek a kombinatorikus analfzisben. I. MTA III Oszt. Ivozl., Vol. 16 (1966), pp. 7-105.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • M. V. Subbarao and A. Verma, Some remarks on a product expansion. An unexplored partition function, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999), pp. 267-283, Kluwer, Dordrecht, 2001.

Links

Crossrefs

Cf. A000110, A011971 (base triangle PE), A078937 (PE^2).
Cf. A007318, A153229, A213170.

Programs

  • Haskell
    a000587 n = a000587_list !! n
a000587_list = 1 : f a007318_tabl [1] where
   f (bs:bss) xs = y : f bss (y : xs) where y = - sum (zipWith (*) xs bs)
-- Reinhard Zumkeller, Mar 04 2014
  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
      add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 28 2016
  • Mathematica
    Table[ -1 * Sum[ (-1)^( k + 1) StirlingS2[ n, k ], {k, 0, n} ], {n, 0, 40} ]
With[{nn=30},CoefficientList[Series[Exp[1-Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 04 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 1 - Exp[x]], {x, 0, n}]]; (* Michael Somos, May 27 2014 *)
a[ n_] := If[ n < 0, 0, With[{m = n + 1}, SeriesCoefficient[ Series[ Nest[ x Factor[ 1 - # /. x -> x / (1 - x)] &, 0, m], {x, 0, m}], {x, 0, m}]]]; (* Michael Somos, May 27 2014 *)
Table[BellB[n, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)
b[1] = 1; k = 1; Flatten[{1, Table[Do[j = k; k -= b[m]; b[m] = j;, {m, 1, n-1}]; b[n] = k; k*(-1)^n, {n, 1, 40}]}] (* Vaclav Kotesovec, Sep 09 2019 *)
  • PARI
    {a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - exp( x + x * O(x^n))), n))}; /* Michael Somos, Mar 14 2011 */
  • PARI
    {a(n) = local(A); if( n<0, 0, n++; A = O(x); for( k=1, n, A = x - x * subst(A, x, x / (1 - x))); polcoeff( A, n))}; /* Michael Somos, Mar 14 2011 */
  • PARI
    Vec(serlaplace(exp(1 - exp(x+O(x^99))))) /* Joerg Arndt, Apr 01 2011 */
  • PARI
    a(n)=round(exp(1)*suminf(k=0,(-1)^k*k^n/k!))
vector(20,n,a(n-1)) \\ Derek Orr, Sep 19 2014 -- a direct approach
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(1 - exp(x)))) \\ Michel Marcus, Sep 19 2014
  • Python
    # The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)] for n > 0.
def A000587_list(n):
    A = [0 for i in range(n)]
    A[n-1] = 1
    R = [1]
    for j in range(0, n):
        A[n-1-j] = -A[n-1]
        for k in range(n-j, n):
            A[k] += A[k-1]
        R.append(A[n-1])
    return R
# Peter Luschny, Apr 18 2011
  • Python
    # Python 3.2 or higher required
from itertools import accumulate
A000587, blist, b = [1,-1], [1], -1
for _ in range(30):
    blist = list(accumulate([b]+blist))
    b = -blist[-1]
    A000587.append(b) # Chai Wah Wu, Sep 19 2014
  • Sage
    expnums(26, -1) # Zerinvary Lajos, May 15 2009

Formula

a(n) = e*Sum_{k>=0} (-1)^k*k^n/k!. - Benoit Cloitre, Jan 28 2003
E.g.f.: exp(1 - e^x).
a(n) = Sum_{k=0..n} (-1)^k S2(n, k), where S2(i, j) are the Stirling numbers of second kind A008277.
G.f.: (x/(1-x))*A(x/(1-x)) = 1 - A(x); the binomial transform equals the negative of the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
With different signs: g.f.: Sum_{k>=0} x^k/Product_{L=1..k} (1 + L*x).
Recurrence: a(n) = -Sum_{i=0..n-1} a(i)*C(n-1, i). - Ralf Stephan, Feb 24 2005
Let P be the lower-triangular Pascal-matrix, PE = exp(P-I) a matrix-exponential in exact integer arithmetic (or PE = lim exp(P)/exp(1) as limit of the exponential); then a(n) = PE^-1 [n,1]. - Gottfried Helms, Apr 08 2007
Take the series 0^n/0! - 1^n/1! + 2^n/2! - 3^n/3! + 4^n/4! + ... If n=0 then the result will be 1/e, where e = 2.718281828... If n=1, the result will be -1/e. If n=2, the result will be 0 (i.e., 0/e). As we continue for higher natural number values of n sequence for the Roa Uppuluri-Carpenter numbers is generated in the numerator, i.e., 1/e, -1/e, 0/e, 1/e, 1/e, -2/e, -9/e, -9/e, 50/e, 267/e, ... . - Peter Collins (pcolins(AT)eircom.net), Jun 04 2007
The sequence (-1)^n*a(n), with general term Sum_{k=0..n} (-1)^(n-k)*S2(n, k), has e.g.f. exp(1-exp(-x)). It also has Hankel transform (-1)^C(n+1,2)*A000178(n) and binomial transform A109747. - Paul Barry, Mar 31 2008
G.f.: 1 / (1 + x / (1 - x / (1 + x / (1 - 2*x / (1 + x / (1 - 3*x / (1 + x / ...))))))). - Michael Somos, May 12 2012
From Sergei N. Gladkovskii, Sep 28 2012 to Feb 07 2014: (Start)
Continued fractions:
G.f.: -1/U(0) where U(k) = x*k - 1 - x + x^2*(k+1)/U(k+1).
G.f.: 1/(U(0)+x) where U(k) = 1 + x - x*(k+1)/(1 + x/U(k+1)).
G.f.: 1+x/G(0) where G(k) = x*k - 1 + x^2*(k+1)/G(k+1).
G.f.: (1 - G(0))/(x+1) where G(k) = 1 - 1/(1-k*x)/(1-x/(x+1/G(k+1) )).
G.f.: 1 + x/(G(0)-x) where G(k) = x*k + 2*x - 1 - x*(x*k+x-1)/G(k+1).
G.f.: G(0)/(1+x), where G(k) = 1-x^2*(k+1)/(x^2*(k+1)+(x*k-1-x)*(x*k-1)/G(k+1)).
(End)
a(n) = B_n(-1), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015
From Mélika Tebni, May 20 2022: (Start)
a(n) = Sum_{k=0..n} (-1)^k*Bell(k)*A129062(n, k).
a(n) = Sum_{k=0..n} (-1)^k*k!*A130191(n, k). (End)

A003422 Left factorials: !n = Sum_{k=0..n-1} k!.

Original entry on oeis.org

0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114, 4037914, 43954714, 522956314, 6749977114, 93928268314, 1401602636314, 22324392524314, 378011820620314, 6780385526348314, 128425485935180314, 2561327494111820314, 53652269665821260314, 1177652997443428940314
Offset: 0

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

Number of {12, 12*, 1*2, 21*}- and {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
a(n) is the number of permutations on [n] that avoid the patterns 2n1 and n12. An occurrence of a 2n1 pattern is a (scattered) subsequence a-n-b with a > b. - David Callan, Nov 29 2007
Also, numbers left over after the following sieving process: At step 1, keep all numbers of the set N = {0, 1, 2, ...}. In step 2, keep only every second number after a(2) = 2: N' = {0, 1, 2, 4, 6, 8, 10, ...}. In step 3, keep every third of the numbers following a(3) = 4, N" = {0, 1, 2, 4, 10, 16, 22, ...}. In step 4, keep every fourth of the numbers beyond a(4) = 10: {0, 1, 2, 4, 10, 34, 58, ...}, and so on. - M. F. Hasler, Oct 28 2010
If s(n) is a second-order recurrence defined as s(0) = x, s(1) = y, s(n) = n*(s(n - 1) - s(n - 2)), n > 1, then s(n) = n*y - n*a(n - 1)*x. - Gary Detlefs, May 27 2012
a(n) is the number of lists of {1, ..., n} with (1st element) = (smallest element) and (k-th element) <> (k-th smallest element) for k > 1, where a list means an ordered subset. a(4) = 10 because we have the lists: [1], [2], [3], [4], [1, 3, 2], [1, 4, 2], [1, 4, 3], [2, 4, 3], [1, 3, 4, 2], [1, 4, 2, 3]. Cf. A000262. - Geoffrey Critzer, Oct 04 2012
Consider a tree graph with 1 vertex. Add an edge to it with another vertex. Now add 2 edges with vertices to this vertex, and then 3 edges to each open vertex of the tree (not the first one!), and the next stage is to add 4 edges, and so on. The total number of vertices at each stage give this sequence (see example). - Jon Perry, Jan 27 2013
Additive version of the superfactorials A000178. - Jon Perry, Feb 09 2013
Repunits in the factorial number system (see links). - Jon Perry, Feb 17 2013
Whether n|a(n) only for 1 and 2 remains an open problem. A published 2004 proof was retracted in 2011. This is sometimes known as Kurepa's conjecture. - Robert G. Wilson v, Jun 15 2013, corrected by Jeppe Stig Nielsen, Nov 07 2015.
!n is not always squarefree for n > 3. Miodrag Zivkovic found that 54503^2 divides !26541. - Arkadiusz Wesolowski, Nov 20 2013
a(n) gives the position of A007489(n) in A227157. - Antti Karttunen, Nov 29 2013
Matches the total domination number of the Bruhat graph from n = 2 to at least n = 5. - Eric W. Weisstein, Jan 11 2019
For the connection with Kurepa trees, see A. Petojevic, The {K_i(z)}{i=1..oo} functions, Rocky Mtn. J. Math., 36 (2006), 1637-1650. - _Aleksandar Petojevic, Jun 29 2018
This sequence converges in the p-adic topology, for every prime number p. - Harry Richman, Aug 13 2024

Examples

			!5 = 0! + 1! + 2! + 3! + 4! = 1 + 1 + 2 + 6 + 24 = 34.
x + 2*x^2 + 4*x^3 + 10*x^4 + 34*x^5 + 154*x^6 + 874*x^7 + 5914*x^8 + 46234*x^9 + ...
From _Arkadiusz Wesolowski_, Aug 06 2012: (Start)
Illustration of initial terms:
.
. o        o         o            o                         o
.          o         o            o                         o
.                   o o          o o                       o o
.                              ooo ooo                   ooo ooo
.                                             oooo oooo oooo oooo oooo oooo
.
. 1        2         4            10                        34
.
(End)
The tree graph. The total number of vertices at each stage is 1, 2, 4, 10, ...
    0 0
    |/
    0-0
   /
0-0
   \
    0-0
    |\
    0 0
- _Jon Perry_, Jan 27 2013

References

  • Richard K. Guy, Unsolved Problems Number Theory, Section B44.
  • D. Kurepa, On the left factorial function !n. Math. Balkanica 1 1971 147-153.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Equals A007489(n-1)+1 for n>=1. Cf. A000142, A014144, A005165.
Twice A014288. See also A049782, A100612.
Cf. A102639, A102411, A102412, A101752, A094216, A094638, A008276, A000166, A000110, A000204, A000045, A000108, A135722, A227157, A000178, A137338, A232845, A357145.

Programs

  • Haskell
    a003422 n = a003422_list !! n
a003422_list = scanl (+) 0 a000142_list
-- Reinhard Zumkeller, Dec 27 2011
  • Maple
    A003422 := proc(n) local k; add(k!,k=0..n-1); end proc:
# Alternative, using the Charlier polynomials A137338:
C := proc(n, x) option remember; if n > 0 then (x-n)*C(n-1, x) - n*C(n-2, x)
elif n = 0 then 1 else 0 fi end: A003422 := n -> (-1)^(n+1)*C(n-1, -1):
seq(A003422(n), n=0..22); # Peter Luschny, Nov 28 2018
# third Maple program:
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+(n-1)!) end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 24 2022
  • Mathematica
    Table[Sum[i!, {i, 0, n - 1}], {n, 0, 20}] (* Stefan Steinerberger, Mar 31 2006 *)
Join[{0}, Accumulate[Range[0, 25]!]] (* Harvey P. Dale, Nov 19 2011 *)
a[0] = 0; a[1] = 1; a[n_] := a[n] = n*a[n - 1] - (n - 1)*a[n - 2]; Array[a, 23, 0] (* Robert G. Wilson v, Jun 15 2013 *)
a[n_] := (-1)^n*n!*Subfactorial[-n-1]-Subfactorial[-1]; Table[a[n] // FullSimplify, {n, 0, 22}] (* Jean-François Alcover, Jan 09 2014 *)
RecurrenceTable[{a[n] == n a[n - 1] - (n - 1) a[n - 2], a[0] == 0, a[1] == 1}, a, {n, 0, 10}] (* Eric W. Weisstein, Jan 11 2019 *)
Range[0, 20]! CoefficientList[Series[(ExpIntegralEi[1] - ExpIntegralEi[1 - x]) Exp[x - 1], {x, 0, 20}], x] (* Eric W. Weisstein, Jan 11 2019 *)
Table[(-1)^n n! Subfactorial[-n - 1] - Subfactorial[-1], {n, 0, 20}] // FullSimplify (* Eric W. Weisstein, Jan 11 2019 *)
Table[(I Pi + ExpIntegralEi[1] + (-1)^n n! Gamma[-n, -1])/E, {n, 0, 20}] // FullSimplify (* Eric W. Weisstein, Jan 11 2019 *)
  • Maxima
    makelist(sum(k!,k,0,n-1), n, 0, 20); /* Stefano Spezia, Jan 11 2019 */
  • PARI
    a003422(n)=sum(k=0,n-1,k!) \\ Charles R Greathouse IV, Jun 15 2011
  • Python
    from itertools import count, islice
def A003422_gen(): # generator of terms
    yield from (0,1)
    c, f = 1, 1
    for n in count(1):
        yield (c:= c + (f:= f*n))
A003422_list = list(islice(A003422_gen(),20)) # Chai Wah Wu, Jun 22 2022
  • Python
    def a(n):
    if n == 0: return 0
    s = f = 1
    for k in range(1, n):
        f *= k
        s += f
    return round(s)
print([a(n) for n in range(24)])  # Peter Luschny, Mar 05 2024

Formula

D-finite with recurrence: a(n) = n*a(n - 1) - (n - 1)*a(n - 2). - Henry Bottomley, Feb 28 2001
Sequence is given by 1 + 1*(1 + 2*(1 + 3*(1 + 4*(1 + ..., terminating in n*(1)...). - Jon Perry, Jun 01 2004
a(n) = Sum_{k=0..n-1} P(n, k) / C(n, k). - Ross La Haye, Sep 20 2004
E.g.f.: (Ei(1) - Ei(1 - x))*exp(-1 + x) where Ei(x) is the exponential integral. - Djurdje Cvijovic and Aleksandar Petojevic, Apr 11 2000
a(n) = Integral_{x = 0..oo} [(x^n - 1)/(x - 1)]*exp(-x) dx. - Gerald McGarvey, Oct 12 2007
A007489(n) = !(n + 1) - 1 = a(n + 1) - 1. - Artur Jasinski, Nov 08 2007. Typos corrected by Antti Karttunen, Nov 29 2013
Starting (1, 2, 4, 10, 34, 154, ...), = row sums of triangle A135722. - Gary W. Adamson, Nov 25 2007
a(n) = a(n - 1) + (n - 1)! for n >= 2. - Jaroslav Krizek, Jun 16 2009
E.g.f. A(x) satisfies the differential equation A'(x) = A(x) + 1/(1 - x). - Vladimir Kruchinin, Jan 19 2011
a(n + 1) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A182386(k) for k = 0, 1, ..., n. - Michael Somos, Apr 27 2012
From Sergei N. Gladkovskii, May 09 2013 to Oct 22 2013: (Start)
Continued fractions:
G.f.: x/(1-x)*Q(0) where Q(k) = 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/Q(k+1))).
G.f.: G(0)*x/(1-x)/2 where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))).
G.f.: 2*x/(1-x)/G(0) where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))).
G.f.: W(0)*x/(1+sqrt(x))/(1-x) where W(k) = 1 + sqrt(x)/(1 - sqrt(x)*(k+1)/(sqrt(x)*(k+1) + 1/W(k+1))).
G.f.: B(x)*(1+x)/(1-x) where B(x) is the g.f. of A153229.
G.f.: x/(1-x) + x^2/(1-x)/Q(0) where Q(k) = 1 - 2*x*(2*k+1) - x^2*(2*k+1)*(2*k+2)/(1 - 2*x*(2*k+2) - x^2*(2*k+2)*(2*k+3)/Q(k+1)).
G.f.: x*(1+x)*B(x) where B(x) is the g.f. of A136580. (End)
a(n) = (-1)^(n+1)*C(n-1, -1) where C(n, x) are the Charlier polynomials (with parameter a=1) as given in A137338. (Evaluation at x = 1 gives A232845.) - Peter Luschny, Nov 28 2018
a(n) = (a(n-3)*(n-2)^2*(n-3)! + a(n-1)^2)/a(n-2) (empirical). - Gary Detlefs, Feb 25 2022
a(n) = signum(n)/b(1,n) with b(i,n) = i - [iMohammed Bouras, Sep 07 2022
Sum_{n>=1} 1/a(n) = A357145. - Amiram Eldar, Oct 01 2022

A005846 Primes of the form k^2 + k + 41.

Original entry on oeis.org

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1847, 1933, 2111, 2203, 2297, 2393, 2591, 2693, 2797
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Note that 41 is the largest of Euler's Lucky numbers (A014556). - Lekraj Beedassy, Apr 22 2004
a(n) = A117530(13, n) for n <= 13: a(1) = A117530(13, 1) = A014556(6) = 41, A117531(13) = 13. - Reinhard Zumkeller, Mar 26 2006
The link to E. Wegrzynowski contains the following incorrect statement: "It is possible to find a polynomial of the form n^2 + n + B that gives prime numbers for n = 0, ..., A, A being any number." It is known that the maximum is A = 39 for B = 41. - Luis Rodriguez (luiroto(AT)yahoo.com), Jun 22 2008
Contrary to the last comment, Mollin's Theorem 2.1 shows that any A is possible if the Prime k-tuples Conjecture is assumed. - T. D. Noe, Aug 31 2009
a(n) can be generated by a recurrence based on the gcd in the type of Eric Rowland and Aldrich Stevens. See the recurrence in PARI under PROG. - Mike Winkler, Oct 02 2013
These primes are not prime in O_(Q(sqrt(-163))). Given p = n^2 + n + 41, we have ((2*n + 1)/2 - sqrt(-163)/2)*((2*n + 1)/2 + sqrt(-163)/2) = p, e.g., 1601 = 39^2 + 39 + 41 = (79/2 - sqrt(-163)/2)*(79/2 + sqrt(-163)/2). - Alonso del Arte, Nov 03 2017
From Peter Bala, Apr 15 2018: (Start)
The polynomial P(n) := n^2 + n + 41 takes distinct prime values for the 40 consecutive integers n = 0 to 39. It follows that the polynomial P(n-40) takes prime values for the 80 consecutive integers n = 0 to 79, consisting of the 40 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n-40) = 4*n^2 - 158*n + 1601 also takes prime values for the 40 consecutive integers n = 0 to 39.
2) The polynomial P(3*n-40) = 9*n^2 - 237*n + 1601 takes prime values for the 27 consecutive integers n = 0 to 26 ( = floor(79/3)). In addition, calculation shows that P(3*n-40) also takes prime values for n from -13 to -1. Equivalently put, the polynomial P(3*n-79) = 9*n^2 - 471*n + 6203 takes prime values for the 40 consecutive integers n = 0 to 39. This result is due to Higgins. Cf. A007635 and A048059. (End)

Examples

			a(39) = 1601 = 39^2 + 39 + 41 is in the sequence because it is prime.
1681 = 40^2 + 40 + 41 is not in the sequence because 1681 = 41*41.

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 225.
  • R. K. Guy, Unsolved Problems Number Theory, Section A1.
  • O. Higgins, Another long string of primes, J. Rec. Math., 14 (1981/2) 185.
  • Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 139, 149.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 115.

Links

Crossrefs

Cf. A048988, A007634, A056561, A002378, A007635.
Intersection of A000040 and A202018; A010051.
Cf. A048059.

Programs

  • GAP
    Filtered(List([0..100],n->n^2+n+41),IsPrime); # Muniru A Asiru, Apr 22 2018
  • Haskell
    a005846 n = a005846_list !! (n-1)
a005846_list = filter ((== 1) . a010051) a202018_list
-- Reinhard Zumkeller, Dec 09 2011
  • Magma
    [a: n in [0..55] | IsPrime(a) where a is n^2+n+ 41]; // Vincenzo Librandi, Apr 24 2018
  • Maple
    for y from 0 to 10 do
U := y^2+y+41;
if isprime(U) = true then print(U) end if ;
end do:
# Matt C. Anderson, Jan 04 2013
  • Mathematica
    Select[Table[n^2 + n + 41, {n, 0, 59}],PrimeQ] (* Alonso del Arte, Dec 08 2011 *)
  • PARI
    for(n=1,1e3,if(isprime(k=n^2+n+41),print1(k", "))) \\ Charles R Greathouse IV, Jul 25 2011
  • PARI
    {k=2; n=1; for(x=1, 100000, f=x^2+x+41; g=x^2+3*x+43; a=gcd(f, g-k); if(a>1, k=k+2); if(a==x+2-k/2, print(n" "a); n++))} \\ Mike Winkler, Oct 02 2013

Formula

a(n) = A056561(n)^2 + A056561(n) + 41.

Extensions

More terms from Henry Bottomley, Jun 26 2000

A005728 Number of fractions in Farey series of order n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 19, 23, 29, 33, 43, 47, 59, 65, 73, 81, 97, 103, 121, 129, 141, 151, 173, 181, 201, 213, 231, 243, 271, 279, 309, 325, 345, 361, 385, 397, 433, 451, 475, 491, 531, 543, 585, 605, 629, 651, 697, 713, 755, 775, 807, 831, 883, 901, 941, 965
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Sometimes called Phi(n).
Leo Moser found an interesting way to generate this sequence, see Gardner.
a(n) is a prime number for nine consecutive values of n: n = 1, 2, 3, 4, 5, 6, 7, 8, 9. - Altug Alkan, Sep 26 2015
Named after the English geologist and writer John Farey, Sr. (1766-1826). - Amiram Eldar, Jun 17 2021

Examples

			a(5)=11 because the fractions are 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1.

References

  • Martin Gardner, The Last Recreations, 1997, chapter 12.
  • Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, a foundation for computer science, Chapter 4.5 - Relative Primality, pages 118 - 120 and Chapter 9 - Asymptotics, Problem 6, pages 448 - 449, Addison-Wesley Publishing Co., Reading, Mass., 1989.
  • William Judson LeVeque, Topics in Number Theory, Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
  • Andrey O. Matveev, Farey Sequences, De Gruyter, 2017, Table 1.7.
  • Leo Moser, Solution to Problem P42, Canadian Mathematical Bulletin, Vol. 5, No. 3 (1962), pp. 312-313.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

For the Farey series see A006842/A006843.
Essentially the same as A049643.
Cf. A002088, A055197, A055201.

Programs

  • GAP
    List([0..60],n->Sum([1..n],i->Phi(i)))+1; # Muniru A Asiru, Jul 31 2018
  • Haskell
    a005728 n = a005728_list
a005728_list = scanl (+) 1 a000010_list
-- Reinhard Zumkeller, Aug 04 2012
  • Magma
    [1] cat [n le 1 select 2 else Self(n-1)+EulerPhi(n): n in [1..60]]; // Vincenzo Librandi, Sep 27 2015
  • Maple
    A005728 := proc(n)
    1+add(numtheory[phi](i),i=1..n) ;
end proc:
seq(A005728(n),n=0..80) ; # R. J. Mathar, Nov 29 2017
  • Mathematica
    Accumulate@ Array[ EulerPhi, 54, 0] + 1
f[n_] := 1 + Sum[ EulerPhi[m], {m, n}]; Array[f, 55, 0] (* or *)
f[n_] := (Sum[ MoebiusMu[m] Floor[n/m]^2, {m, n}] + 3)/2; f[0] = 1; Array[f, 55, 0] (* or *)
f[n_] := n (n + 3)/2 - Sum[f[Floor[n/m]], {m, 2, n}]; f[0] = 1; Array[f, 55, 0] (* Robert G. Wilson v, Sep 26 2015 *)
a[n_] := If[n == 0, 1, FareySequence[n] // Length];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 16 2022 *)
  • PARI
    a(n)=1+sum(k=1,n,eulerphi(k)) \\ Charles R Greathouse IV, Jun 03 2013
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A005728(n): # based on second formula in A018805
    if n == 0:
        return 1
    c, j = -2, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(2*A005728(k1)-3)
        j, k1 = j2, n//j2
    return (n*(n-1)-c+j)//2 # Chai Wah Wu, Mar 24 2021

Formula

a(n) = 1 + Sum_{i=1..n} phi(i).
a(n) = n*(n+3)/2 - Sum_{k=2..n} a(floor(n/k)). - David W. Wilson, May 25 2002
a(n) = a(n-1) + phi(n) with a(0) = 1. - Arkadiusz Wesolowski, Oct 13 2012
a(n) = 1 + A002088(n). - Robert G. Wilson v, Sep 26 2015
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