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.

Showing 1-6 of 6 results.

A002450 a(n) = (4^n - 1)/3.

Original entry on oeis.org

0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, 1398101, 5592405, 22369621, 89478485, 357913941, 1431655765, 5726623061, 22906492245, 91625968981, 366503875925, 1466015503701, 5864062014805, 23456248059221, 93824992236885, 375299968947541
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For n > 0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3) = 21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman, Dec 18 2001
a(n) is composite for all n > 2 and has factors x, (3*x + 2*(-1)^n) where x belongs to A001045. In binary the terms greater than 0 are 1, 101, 10101, 1010101, etc. - John McNamara, Jan 16 2002
Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - R. H. Hardin, Mar 16 2002
The Collatz-function iteration started at a(n), for n >= 1, will end at 1 after 2*n+1 steps. - Labos Elemer, Sep 30 2002 [corrected by Wolfdieter Lang, Aug 16 2021]
Second binomial transform of A001045. - Paul Barry, Mar 28 2003
All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast, Apr 10 2003
Also sum of squares of divisors of 2^(n-1): a(n) = A001157(A000079(n-1)), for n > 0. - Paul Barry, Apr 11 2003
Binomial transform of A000244 (with leading zero). - Paul Barry, Apr 11 2003
Number of walks of length 2n between two vertices at distance 2 in the cycle graph C_6. For n = 2 we have for example 5 walks of length 4 from vertex A to C: ABABC, ABCBC, ABCDC, AFABC and AFEDC. - Herbert Kociemba, May 31 2004
Also number of walks of length 2n + 1 between two vertices at distance 3 in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004
a(n+1) is the number of steps that are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to mark one vertex on the lattice (compare A080674). - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 13 2005
a(n+1) is the sum of square divisors of 4^n. - Paul Barry, Oct 13 2005
a(n+1) is the decimal number generated by the binary bits in the n-th generation of the Rule 250 elementary cellular automaton. - Eric W. Weisstein, Apr 08 2006
a(k) = [M^k]2,1, where M is the 3 X 3 matrix defined as follows: M = [1, 1, 1; 1, 3, 1; 1, 1, 1]. - _Simone Severini, Jun 11 2006
a(n-1) / a(n) = percentage of wasted storage if a single image is stored as a pyramid with a each subsequent higher resolution layer containing four times as many pixels as the previous layer. n is the number of layers. - Victor Brodsky (victorbrodsky(AT)gmail.com), Jun 15 2006
k is in the sequence if and only if C(4k + 1, k) (A052203) is odd. - Paul Barry, Mar 26 2007
This sequence also gives the number of distinct 3-colorings of the odd cycle C(2*n - 1). - Keith Briggs, Jun 19 2007
All numbers of the form m*4^m + (4^m-1)/3 have the property that they are sums of two squares and also their indices are the sum of two squares. This follows from the identity m*4^m + (4^m-1)/3 = 4(4(..4(4m + 1) + 1) + 1) + 1 ..) + 1. - Artur Jasinski, Nov 12 2007
For n > 0, terms are the numbers that, in base 4, are repunits: 1_4, 11_4, 111_4, 1111_4, etc. - Artur Jasinski, Sep 30 2008
Let A be the Hessenberg matrix of order n, defined by: A[1, j] = 1, A[i, i] := 5, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n) = charpoly(A,1). - Milan Janjic, Jan 27 2010
This is the sequence A(0, 1; 3, 4; 2) = A(0, 1; 4, 0; 1) of the family of sequences [a, b : c, d : k] considered by G. Detlefs, and treated as A(a, b; c, d; k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
6*a(n) + 1 is every second Mersenne number greater than or equal to M3, hence all Mersenne primes greater than M2 must be a 6*a(n) + 1 of this sequence. - Roderick MacPhee, Nov 01 2010
Smallest number having alternating bit sum n. Cf. A065359.
For n = 1, 2, ..., the last digit of a(n) is 1, 5, 1, 5, ... . - Washington Bomfim, Jan 21 2011
Rule 50 elementary cellular automaton generates this sequence. This sequence also appears in the second column of array in A173588. - Paul Muljadi, Jan 27 2011
Sequence found by reading the line from 0, in the direction 0, 5, ... and the line from 1, in the direction 1, 21, ..., in the square spiral whose edges are the Jacobsthal numbers A001045 and whose vertices are the numbers A000975. These parallel lines are two semi-diagonals in the spiral. - Omar E. Pol, Sep 10 2011
a(n), n >= 1, is also the inverse of 3, denoted by 3^(-1), Modd(2^(2*n - 1)). For Modd n see a comment on A203571. E.g., a(2) = 5, 3 * 5 = 15 == 1 (Modd 8), because floor(15/8) = 1 is odd and -15 == 1 (mod 8). For n = 1 note that 3 * 1 = 3 == 1 (Modd 2) because floor(3/2) = 1 and -3 == 1 (mod 2). The inverse of 3 taken Modd 2^(2*n) coincides with 3^(-1) (mod 2^(2*n)) given in A007583(n), n >= 1. - Wolfdieter Lang, Mar 12 2012
If an AVL tree has a leaf at depth n, then the tree can contain no more than a(n+1) nodes total. - Mike Rosulek, Nov 20 2012
Also, this is the Lucas sequence V(5, 4). - Bruno Berselli, Jan 10 2013
Also, for n > 0, a(n) is an odd number whose Collatz trajectory contains no odd number other than n and 1. - Jayanta Basu, Mar 24 2013
Sum_{n >= 1} 1/a(n) converges to (3*(log(4/3) - QPolyGamma[0, 1, 1/4]))/log(4) = 1.263293058100271... = A321873. - K. G. Stier, Jun 23 2014
Consider n spheres in R^n: the i-th one (i=1, ..., n) has radius r(i) = 2^(1-i) and the coordinates of its center are (0, 0, ..., 0, r(i), 0, ..., 0) where r(i) is in position i. The coordinates of the intersection point in the positive orthant of these spheres are (2/a(n), 4/a(n), 8/a(n), 16/a(n), ...). For example in R^2, circles centered at (1, 0) and (0, 1/2), and with radii 1 and 1/2, meet at (2/5, 4/5). - Jean M. Morales, May 19 2015
From Peter Bala, Oct 11 2015: (Start)
a(n) gives the values of m such that binomial(4*m + 1,m) is odd. Cf. A003714, A048716, A263132.
2*a(n) = A020988(n) gives the values of m such that binomial(4*m + 2, m) is odd.
4*a(n) = A080674(n) gives the values of m such that binomial(4*m + 4, m) is odd. (End)
Collatz Conjecture Corollary: Except for powers of 2, the Collatz iteration of any positive integer must eventually reach a(n) and hence terminate at 1. - Gregory L. Simay, May 09 2016
Number of active (ON, black) cells at stage 2^n - 1 of the two-dimensional cellular automaton defined by "Rule 598", based on the 5-celled von Neumann neighborhood. - Robert Price, May 16 2016
From Luca Mariot and Enrico Formenti, Sep 26 2016: (Start)
a(n) is also the number of coprime pairs of polynomials (f, g) over GF(2) where both f and g have degree n + 1 and nonzero constant term.
a(n) is also the number of pairs of one-dimensional binary cellular automata with linear and bipermutive local rule of neighborhood size n+1 giving rise to orthogonal Latin squares of order 2^m, where m is a multiple of n. (End)
Except for 0, 1 and 5, all terms are Brazilian repunits numbers in base 4, and so belong to A125134. For n >= 3, all these terms are composite because a(n) = {(2^n-1) * (2^n + 1)}/3 and either (2^n - 1) or (2^n + 1) is a multiple of 3. - Bernard Schott, Apr 29 2017
Given the 3 X 3 matrix A = [2, 1, 1; 1, 2, 1; 1, 1, 2] and the 3 X 3 unit matrix I_3, A^n = a(n)(A - I_3) + I_3. - Nicolas Patrois, Jul 05 2017
The binary expansion of a(n) (n >= 1) consists of n 1's alternating with n - 1 0's. Example: a(4) = 85 = 1010101_2. - Emeric Deutsch, Aug 30 2017
a(n) (n >= 1) is the viabin number of the integer partition [n, n - 1, n - 2, ..., 2, 1] (for the definition of viabin number see comment in A290253). Example: a(4) = 85 = 1010101_2; consequently, the southeast border of the Ferrers board of the corresponding integer partition is ENENENEN, where E = (1, 0), N = (0, 1); this leads to the integer partition [4, 3, 2, 1]. - Emeric Deutsch, Aug 30 2017
Numbers whose binary and Gray-code representations are both palindromes (i.e., intersection of A006995 and A281379). - Amiram Eldar, May 17 2021
Starting with n = 1 the sequence satisfies {a(n) mod 6} = repeat{1, 5, 3}. - Wolfdieter Lang, Jan 14 2022
Terms >= 5 are those q for which the multiplicative order of 2 mod q is floor(log_2(q)) + 2 (and which is 1 more than the smallest possible order for any q). - Tim Seuré, Mar 09 2024
The order of 2 modulo a(n) is 2*n for n >= 2. - Joerg Arndt, Mar 09 2024

Examples

			Apply Collatz iteration to 9: 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1.
Apply Collatz iteration to 27: 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1. [Corrected by _Sean A. Irvine_ at the suggestion of Stephen Cornelius, Mar 04 2024]
a(5) = (4^5 - 1)/3 = 341 = 11111_4 = {(2^5 - 1) * (2^5 + 1)}/3 = 31 * 33/3 = 31 * 11. - _Bernard Schott_, Apr 29 2017

References

  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
  • 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

Partial sums of powers of 4, A000302.
When converted to binary, this gives A094028.
Subsequence of A003714.
Primitive factors: A129735.
Cf. A000225, A002446, A006995, A007583, A018215, A020988, A024036, A047849, A048716, A080355, A080674, A112627, A113860, A139391, A155701, A156605, A160967, A163834, A163868, A178415, A263132, A281379, A347834.

Programs

  • GAP
    List([0..25], n -> (4^n-1)/3); # Muniru A Asiru, Feb 18 2018
  • Haskell
    a002450 = (`div` 3) . a024036
a002450_list = iterate ((+ 1) . (* 4)) 0
-- Reinhard Zumkeller, Oct 03 2012
  • Magma
    [ (4^n-1)/3: n in [0..25] ]; // Klaus Brockhaus, Oct 28 2008
  • Magma
    [n le 2 select n-1 else 5*Self(n-1)-4*Self(n-2): n in [1..70]]; // Vincenzo Librandi, Jun 13 2015
  • Maple
    [seq((4^n-1)/3,n=0..40)];
A002450:=1/(4*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation, dropping the initial zero
  • Mathematica
    Table[(4^n - 1)/3, {n, 0, 127}] (* Vladimir Joseph Stephan Orlovsky, Sep 29 2008 *)
LinearRecurrence[{5, -4}, {0, 1}, 30] (* Harvey P. Dale, Jun 23 2013 *)
  • Maxima
    makelist((4^n-1)/3, n, 0, 30); /* Martin Ettl, Nov 05 2012 */
  • PARI
    a(n) = (4^n-1)/3;
  • PARI
    my(z='z+O('z^40)); Vec(z/((1-z)*(1-4*z))) \\ Altug Alkan, Oct 11 2015
  • Python
    def A002450(n): return ((1<<(n<<1))-1)//3 # Chai Wah Wu, Jan 29 2023
  • Scala
    ((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * )).scanLeft(0: BigInt)( + ) // Alonso del Arte, Sep 17 2019

Formula

From Wolfdieter Lang, Apr 24 2001: (Start)
a(n+1) = Sum_{m = 0..n} A060921(n, m).
G.f.: x/((1-x)*(1-4*x)). (End)
a(n) = Sum_{k = 0..n-1} 4^k; a(n) = A001045(2*n). - Paul Barry, Mar 17 2003
E.g.f.: (exp(4*x) - exp(x))/3. - Paul Barry, Mar 28 2003
a(n) = (A007583(n) - 1)/2. - N. J. A. Sloane, May 16 2003
a(n) = A000975(2*n)/2. - N. J. A. Sloane, Sep 13 2003
a(n) = A084160(n)/2. - N. J. A. Sloane, Sep 13 2003
a(n+1) = 4*a(n) + 1, with a(0) = 0. - Philippe Deléham, Feb 25 2004
a(n) = Sum_{i = 0..n-1} C(2*n - 1 - i, i)*2^i. - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
a(n+1) = Sum_{k = 0..n} binomial(n+1, k+1)*3^k. - Paul Barry, Aug 20 2004
a(n) = center term in M^n * [1 0 0], where M is the 3 X 3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 0 0] = [A007583(n-1) a(n) A007583(n-1)]. E.g., a(4) = 85 since M^4 * [1 0 0] = [43 85 43] = [A007583(3) a(4) A007583(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = Sum_{k = 0..n, j = 0..n} C(n, j)*C(j, k)*A001045(j - k). - Paul Barry, Feb 15 2005
a(n) = Sum_{k = 0..n} C(n, k)*A001045(n-k)*2^k = Sum_{k = 0..n} C(n, k)*A001045(k)*2^(n-k). - Paul Barry, Apr 22 2005
a(n) = A125118(n, 3) for n > 2. - Reinhard Zumkeller, Nov 21 2006
a(n) = Sum_{k = 0..n} 2^(n - k)*A128908(n, k), n >= 1. - Philippe Deléham, Oct 19 2008
a(n) = Sum_{k = 0..n} A106566(n, k)*A100335(k). - Philippe Deléham, Oct 30 2008
If we define f(m, j, x) = Sum_{k = j..m} binomial(m, k)*stirling2(k, j)*x^(m - k) then a(n-1) = f(2*n, 4, -2), n >= 2. - Milan Janjic, Apr 26 2009
a(n) = A014551(n) * A001045(n). - R. J. Mathar, Jul 08 2009
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3) = 5*a(n-1) - 4*a(n-2), a(0) = 0, a(1) = 1, a(2) = 5. - Wolfdieter Lang, Oct 18 2010
a(0) = 0, a(n+1) = a(n) + 2^(2*n). - Washington Bomfim, Jan 21 2011
A036555(a(n)) = 2*n. - Reinhard Zumkeller, Jan 28 2011
a(n) = Sum_{k = 1..floor((n+2)/3)} C(2*n + 1, n + 2 - 3*k). - Mircea Merca, Jun 25 2011
a(n) = Sum_{i = 1..n} binomial(2*n + 1, 2*i)/3. - Wesley Ivan Hurt, Mar 14 2015
a(n+1) = 2^(2*n) + a(n), a(0) = 0. - Ben Paul Thurston, Dec 27 2015
a(k*n)/a(n) = 1 + 4^n + ... + 4^((k-1)*n). - Gregory L. Simay, Jun 09 2016
Dirichlet g.f.: (PolyLog(s, 4) - zeta(s))/3. - Ilya Gutkovskiy, Jun 26 2016
A000120(a(n)) = n. - André Dalwigk, Mar 26 2018
a(m) divides a(m*n), in particular: a(2*n) == 0 (mod 5), a(3*n) == 0 (mod 3*7), a(5*n) == 0 (mod 11*31), etc. - M. F. Hasler, Oct 19 2018
a(n) = 4^(n-1) + a(n-1). - Bob Selcoe, Jan 01 2020
a(n) = A178415(1, n) = A347834(1, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021
a(n) = A000225(2*n)/3. - John Keith, Jan 22 2022
a(n) = A080674(n) + 1 = A047849(n) - 1 = A163834(n) - 2 = A155701(n) - 3 = A163868(n) - 4 = A156605(n) - 7. - Ray Chandler, Jun 16 2023
From Peter Bala, Jul 23 2025: (Start)
The following are examples of telescoping products. Cf. A016153:
Product_{k = 1..2*n} 1 + 2^k/a(k+1) = a(n+1)/A007583(n) = (4^(n+1) - 1)/(2*4^n + 1).
Hence, Product_{k >= 1} 1 + 2^k/a(k+1) = 2.
Product_{k >= 1} 1 - 2^k/a(k+1) = 2/5, since 1 - 2^n/a(n+1) = b(n)/b(n-1), where b(n) = 2 - 3/(1 - 2^(n+1)).
Product_{k >= 1} 1 + (-2)^k/a(k+1) = 2/3, since 1 + (-2)^n/a(n+1) = c(n)/c(n-1), where c(n) = 2 - 1/(1 + (-2)^(n+1)).
Product_{k >= 1} 1 - (-2)^k/a(k+1) = 6/5, since 1 - (-2)^n/a(n+1) = d(n)/d(n-1), where d(n) = 2 - 1/(1 - (-2)^(n+1)). (End)

A000918 a(n) = 2^n - 2.

Original entry on oeis.org

-1, 0, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182, 34359738366, 68719476734, 137438953470
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For n > 1, a(n) is the expected number of tosses of a fair coin to get n-1 consecutive heads. - Pratik Poddar, Feb 04 2011
For n > 2, Sum_{k=1..a(n)} (-1)^binomial(n, k) = A064405(a(n)) + 1 = 0. - Benoit Cloitre, Oct 18 2002
For n > 0, the number of nonempty proper subsets of an n-element set. - Ross La Haye, Feb 07 2004
Numbers j such that abs( Sum_{k=0..j} (-1)^binomial(j, k)*binomial(j + k, j - k) ) = 1. - Benoit Cloitre, Jul 03 2004
For n > 2 this formula also counts edge-rooted forests in a cycle of length n. - Woong Kook (andrewk(AT)math.uri.edu), Sep 08 2004
For n >= 1, conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4, 5, 6 and 7 as a digit. - Alexandre Wajnberg, Apr 25 2005
Beginning with a(2) = 2, these are the partial sums of the subsequence of A000079 = 2^n beginning with A000079(1) = 2. Hence for n >= 2 a(n) is the smallest possible sum of exactly one prime, one semiprime, one triprime, ... and one product of exactly n-1 primes. A060389 (partial sums of the primorials, A002110, beginning with A002110(1)=2) is the analog when all the almost primes must also be squarefree. - Rick L. Shepherd, May 20 2005
From the second term on (n >= 1), the binary representation of these numbers is a 0 preceded by (n - 1) 1's. This pattern (0)111...1110 is the "opposite" of the binary 2^n+1: 1000...0001 (cf. A000051). - Alexandre Wajnberg, May 31 2005
The numbers 2^n - 2 (n >= 2) give the positions of 0's in A110146. Also numbers k such that k^(k + 1) = 0 mod (k + 2). - Zak Seidov, Feb 20 2006
Partial sums of A155559. - Zerinvary Lajos, Oct 03 2007
Number of surjections from an n-element set onto a two-element set, with n >= 2. - Mohamed Bouhamida, Dec 15 2007
It appears that these are the numbers n such that 3*A135013(n) = n*(n + 1), thus answering Problem 2 on the Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993 (see link Japanese Mathematical Olympiad).
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is a proper subset of y or y is a proper subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009
The permutohedron Pi_n has 2^n - 2 facets [Pashkovich]. - Jonathan Vos Post, Dec 17 2009
First differences of A005803. - Reinhard Zumkeller, Oct 12 2011
For n >= 1, a(n + 1) is the smallest even number with bit sum n. Cf. A069532. - Jason Kimberley, Nov 01 2011
a(n) is the number of branches of a complete binary tree of n levels. - Denis Lorrain, Dec 16 2011
For n>=1, a(n) is the number of length-n words on alphabet {1,2,3} such that the gap(w)=1. For a word w the gap g(w) is the number of parts missing between the minimal and maximal elements of w. Generally for words on alphabet {1,2,...,m} with g(w)=g>0 the e.g.f. is Sum_{k=g+2..m} (m - k + 1)*binomial((k - 2),g)*(exp(x) - 1)^(k - g). a(3)=6 because we have: 113, 131, 133, 311, 313, 331. Cf. A240506. See the Heubach/Mansour reference. - Geoffrey Critzer, Apr 13 2014
For n > 0, a(n) is the minimal number of internal nodes of a red-black tree of height 2*n-2. See the Oct 02 2015 comment under A027383. - Herbert Eberle, Oct 02 2015
Conjecture: For n>0, a(n) is the least m such that A007814(A000108(m)) = n-1. - L. Edson Jeffery, Nov 27 2015
Actually this follows from the procedure for determining the multiplicity of prime p in C(n) given in A000108 by Franklin T. Adams-Watters: For p=2, the multiplicity is the number of 1 digits minus 1 in the binary representation of n+1. Obviously, the smallest k achieving "number of 1 digits" = k is 2^k-1. Therefore C(2^k-2) is divisible by 2^(k-1) for k > 0 and there is no smaller m for which 2^(k-1) divides C(m) proving the conjecture. - Peter Schorn, Feb 16 2020
For n >= 0, a(n) is the largest number you can write in bijective base-2 (a.k.a. the dyadic system, A007931) with n digits. - Harald Korneliussen, May 18 2019
The terms of this sequence are also the sum of the terms in each row of Pascal's triangle other than the ones. - Harvey P. Dale, Apr 19 2020
For n > 1, binomial(a(n),k) is odd if and only if k is even. - Charlie Marion, Dec 22 2020
For n >= 2, a(n+1) is the number of n X n arrays of 0's and 1's with every 2 X 2 square having density exactly 2. - David desJardins, Oct 27 2022
For n >= 1, a(n+1) is the number of roots of unity in the unique degree-n unramified extension of the 2-adic field Q_2. Note that for each p, the unique degree-n unramified extension of Q_p is the splitting field of x^(p^n) - x, hence containing p^n - 1 roots of unity for p > 2 and 2*(2^n - 1) for p = 2. - Jianing Song, Nov 08 2022

Examples

			a(4) = 14 because the 14 = 6 + 4 + 4 rationals (in lowest terms) for n = 3 are (see the Jun 14 2017 formula above): 1/2, 1, 3/2, 2, 5/2, 3; 1/4, 3/4, 5/4, 7/4; 1/8, 3/8, 5/8, 7/8. - _Wolfdieter Lang_, Jun 14 2017

References

  • H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
  • Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, Addison-Wesley, 2004, p. 134. - Mohammad K. Azarian, Oct 27 2011
  • S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009 page 86, Exercise 3.16.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
  • 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).
  • A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Links

Crossrefs

Row sums of triangle A026998.
Cf. A000919, A001117, A001118, A095121, A110146.
Cf. A033484, A000225, A000325, A095151, A125128.
Cf. A058809 (3^n-3, n>0).

Programs

  • Haskell
    a000918 = (subtract 2) . (2 ^)
a000918_list = iterate ((subtract 2) . (* 2) . (+ 2)) (- 1)
-- Reinhard Zumkeller, Apr 23 2013
  • Magma
    [2^n - 2: n in [0..40]]; // Vincenzo Librandi, Jun 23 2011
  • Maple
    seq(2^n-2,n=0..20) ;
  • Mathematica
    Table[2^n - 2, {n, 0, 29}] (* Alonso del Arte, Dec 01 2012 *)
  • PARI
    a(n)=2^n-2 \\ Charles R Greathouse IV, Jun 16 2011
  • Python
    def A000918(n): return (1<Chai Wah Wu, Jun 10 2025

Formula

a(n) = 2*A000225(n-1).
G.f.: 1/(1 - 2*x) - 2/(1 - x), e.g.f.: (e^x - 1)^2 - 1. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
For n >= 1, a(n) = A008970(n + 1, 2). - Philippe Deléham, Feb 21 2004
G.f.: (3*x - 1)/((2*x - 1)*(x - 1)). - Simon Plouffe in his 1992 dissertation for the sequence without the leading -1
a(n) = 2*a(n - 1) + 2. - Alexandre Wajnberg, Apr 25 2005
a(n) = A000079(n) - 2. - Omar E. Pol, Dec 16 2008
a(n) = A058896(n)/A052548(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = A164874(n - 1, n - 1) for n > 1. - Reinhard Zumkeller, Aug 29 2009
a(n) = A173787(n,1); a(n) = A028399(2*n)/A052548(n), n > 0. - Reinhard Zumkeller, Feb 28 2010
a(n + 1) = A027383(2*n - 1). - Jason Kimberley, Nov 02 2011
G.f.: U(0) - 1, where U(k) = 1 + x/(2^k + 2^k/(x - 1 - x^2*2^(k + 1)/(x*2^(k + 1) - (k + 1)/U(k + 1) ))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n+1) is the sum of row n in triangle A051601. - Reinhard Zumkeller, Aug 05 2013
a(n+1) = A127330(n,0). - Reinhard Zumkeller, Nov 16 2013
a(n) = Sum_{k=1..n-1} binomial(n, k) for n > 0. - Dan McCandless, Nov 14 2015
From Miquel Cerda, Aug 16 2016: (Start)
a(n) = A000225(n) - 1.
a(n) = A125128(n-1) - A000325(n).
a(n) = A095151(n) - A125128(n) - 1. (End)
a(n+1) = 2*(n + Sum_{j=1..n-1} (n-j)*2^(j-1)), n >= 1. This is the number of the rationals k/2, k = 1..2*n for n >= 1 and (2*k+1)/2^j for j = 2..n, n >= 2, and 2*k+1 < n-(j-1). See the example for n = 3 below. Motivated by the proposal A287012 by Mark Rickert. - Wolfdieter Lang, Jun 14 2017

Extensions

Maple programs fixed by Vaclav Kotesovec, Dec 13 2014

A052548 a(n) = 2^n + 2.

Original entry on oeis.org

3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

The most "compact" sequence that satisfies Bertrand's Postulate. Begin with a(1) = 3 = n, then 2n - 2 = 4 = n_1, 2n_1 - 2 = 6 = n_2, 2n_2 - 2 = 10, etc. = a(n), hence there is guaranteed to be at least one prime between successive members of the sequence. - Andrew S. Plewe, Dec 11 2007
Number of 2-sided prudent polygons of area n, for n>0, see Beaton, p. 5. - Jonathan Vos Post, Nov 30 2010

Links

Crossrefs

Apart from initial term, same as A056469.
Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774, A079004, A058481, A100585, A100586, A058896, A000918, A173786.
Cf. also A000079, A000051, A100314.

Programs

  • Haskell
    a052548 = (+ 2) . a000079
a052548_list = iterate ((subtract 2) . (* 2)) 3
-- Reinhard Zumkeller, Sep 05 2015
  • Magma
    [2^n + 2: n in [0..35]]; // Vincenzo Librandi, Apr 29 2011
  • Maple
    spec := [S,{S=Union(Sequence(Union(Z,Z)),Sequence(Z),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    2^Range[0,40]+2 (* Harvey P. Dale, Jun 26 2012 *)
  • PARI
    a(n)=1<Charles R Greathouse IV, Nov 20 2011

Formula

G.f.: (3-5*x)/((1-2*x)*(1-x)) = (3-5*x)/(1 - 3*x + 2*x^2) = 2/(1-x) + 1/(1-2*x).
a(0)=3, a(1)=4, a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = A058896(n)/A000918(n), for n>0. - Reinhard Zumkeller, Feb 14 2009
a(n) = A173786(n,1), for n>0. - Reinhard Zumkeller, Feb 28 2010
a(n)*A000918(n) = A028399(2*n), for n>0. - Reinhard Zumkeller, Feb 28 2010
a(0)=3, a(n) = 2*a(n-1) - 2. - Vincenzo Librandi, Aug 06 2010
E.g.f.: (2 + exp(x))*exp(x). - Ilya Gutkovskiy, Aug 16 2016

Extensions

More terms from James Sellers, Jun 06 2000

A141725 a(n) = 4^(n+1) - 3.

Original entry on oeis.org

1, 13, 61, 253, 1021, 4093, 16381, 65533, 262141, 1048573, 4194301, 16777213, 67108861, 268435453, 1073741821, 4294967293, 17179869181, 68719476733, 274877906941, 1099511627773, 4398046511101, 17592186044413, 70368744177661
Offset: 0

Views

Author

Paul Curtz, Sep 13 2008

Keywords

Comments

Inverse binomial transform yields A003946 with A003946(1)=4 deleted. - R. J. Mathar, Sep 13 2008
Starting with n=1, binary numbers of the form 1X01 where X is an odd number of 1's. - Brad Clardy, Mar 22 2011
Column 4 of A193871. - Omar E. Pol, Aug 22 2011
The Sierpinski square curve is a representation of this sequence, where a(n) is the number squares filled by the Sierpinski (space-filling) square curve. The square footprint expands at a rate of (2^n-1)^2 (A000225)^2. The number of nodes per iteration grows at a rate of (4^n-1)/3 (A002450). See illustration in links. - John Elias, Jul 25 2022

Links

Crossrefs

Cf. A001045, A036563.

Programs

Formula

a(n) = 10*A001045(2*n) + A001045(2*n+1).
a(n) = 4*a(n-1) + 9 for n > 0, a(0) = 1.
a(n) = A036563(2*n+2).
G.f.: (1 + 8*x)/((1 - x)*(1 - 4*x)). - R. J. Mathar, Sep 13 2008
a(n) = 4^n - 3, with offset 1. - Omar E. Pol, Aug 22 2011
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1, a(0) = 1, a(1) = 13. - Harvey P. Dale, Sep 25 2011
E.g.f.: exp(4*x) - 3*exp(x). - Elmo R. Oliveira, Nov 15 2023

Extensions

Edited by N. J. A. Sloane, Sep 13 2008
More terms from R. J. Mathar, Sep 13 2008

A304330 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 12, 0, 1, 60, 360, 0, 1, 252, 5040, 20160, 0, 1, 1020, 52920, 604800, 1814400, 0, 1, 4092, 506880, 12640320, 99792000, 239500800, 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600, 0, 1, 65532, 42653520, 3952428480, 111567456000, 1264085222400, 6102480384000, 10461394944000
Offset: 0

Views

Author

Peter Luschny, May 11 2018

Keywords

Examples

			Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1,    12;
  [3] 0, 1,    60,     360;
  [4] 0, 1,   252,    5040,     20160;
  [5] 0, 1,  1020,   52920,    604800,    1814400;
  [6] 0, 1,  4092,  506880,  12640320,   99792000,   239500800;
  [7] 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600;

Links

Crossrefs

Row sums are A100872, T(n,2) = A058896, T(n,n) = A002674, T(n,n-1)= A091032.
Cf. A304334, A304336.

Programs

  • Maple
    T := (n, k) -> add((-1)^j*binomial(2*k,j)*(k-j)^(2*n), j=0..k):
for n from 0 to 8 do seq(T(n, k), k=0..n) od;
  • PARI
    T(n, k) = sum(j=0, k, (-1)^j*binomial(2*k, j)*(k - j)^(2*n)); \\ Michel Marcus, Aug 03 2025

A277799 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 1, 12, 1, 60, 1, 252, 1, 1020, 1, 4092, 1, 16380, 1, 65532, 1, 262140, 1, 1048572, 1, 4194300, 1, 16777212, 1, 67108860, 1, 268435452, 1, 1073741820, 1, 4294967292, 1, 17179869180, 1, 68719476732, 1, 274877906940, 1, 1099511627772, 1, 4398046511100, 1
Offset: 0

Views

Author

Robert Price, Oct 31 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Rule numbers 1, 9, 17, 25, 257, 265, 273 and 281 all generate this sequence.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A277797, A277798, A277800.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=1; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],2], {i,1,stages-1}]

Formula

Conjectures from Colin Barker, Nov 01 2016: (Start)
G.f.: (1 - 4*x^2 + 12*x^3) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
a(n) = (-3/2-(-2)^n+(5*(-1)^n)/2+2^n). (End)
Showing 1-6 of 6 results.

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