A058368 -id:A058368 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

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

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.

Indranil Ghosh, Table of n, a(n) for n = 1..4780 (terms 1..500 computed by N. J. A. Sloane)
Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
Arno Berger and Theodore P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
J. Brown and R. L. Duncan, Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences, Fibonacci Quarterly 8 (1970) 482-486.
Enrico Di Cera and Yong Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3
Sergio Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
Scott Garrabrant and Igor Pak, Counting with irrational tiles, arXiv:1407.8222 [math.CO], 2014.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Sarah H. Holliday and Takao Komatsu, On the Sum of Reciprocal Generalized Fibonacci Numbers, Integers. Volume 11, Issue 4, Pages 441-455.
R. Jovanovic, First 70 Lucas numbers
Blair Kelly, Factorizations of Lucas numbers
Tanya Khovanova, Recursive Sequences
Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
Ron Knott, The Lucas Numbers in Pascal's Triangle.
Kantaphon Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1.
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
Mathilde Noual, Dynamics in parallel of double Boolean automata circuits, arXiv:1011.3930 [cs.DM], 2010. - _N. J. A. Sloane_, Jul 07 2012
Mathilde Noual, Dynamics of Circuits and Intersecting Circuits, in Language and Automata Theory and Applications, Lecture Notes in Computer Science, 2012, Volume 7183/2012, 433-444. - _N. J. A. Sloane_, Jul 07 2012
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012.
José L. Ramírez and Gustavo N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).
Mark A. Shattuck and Carl G. Wagner, Periodicity and Parity Theorems for a Statistic on r-Mino Arrangements, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.6.
N. J. A. Sloane, Illustration of initial terms: the Lucas tree
Zdzisław Skupień, Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets, Discussiones Mathematicae Graph Theory. Volume 33, Issue 1, Pages 217-229, ISSN (Print) 2083-5892, DOI: 10.7151/dmgt.1658, April 2013.
Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Lucas n-Step Number
Richard J. Yanco, Letter and Email to N. J. A. Sloane, 1994
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).
Index entries for sequences related to Benford's law

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

a000204 n = a000204_list !! n
a000204_list = 1 : 3 : zipWith (+) a000204_list (tail a000204_list)
-- Reinhard Zumkeller, Dec 18 2011

[Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 17 2017

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

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 *)

A000204(n)=fibonacci(n+1)+fibonacci(n-1) \\ Michael B. Porter, Nov 05 2009

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

[(i:=-1)+(j:=2)] + [(j:=i+j)+(i:=j-i) for  in range(100)] # _Jwalin Bhatt, Apr 02 2025

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

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

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

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

Plouffe Maple line edited by N. J. A. Sloane, May 13 2008

A001609 a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).

Original entry on oeis.org

1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575, 2033590, 2980371, 4367946, 6401536, 9381907

Offset: 1

N. J. A. Sloane

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.
The sequence defined by {a(n) - 1} plays a role for the computation of A065414, A146486, A146487, and A146488 equivalent to the role of A001610 for A005596, A146482, A146483 and A146484, see the variable a_{2,n} in arXiv:0903.2514. - R. J. Mathar, Mar 28 2009
Except for n = 2, a(n) is the number of digits in n-th term of A049064. This can be derived form the T. Sillke link below. - Jianing Song, Apr 28 2019

			G.f. = x + x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 15*x^7 + 21*x^8 + ...

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).

Indranil Ghosh, Table of n, a(n) for n = 1..6012 (terms 1..500 from T. D. Noe)
Ignacio Cascudo, On squares of cyclic codes, arXiv:1703.01267 [cs.IT], 2017. See Theorem 6.1/Table 1.
Johann Cigler, Some remarks on generalized Fibonacci and Lucas polynomials, arXiv:1912.06651 [math.CO], 2019.
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Daniel C. Fielder, Errata:Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 91.
Matthew Macauley, Jon McCammond, Henning S. Mortveit, Dynamics groups of asynchronous cellular automata, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.
M. Newman and D. Shanks, On a sequence arising in series for pi, Math. Comp., 42 (1984), 199-217 (see Eq. 29).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Souvik Roy, Nazim Fatès, and Sukanta Das, Reversibility of Elementary Cellular Automata with fully asynchronous updating: an analysis of the rules with partial recurrence, hal-04456320 [nlin.CG], [cs], 2024. See p. 19.
T. Sillke, The binary form of Conway's sequence
Z. Skupien, Sparse Hamiltonian 2-decompositions together with exact count of numerous Hamiltonian cycles, Discr. Math., 309 (2009), 6382-6390. - _N. J. A. Sloane_, Feb 12 2010
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000204, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A000930, A218439.

Cf. also A049064, A049194.

I:=[1,1,4]; [n le 3 select I[n] else Self(n-1)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Jun 28 2015

A001609:=-(1+3*z**2)/(-1+z+z**3); # Simon Plouffe in his 1992 dissertation
f:= gfun:-rectoproc({a(n) = a(n-1) + a(n-3), a(1)=1,a(2)=1,a(3)=4},a(n),remember):
map(f, [$1..100]); # Robert Israel, Jun 29 2015

Table[Tr[MatrixPower[{{0, 0, 1}, {1, 0, 0}, {0, 1, 1}}, n]], {n, 1, 60}] (* Artur Jasinski, Jan 10 2007 *)
Table[ HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -(n/3)}, {1/2 - n/2, 1 - n/2}, -(27/4)], {n, 20}] (* Alexander R. Povolotsky, Nov 21 2008 *)
a[1] = a[2] = 1; a[3] = 4; m = 3; a[n_] := 1 + n*Sum [Binomial [n - 1 - (m - 1)*i, i - 1]/i, {i, n/m}] A001609 = Table[a[n], {n, 100}] (* Zak Seidov, Nov 21 2008 *)
LinearRecurrence[{1, 0, 1}, {1, 1, 4}, 50] (* Vincenzo Librandi, Jun 28 2015 *)

{a(n) = if( n<1, n=-n; polcoeff( (3 + x^2) / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( x * (1 + 3*x^2) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, Aug 15 2016 */

G.f.: x*(1 + 3*x^2)/(1 - x - x^3).
a(n) = trace of successive powers of matrix ({{0,0,1},{1,0,0},{0,1,1}})^n. - Artur Jasinski, Jan 10 2007
a(n) = A000930(n) + 3*A000930(n-2). - R. J. Mathar, Nov 16 2007
Logarithmic derivative of Narayana's cows sequence A000930. - Paul D. Hanna, Oct 28 2012
a(n) = w1^n + w2^n + w3^n, where w1,w2,w3 are the roots of the cubic: (-1 - x^2 + x^3), see A092526. - Gerry Martens, Jun 27 2015

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
More terms from Michael Somos, Oct 03 2002
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A014097 a(n) = a(n-1)+a(n-4).

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 8, 13, 19, 26, 34, 47, 66, 92, 126, 173, 239, 331, 457, 630, 869, 1200, 1657, 2287, 3156, 4356, 6013, 8300, 11456, 15812, 21825, 30125, 41581, 57393, 79218, 109343, 150924, 208317, 287535

Offset: 1

David Broadhurst

Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 4 sites wide.

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.

Indranil Ghosh, Table of n, a(n) for n = 1..7130
D. J. Broadhurst, Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams, arXiv:hep-th/9612012, 1996.
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Cf. A020999, A000204, A001609, A000079, A003269, A003520, A005708, A005709, A005710.

LinearRecurrence[{1,0,0,1},{1,1,1,5},40] (* Harvey P. Dale, Mar 06 2016 *)

a(n):=sum(binomial(n-3*j,n-4*j)*n/(n-3*j),j,0,(n-1)/3); /* Vladimir Kruchinin, Mar 25 2016 */

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,1]^(n-1)*[1;1;1;5])[1,1] \\ Charles R Greathouse IV, Sep 09 2016

G.f.: -x*(1+4*x^3)/(-1+x+x^4). a(n)= 4*A003269(n)-3*A003269(n-1). - R. J. Mathar, Nov 16 2007
a(n) = Sum_{j=0..(n-1)/3}(binomial(n-3*j,n-4*j)*n/(n-3*j)). - Vladimir Kruchinin, Mar 25 2016
From Greg Dresden, Aug 23 2019: (Start)
a(n) = r1^n + r2^n + r3^n + r4^n, where {r1,r2,r3,r4} are the four roots of x^4-x^3-1=0, see A086106, A230151.
a(n) = round(r^n) for n>21 and r the positive real root of x^4-x^3-1.
(End)

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

A058364 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 9 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 39, 51, 64, 78, 93, 109, 126, 144, 172, 211, 262, 326, 404, 497, 606, 732, 876, 1048, 1259, 1521, 1847, 2251, 2748, 3354, 4086, 4962, 6010, 7269, 8790, 10637, 12888, 15636, 18990, 23076, 28038

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

			a(9) = 10 because there is one way to put zero molecule to the necklace and 9 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-8*i, i-1)/i, i=1..n/9). a(n) = a(n-1) + a(n-9), a(n) = 1 for n = 1..8, a(9) = 10. generating function = (x+9*x^9)/(1-x-x^9).

A058365 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 8 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 10, 11, 12, 13, 14, 15, 16, 25, 35, 46, 58, 71, 85, 100, 116, 141, 176, 222, 280, 351, 436, 536, 652, 793, 969, 1191, 1471, 1822, 2258, 2794, 3446, 4239, 5208, 6399, 7870, 9692, 11950, 14744, 18190, 22429, 27637, 34036, 41906

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

			a(8) = 9 because there is one way to put zero molecule to the necklace and 8 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-7*i, i-1)/i, i=1..n/8). a(n) = a(n-1) + a(n-8), a(n) = 1 for n = 1..7, a(8) = 9. generating function = (x+8*x^8)/(1-x-x^8).

A058366 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 7 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 9, 10, 11, 12, 13, 14, 22, 31, 41, 52, 64, 77, 91, 113, 144, 185, 237, 301, 378, 469, 582, 726, 911, 1148, 1449, 1827, 2296, 2878, 3604, 4515, 5663, 7112, 8939, 11235, 14113, 17717, 22232, 27895, 35007, 43946, 55181, 69294, 87011

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

			a(7) = 8 because there is one way to put zero molecule to the necklace and 7 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-6*i, i-1)/i, i=1..n/7). a(n) = a(n-1) + a(n-7), a(n) = 1 for n = 1..6, a(7) = 8. generating function = (x+7*x^7)/(1-x-x^7).

A058367 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 6 sites wide.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 8, 9, 10, 11, 12, 19, 27, 36, 46, 57, 69, 88, 115, 151, 197, 254, 323, 411, 526, 677, 874, 1128, 1451, 1862, 2388, 3065, 3939, 5067, 6518, 8380, 10768, 13833, 17772, 22839, 29357, 37737, 48505, 62338, 80110, 102949, 132306, 170043, 218548

Offset: 1

Yong Kong (ykong(AT)curagen.com), Dec 17 2000

nonn

			a(6) = 7 because there is one way to put zero molecule to the necklace and 6 ways to put one molecule.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799.

Cf. A000079, A003269, A003520, A005708, A005709, A005710.

a(n) = 1 + n*sum(binomial(n-1-5*i, i-1)/i, i=1..n/6). a(n) = a(n-1) + a(n-6), a(n) = 1 for n = 1..5, a(6) = 7. generating function = (x+6*x^6)/(1-x-x^6).

A146492 Decimal expansion of Product_{n>=2} (1 - 1/(n^4*(n-1))).

Original entry on oeis.org

9, 2, 9, 8, 3, 8, 4, 7, 3, 9, 5, 4, 3, 4, 6, 8, 5, 2, 2, 3, 8, 3, 1, 8, 4, 6, 9, 5, 3, 4, 5, 5, 3, 5, 4, 8, 9, 4, 4, 9, 0, 8, 3, 0, 5, 4, 8, 2, 2, 5, 3, 6, 3, 5, 2, 3, 6, 7, 5, 7, 4, 9, 7, 0, 6, 9, 7, 3, 5, 3, 7, 8, 0, 0, 2, 2, 5, 8, 0, 8, 2, 2, 3, 2, 2, 2, 3, 4, 5, 7, 6, 6, 4, 0, 2, 7, 0, 3, 1, 2, 0, 2, 4, 1, 5

Offset: 0

R. J. Mathar, Feb 13 2009

Product of Artin's constant of rank 4 and the equivalent almost-prime products.

			0.9298384739543468522383.. = (1-1/16)*(1-1/162)*(1-1/768)*(1-1/2500)*..

Cf. A065416.

r := 4 : ni := fsolve( (n+1)^r*n-1,n,complex) : 1.0/mul(GAMMA(1-d),d=ni) ; # R. J. Mathar, Feb 20 2009

g[k_] := Gamma[Root[-5 + 11# - 6#^2 + #^3 & , k]]; RealDigits[Cosh[(Sqrt[3] Pi)/2]/(Pi g[1] g[2] g[3]) // Re, 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits[Re[N[Product[1 - 1/(n^4 (n - 1)), {n, 2, Infinity}], 110]]] (* Bruno Berselli, Apr 02 2013 *)

The logarithm is -Sum_{s>=2} Sum_{j=1..floor(s/(1+r))} binomial(s-r*j-1, j-1)*(1-Zeta(s))/j at r = 4.
s*Sum_{j=1..floor(s/5)} binomial(s-4j-1, j-1)/j = A058368(s)-1.
Equals 1/Product_{k=1..5} Gamma(1-x_k), where x_k are the 5 roots of the polynomial x*(x+1)^4-1. [R. J. Mathar, Feb 20 2009]

More terms from Jean-François Alcover, Feb 12 2013

cp's OEIS Frontend

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

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A001609 a(1) = a(2) = 1, a(3) = 4; thereafter a(n) = a(n-1) + a(n-3).

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A014097 a(n) = a(n-1)+a(n-4).

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A058364 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 9 sites wide.

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A058365 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 8 sites wide.

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A058366 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 7 sites wide.

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A058367 Number of ways to cover (without overlapping) a ring lattice (necklace) of n sites with molecules that are 6 sites wide.

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