0,3

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 6 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(6). - Cino Hilliard, Sep 25 2005

For n>=2, number of ordered partitions of n-1 into parts of sizes 1 and 2 where there are two types of 1 (singletons) and five types of 2 (twins). For example, the number of possible configurations of families of n-1 male (M) and female (F) offspring considering only single births and twins, where the birth order of M/F/pair-of-twins is considered and there are five types of twins; namely, both F (identical twins), both F (fraternal twins), both M (identical), both M (fraternal), or one F and one M - where birth order within a pair of twins itself is disregarded. In particular, for a(3)=9, two children could be either: (1) F, then M; (2) M, then F; (3) F,F; (4) M,M; (5) F,F identical twins; (6) F,F fraternal twins; (7) M,M identical twins; (8) M,M fraternal twins; or (9) M,F twins (emphasizing that birth order is irrelevant here when children are the same gender, when two children are within the same pair of twins and when pairs of twins have both the same gender(s) and identical-vs-fraternal characteristics). - Rick L. Shepherd, Sep 19 2004

Pisano period lengths: 1, 2, 3, 4, 4, 6, 24, 8, 3, 4, 120, 12, 56, 24, 12, 16, 288, 6, 18, 4, ... . - R. J. Mathar, Aug 10 2012

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

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. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12 [Annotated scanned copy]

Index entries for linear recurrences with constant coefficients, signature (2,5).

From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start)

a(2*n+1) = 5*a(n)^2 + a(n+1)^2.

6*a(2*n+1) = 5*A002533(n)^2 + A002533(n+1)^2. (End)

From Paul Barry, Sep 20 2003: (Start)

G.f.: x/(1-2*x-5*x^2).

E.g.f.: exp(x)*sinh(sqrt(6)*x)/sqrt(6).

a(n) = ((1+sqrt(6))^n - (1-sqrt(6))^n)/(2*sqrt(6)). (End)

a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*6^k. - Paul Barry, Sep 29 2004

G.f.: G(0)*x/(2*(1-x)), where G(k)= 1 + 1/(1 - x*(6*k-1)/(x*(6*k+5) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

From Peter Bala, Jul 06 2025: (Start)

For n >= 0, a(n+1) = (2^n) * Sum_{k = 0..floor(n/2)} binomial(n-k, k) * (5/4)^k.

For n >= 2, a(n) = 2^(n-1) * hypergeom([1 - (1/2)*n, 1/2 - (1/2)*n], [1 - n], -5).

Sum_{n >= 1} (-5)^n/(a(n)*a(n+1)) = -(sqrt(6) - 1).

Sum_{n >= 1} 5^n/(a(n)*a(n+2)) = 5/4; Sum_{n >= 1} 5^n/(a(n)*a(n+4)) = 755/7056.

G.f. A(x) = x*exp(Sum_{n >= 1} a(2*n)/a(n)*x^n/n) = x + 2*x^2 + 9*x^3 + 28*x^4 + .... (End)

G.f. = x + 2*x^2 + 9*x^3 + 28*x^4 + 101*x^5 + 342*x^6 + 1189*x^7 + ...

A002532:=-z/(-1+2*z+5*z**2); # Conjectured by Simon Plouffe in his 1992 dissertation

# Alternative:

seq(simplify(2^(n-1) * hypergeom([1 - (1/2)*n, 1/2 - (1/2)*n], [1 - n], -5)), n = 2..25); # Peter Bala, Jul 06 2025

Expand[Table[((1 + Sqrt[6])^n - (1 - Sqrt[6])^n)/(2Sqrt[6]), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *)

a[n_]:=(MatrixPower[{{1, 2}, {1, -3}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

LinearRecurrence[{2, 5}, {0, 1}, 30] (* Harvey P. Dale, Nov 03 2011 *)

(SageMath) from sage.combinat.sloane_functions import recur_gen2; it = recur_gen2(0, 1, 2, 5); [next(it) for i in range(30)] # Zerinvary Lajos, Jun 25 2008

(SageMath) [lucas_number1(n, 2, -5) for n in range(0, 26)] # Zerinvary Lajos, Apr 22 2009

(Magma) [Floor(((1+Sqrt(6))^n-(1-Sqrt(6))^n)/(2*Sqrt(6))): n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

(PARI) Vec(1/(1-2*x-5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 17 2012

(Magma) [n le 2 select n-1 else 2*Self(n-1) + 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018

Cf. A015581 (similar application, but no distinguishing identical vs. fraternal twins).

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

Sequence in context: A026087 A109188 A248437 * A360023 A098518 A128239

Adjacent sequences: A002529 A002530 A002531 * A002533 A002534 A002535

nonn,easy

More terms from Rick L. Shepherd, Sep 19 2004

approved