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A002532 a(n) = 2*a(n-1) + 5*a(n-2), a(0) = 0, a(1) = 1.

%I A002532 M1923 N0758 #84 Jul 13 2025 11:06:36
%S A002532 0,1,2,9,28,101,342,1189,4088,14121,48682,167969,579348,1998541,
%T A002532 6893822,23780349,82029808,282961361,976071762,3366950329,11614259468,
%U A002532 40063270581,138197838502,476712029909,1644413252328,5672386654201,19566839570042,67495612411089
%N A002532 a(n) = 2*a(n-1) + 5*a(n-2), a(0) = 0, a(1) = 1.
%C A002532 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
%C A002532 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
%C A002532 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
%D A002532 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
%D A002532 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002532 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002532 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.
%H A002532 Vincenzo Librandi, <a href="/A002532/b002532.txt">Table of n, a(n) for n = 0..1000</a>
%H A002532 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A002532 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A002532 Albert Tarn, <a href="/A001333/a001333_1.pdf">Approximations to certain square roots and the series of numbers connected therewith</a>, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12 [Annotated scanned copy]
%H A002532 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,5).
%F A002532 From Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003: (Start)
%F A002532 a(2*n+1) = 5*a(n)^2 + a(n+1)^2.
%F A002532 6*a(2*n+1) = 5*A002533(n)^2 + A002533(n+1)^2. (End)
%F A002532 From _Paul Barry_, Sep 20 2003: (Start)
%F A002532 G.f.: x/(1-2*x-5*x^2).
%F A002532 E.g.f.: exp(x)*sinh(sqrt(6)*x)/sqrt(6).
%F A002532 a(n) = ((1+sqrt(6))^n - (1-sqrt(6))^n)/(2*sqrt(6)). (End)
%F A002532 a(n) = Sum_{k=0..n} binomial(n, 2*k+1)*6^k. - _Paul Barry_, Sep 29 2004
%F A002532 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
%F A002532 From _Peter Bala_, Jul 06 2025: (Start)
%F A002532 For n >= 0, a(n+1) = (2^n) * Sum_{k = 0..floor(n/2)} binomial(n-k, k) * (5/4)^k.
%F A002532 For n >= 2, a(n) = 2^(n-1) * hypergeom([1 - (1/2)*n, 1/2 - (1/2)*n], [1 - n], -5).
%F A002532 Sum_{n >= 1} (-5)^n/(a(n)*a(n+1)) = -(sqrt(6) - 1).
%F A002532 Sum_{n >= 1} 5^n/(a(n)*a(n+2)) = 5/4; Sum_{n >= 1} 5^n/(a(n)*a(n+4)) = 755/7056.
%F A002532 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)
%e A002532 G.f. = x + 2*x^2 + 9*x^3 + 28*x^4 + 101*x^5 + 342*x^6 + 1189*x^7 + ...
%p A002532 A002532:=-z/(-1+2*z+5*z**2); # Conjectured by _Simon Plouffe_ in his 1992 dissertation
%p A002532 # second program
%p A002532 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
%t A002532 Expand[Table[((1 + Sqrt[6])^n - (1 - Sqrt[6])^n)/(2Sqrt[6]), {n, 0, 25}]] (* _Zerinvary Lajos_, Mar 22 2007 *)
%t A002532 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 *)
%t A002532 LinearRecurrence[{2,5},{0,1},30] (* _Harvey P. Dale_, Nov 03 2011 *)
%o A002532 (Sage) 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
%o A002532 (Sage) [lucas_number1(n,2,-5) for n in range(0, 26)] # _Zerinvary Lajos_, Apr 22 2009
%o A002532 (Magma) [Floor(((1+Sqrt(6))^n-(1-Sqrt(6))^n)/(2*Sqrt(6))): n in [0..30]]; // _Vincenzo Librandi_, Aug 15 2011
%o A002532 (PARI) Vec(1/(1-2*x-5*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Apr 17 2012
%o A002532 (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
%Y A002532 Cf. A015581 (similar application, but no distinguishing identical vs. fraternal twins).
%Y A002532 The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
%K A002532 nonn,easy
%O A002532 0,3
%A A002532 _N. J. A. Sloane_
%E A002532 More terms from _Rick L. Shepherd_, Sep 19 2004