%I A002532 M1923 N0758
%S A002532 0,1,2,9,28,101,342,1189,4088,14121,48682,167969,579348,1998541,6893822,
%T A002532 23780349,82029808,282961361,976071762,3366950329,11614259468,
%U A002532 40063270581,138197838502,476712029909,1644413252328,5672386654201
%N A002532 a(n) = 2a(n-1) + 5a(n-2).
%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 (hillcino368(AT)gmail.com), 
               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-frat ernal characteristics). 
               - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 19 2004
%D A002532 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002532 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002532 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A002532 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%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 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002532 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A002532 a(2n+1)=5a(n)^2+a(n+1)^2. 6a(2n+1)=5*A002533(n)^2+A002533(n+1)^2. - Mario 
               Catalani (mario.catalani(AT)unito.it), Jun 14 2003
%F A002532 G.f.: x/(1-2x-5x^2); E.g.f. : exp(x)sinh(sqrt(6)x)/sqrt(6); a(n)=((1+sqrt(6))^n-(1-sqrt(6))^n)/
               (2sqrt(6)). - Paul Barry (pbarry(AT)wit.ie), Sep 20 2003
%F A002532 a(n)=sum{k=0..n, binomial(n, 2k+1)6^k}. - Paul Barry (pbarry(AT)wit.ie), 
               Sep 29 2004
%p A002532 A002532:=-z/(-1+2*z+5*z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
%p A002532 sage: from sage.combinat.sloane_functions import recur_gen2 sage: it 
               = recur_gen2(0,1,2,5) sage: [it.next() for i in range(30)] - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%t A002532 Expand[Table[((1 + Sqrt[6])^n - (1 - Sqrt[6])^n)/(2Sqrt[6]), {n, 0, 25}]] 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2007
%o A002532 (Other) sage: [lucas_number1(n,2,-5) for n in xrange(0, 26)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%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.
%Y A002532 Sequence in context: A058877 A026087 A109188 this_sequence A098518 A086511 
               A138912
%Y A002532 Adjacent sequences: A002529 A002530 A002531 this_sequence A002533 A002534 
               A002535
%K A002532 nonn
%O A002532 0,3
%A A002532 N. J. A. Sloane (njas(AT)research.att.com).
%E A002532 More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 19 
               2004