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 (hillcino368(AT)gmail.com), 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-frat ernal characteristics). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 19 2004

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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

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.

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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

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

a(n)=sum{k=0..n, binomial(n, 2k+1)6^k}. - Paul Barry (pbarry(AT)wit.ie), Sep 29 2004

A002532:=-z/(-1+2*z+5*z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

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

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

(Other) sage: [lucas_number1(n, 2, -5) for n in xrange(0, 26)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

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: A058877 A026087 A109188 this_sequence A098518 A086511 A138912

Adjacent sequences: A002529 A002530 A002531 this_sequence A002533 A002534 A002535

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 19 2004