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

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.

Index entries for sequences related to linear recurrences with constant coefficients

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.

A002533(n)/A002532(n), n>0, converges to sqrt(6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

G.f.: (1-x)/(1-2x-5x^2). a(n)=(1/2)[(1+sqrt(6))^n+(1-sqrt(6))^n]. a(n)/A083694(n) converges to sqrt(3/2). a(n)/A083695(n) converges to sqrt(2/3). a(n)=a(n-1)+3*A083694(n-1), a(n)=a(n-1)+2*A083695(n-1), n>0. - Mario Catalani (mario.catalani(AT)unito.it), May 03 2003

Binomial transform of expansion of cosh(sqrt(6)x) (A000400, with interpolated zeros). E.g.f.: exp(x)cosh(sqrt(6)x) - Paul Barry (pbarry(AT)wit.ie), May 09 2003

a(2n+1)=2a(n)a(n+1)-(-5)^n. a(n)^2-6*A002532(n)^2=(-5)^n. - Mario Catalani (mario.catalani(AT)unito.it), Jun 14 2003

a(n)=sum{k=0..floor(n/2), binomial(n, 2k)6^k } - Paul Barry (pbarry(AT)wit.), Jul 25 2004

a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*6^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007

A002533:=(-1+z)/(-1+2*z+5*z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

(Other) sage: [lucas_number2(n, 2, -5)/2 for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]

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: A155463 A005516 A152008 this_sequence A111011 A144723 A062551

Adjacent sequences: A002530 A002531 A002532 this_sequence A002534 A002535 A002536

nonn

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