Search: id:A002533
Results 1-1 of 1 results found.

%I A002533 M4369 N1834
%S A002533 1,1,7,19,73,241,847,2899,10033,34561,119287,411379,1419193,4895281,
%T A002533 16886527,58249459,200931553,693110401,2390878567,8247309139,
%U A002533 28449011113
%N A002533 a(n) = 2a(n-1) + 5a(n-2).
%C A002533 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
%D A002533 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002533 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002533 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 A002533 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%D A002533 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 A002533 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002533 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 A002533 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 A002533 A002533(n)/A002532(n), n>0, converges to sqrt(6). - Mario Catalani (mario.catalani(AT)unito.it), 
               Apr 22 2003
%F A002533 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
%F A002533 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
%F A002533 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
%F A002533 a(n)=sum{k=0..floor(n/2), binomial(n, 2k)6^k } - Paul Barry (pbarry(AT)wit.), 
               Jul 25 2004
%F A002533 a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*6^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 26 2007
%p A002533 A002533:=(-1+z)/(-1+2*z+5*z**2); [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%o A002533 (Other) sage: [lucas_number2(n,2,-5)/2 for n in xrange(0, 21)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A002533 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 A002533 Sequence in context: A155463 A005516 A152008 this_sequence A111011 A144723 
               A062551
%Y A002533 Adjacent sequences: A002530 A002531 A002532 this_sequence A002534 A002535 
               A002536
%K A002533 nonn
%O A002533 0,3
%A A002533 N. J. A. Sloane (njas(AT)research.att.com).

Search completed in 0.002 seconds