Search: id:A026150
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%I A026150
%S A026150 1,1,4,10,28,76,208,568,1552,4240,11584,31648,86464,236224,
%T A026150 645376,1763200,4817152,13160704,35955712,98232832,268377088,
%U A026150 733219840,2003193856,5472827392,14952042496,40849739776
%N A026150 a(0) = a(1) = 1; a(n+2) = 2*a(n+1) + 2*a(n).
%C A026150 a(n+1)/A002605(n) converges to sqrt(3). - Mario Catalani (mario.catalani(AT)unito.it), 
               Apr 22 2003
%C A026150 Binomial transform of expansion of cosh(sqrt(3)x) (A000244 with interpolated 
               zeros); inverse binomial transform of A001075 . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Jul 04 2005
%C A026150 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 3 times the bottom to get the new top. The limit of the 
               sequence of fractions is sqrt(3). - Cino Hilliard (hillcino368(AT)gmail.com), 
               Sep 25 2005
%C A026150 [1,3; 1,1]^n * [1,0] = [A026150(n), A002605(n)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 21 2008
%C A026150 (1+sqrt(3))^n = A026150(n) + A002605(n)*(sqrt(3)). - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Mar 21 2008
%C A026150 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008: 
               (Start)
%C A026150 Equals right border of triangle A143908.
%C A026150 Starting (1, 4, 10, 28,...) = row sums of triangle A143908 and INVERT 
               transform of (1, 3, 3, 3,...). (End)
%D A026150 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%H A026150 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A026150 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A026150 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/
               poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, 
               Melbourne, 2002.
%H A026150 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=1052">
               Encyclopedia of Combinatorial Structures 1052</a>
%H A026150 A. Burstein, S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/
               math.CO/0310379">Independent sets in certain classes of (almost) 
               regular graphs</a>
%F A026150 a(n)=(1/2)*((1+sqrt(3))^n+(1-sqrt(3))^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Oct 28 2002
%F A026150 G.f.: (1-x)/(1-2x-2x^2).
%F A026150 a(n)=a(n-1)+A083337(n-1). A083337(n)/a(n) converges to sqrt(3). - Mario 
               Catalani (mario.catalani(AT)unito.it), Apr 29 2003
%F A026150 a(n)=sum{k=0..floor(n/2), C(n, 2k)3^k }; E.g.f.: exp(x)cosh(sqrt(3)x). 
               - Paul Barry (pbarry(AT)wit.ie), May 15 2003
%F A026150 a(n+1)/a(n) converges to 1+sqrt(3) = 2.732050807568877293.... - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Jul 03 2005
%F A026150 Inverse binomial transform of A001075: (1, 2, 7, 26, 97, 362,...). - 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007
%F A026150 Starting (1, 4, 10, 28, 76,...), = binomial transform of [1, 3, 3, 9, 
               9, 27, 27, 81, 81,...]; and inverse binomial transform of A001834: 
               (1, 5, 19, 71, 265,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 30 2007
%F A026150 a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*3^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 26 2007
%F A026150 a(n) = upper left and lower right terms of [1,1; 3,1]^n. (1+sqrt(3))^n 
               = a(n) + A083337(n)/(sqrt(3)). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 12 2008
%F A026150 a(n)=A080040(n)/2. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 
               19 2008]
%p A026150 with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, 
               Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, 
               Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, 
               Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL2,ZL2,ZL2), b=ZL1], 
               ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, 
               end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, 
               mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, 
               {Q}, unlabelled], size=n)/3, n=2..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 08 2008
%t A026150 Expand[Table[((1 + Sqrt[3])^n + (1 - Sqrt[3])^n)/(2), {n, 0, 30}]] - 
               Artur Jasinski (grafix(AT)csl.pl), Dec 10 2006
%o A026150 (PARI) a(n)=if(n<0,0,real((1+quadgen(12))^n))
%o A026150 sage: from sage.combinat.sloane_functions import recur_gen2 sage: it 
               = recur_gen2(1,1,2,2) sage: [it.next() for i in range(30)] - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%o A026150 (Other) sage: [lucas_number2(n,2,-2)/2 for n in xrange(0, 26)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A026150 First differences of A002605.
%Y A026150 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 A026150 Cf. A001075.
%Y A026150 Cf. A001834.
%Y A026150 Cf. A083337.
%Y A026150 Cf. A002605.
%Y A026150 A143908 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008]
%Y A026150 Sequence in context: A026534 A111308 A121302 this_sequence A026123 A091468 
               A103457
%Y A026150 Adjacent sequences: A026147 A026148 A026149 this_sequence A026151 A026152 
               A026153
%K A026150 nonn
%O A026150 0,3
%A A026150 N. J. A. Sloane (njas(AT)research.att.com).

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