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A026150
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a(0) = a(1) = 1; a(n+2) = 2*a(n+1) + 2*a(n).
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42
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1, 1, 4, 10, 28, 76, 208, 568, 1552, 4240, 11584, 31648, 86464, 236224, 645376, 1763200, 4817152, 13160704, 35955712, 98232832, 268377088, 733219840, 2003193856, 5472827392, 14952042496, 40849739776
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n+1)/A002605(n) converges to sqrt(3). - Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
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
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
[1,3; 1,1]^n * [1,0] = [A026150(n), A002605(n)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2008
(1+sqrt(3))^n = A026150(n) + A002605(n)*(sqrt(3)). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2008
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008: (Start)
Equals right border of triangle A143908.
Starting (1, 4, 10, 28,...) = row sums of triangle A143908 and INVERT transform of (1, 3, 3, 3,...). (End)
a(n) is the number of compositions of n when there are 1 type of 1 and 3 types of other natural numbers. [From Milan R. Janjic (agnus(AT)blic.net), Aug 13 2010]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 15 2010: (Start)
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 85, 277, 337 and 340, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A002605 (without the leading 0).
(End)
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REFERENCES
| John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1052
A. Burstein, S. Kitaev and T. Mansour, Independent sets in certain classes of (almost) regular graphs
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,2).
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FORMULA
| a(n)=(1/2)*((1+sqrt(3))^n+(1-sqrt(3))^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 28 2002
G.f.: (1-x)/(1-2x-2x^2).
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
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
a(n+1)/a(n) converges to 1+sqrt(3) = 2.732050807568877293.... - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 03 2005
Inverse binomial transform of A001075: (1, 2, 7, 26, 97, 362,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007
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
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*3^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007
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
a(n)=A080040(n)/2. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2008]
If p[1]=1, and p[i]=3, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. [From Milan R. Janjic (agnus(AT)blic.net), Apr 29 2010]
a(n) = 2 * A052945(n-1) - Vladimir Joseph Stephan Orlovsky, Mar 24 2011
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MAPLE
| 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
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MATHEMATICA
| Expand[Table[((1 + Sqrt[3])^n + (1 - Sqrt[3])^n)/(2), {n, 0, 30}]] - Artur Jasinski (grafix(AT)csl.pl), Dec 10 2006
LinearRecurrence[{2, 2}, {1, 1}, 30]
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PROG
| (PARI) a(n)=if(n<0, 0, real((1+quadgen(12))^n))
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
(Other) sage: [lucas_number2(n, 2, -2)/2 for n in xrange(0, 26)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
(Haskell)
a026150 n = a026150_list !! n
a026150_list = 1 : 1 : map (* 2) (zipWith (+) a026150_list (tail
a026150_list))
-- Reinhard Zumkeller, Oct 15 2011
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CROSSREFS
| First differences of A002605.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A001075, A001834, A083337, A002605.
A143908 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008]
Cf. A028859, A030195, A106435, A108898, A125145.
Sequence in context: A203293 A111308 A121302 * A026123 A091468 A103457
Adjacent sequences: A026147 A026148 A026149 * A026151 A026152 A026153
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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