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A084326
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a(0)=0, a(1)=1, a(n)=6a(n-1)-4a(n-2).
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6
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0, 1, 6, 32, 168, 880, 4608, 24128, 126336, 661504, 3463680, 18136064, 94961664, 497225728, 2603507712, 13632143360, 71378829312, 373744402432, 1956951097344, 10246728974336, 53652569456640, 280928500842496
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Binomial transform of A001076. - Paul Barry (pbarry(AT)wit.ie), Aug 25 2003
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (6,-4)
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FORMULA
| a(n)=(1/2)*sum(k=0, n, binomial(n, k)*F(3*k)) where F(k) denotes the k-th Fibonacci number.
a(n)=sqrt(5)((3+sqrt(5))^n-(3-sqrt(5))^n)/10. - Paul Barry (pbarry(AT)wit.ie), Aug 25 2003
a(n)=Sum(C(n, 2k+1)5^k 3^(n-2k-1), k=0, .., Floor[(n-1)/2]). a(n)=2^(n-1)F(2n). - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
a(n) = rightmost term in M^n * [1 0] where M = the 2X2 matrix [5 1 / 1 1]. The characteristic polynomial of M = x^2 - 6x + 4. a(n)/a(n-1) tends to (3 + sqrt(5)), a root of the polynomial and an eigenvalue of M. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 16 2004
a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)F(j+k)/2}}. - Paul Barry (pbarry(AT)wit.ie), Feb 14 2005
G.f.: x/(1-6x+4x^2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 09 2008]
If p[i]=(4^i-1)/3, and if A is the 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), May 08 2010]
a(n) = 5a(n - 1) + a(n - 2) + a(n - 3) + ... + a(1) + 1. [From Gary W. Adamson, Feb 18 2011]
a(n) = 2^(n-1)*A001906(n). - R. J. Mathar, Apr 03 2011
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EXAMPLE
| a(5) = 6 * a(4) - 4 * a(3) = 6*168 - 4*32 = 880.
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MATHEMATICA
| Join[{a = 0, b = 1}, Table[c = 6 * b - 4 * a; a = b; b = c, {n, 60}]] (* From Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
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PROG
| (PARI) a(n)=(1/2)*sum(k=0, n, binomial(n, k)*fibonacci(3*k))
(Other) sage: [lucas_number1(n, 6, 4) for n in xrange(0, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
| Cf. A030191.
Sequence in context: A046714 A129171 A082585 * A199699 A137637 A125190
Adjacent sequences: A084323 A084324 A084325 * A084327 A084328 A084329
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KEYWORD
| nonn,easy
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 21 2003
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