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Search: id:A084326
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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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+0
2
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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; listen)
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OFFSET
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0,3
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COMMENT
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Binomial transform of A001076. - Paul Barry (pbarry(AT)wit.ie), Aug 25 2003
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FORMULA
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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]
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PROGRAM
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(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
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Cf. A030191.
Sequence in context: A046714 A129171 A082585 this_sequence A137637 A125190 A000558
Adjacent sequences: A084323 A084324 A084325 this_sequence A084327 A084328 A084329
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 21 2003
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