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Search: id:A084057
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| A084057 |
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a(n)=2a(n-1)+4a(n-2), a(0)=1, a(1)=1. |
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+0
25
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| 1, 1, 6, 16, 56, 176, 576, 1856, 6016, 19456, 62976, 203776, 659456, 2134016, 6905856, 22347776, 72318976, 234029056, 757334016, 2450784256, 7930904576, 25664946176, 83053510656, 268766806016, 869747654656, 2814562533376
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OFFSET
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0,3
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COMMENT
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Inverse binomial transform of A001077. Binomial transform of expansion of cosh(sqrt(5)x) (1,0,5,0,25,...).
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 5 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(5). - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005
Equals right border of triangle A143969. (1, 6, 16, 56,...) = row sums of triangle A143969 and INVERT transform of (1, 5, 5, 5,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008]
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REFERENCES
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John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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a(n)=((1+sqrt(5))^n+(1-sqrt(5))^n)/2; G.f.:(1-x)/(1-2x-4x^2); E.g.f.: exp(x)cosh(sqrt(5)x).
a(2n+1)=2a(n)a(n+1)-(-4)^n. - Mario Catalani (mario.catalani(AT)unito.it), Jun 13 2003
a(n)=sum{k=0..floor(n/2), binomial(n, 2k)5^k }. - Paul Barry (pbarry(AT)wit.ie), Jul 25 2004
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*5^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007
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PROGRAM
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sage: from sage.combinat.sloane_functions import recur_gen2b sage: it = recur_gen2b(1, 1, 2, 4, lambda n: 0) sage: [it.next() for i in xrange(1, 26)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008
(Other) sage: [lucas_number2(n, 2, -4)/2 for n in xrange(0, 26)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
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CROSSREFS
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Cf. A046717, A002533.
Equals (1/2) A087131.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
A143969 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008]
Sequence in context: A026086 A032282 A163302 this_sequence A091649 A125628 A078672
Adjacent sequences: A084054 A084055 A084056 this_sequence A084058 A084059 A084060
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), May 10 2003
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