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Search: id:A083098
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| 1, 1, 8, 22, 92, 316, 1184, 4264, 15632, 56848, 207488, 756064, 2757056, 10050496, 36643328, 133589632, 487039232, 1775616256, 6473467904, 23600633344, 86042074112, 313687948288, 1143628341248, 4169384372224, 15200538791936
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n+1)=a(n)+7*A083099(n-1), a(n+1)/A083099(n) converges to sqrt(7).
Binomial transform of expansion of cosh(sqrt(7)x) (A000420 with interpolated zeros : 1, 0, 7, 0, 49, 0, 343, 0, ...).
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 7 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(7). - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005
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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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FORMULA
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G.f.: (1-x)/(1-2x-6x^2).
a(n)=(1+sqrt(7))^n/2+(1-sqrt(7))^n/2 with e.g.f. exp(x)cosh(sqrt(7)x).
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*7^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007
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MATHEMATICA
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CoefficientList[Series[(1+6x)/(1-2x-6x^2), {x, 0, 25}], x]
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PROGRAM
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(Other) sage: [lucas_number2(n, 2, -6)/2 for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
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CROSSREFS
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The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Sequence in context: A058404 A126362 A140418 this_sequence A033456 A026593 A131622
Adjacent sequences: A083095 A083096 A083097 this_sequence A083099 A083100 A083101
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KEYWORD
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easy,nonn
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
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