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A015530
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Expansion of x/(1-4x-3x^2).
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18
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0, 1, 4, 19, 88, 409, 1900, 8827, 41008, 190513, 885076, 4111843, 19102600, 88745929, 412291516, 1915403851, 8898489952, 41340171361, 192056155300, 892245135283, 4145149007032, 19257331433977, 89464772757004
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Let b(1)=1, b(k)=floor(b(k-1))+3/b(k-1); then for n>1, b(n)=a(n)/a(n-1). - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 09 2002
In general, x/(1-a*x-b*x^2) has a(n)=sum{k=0..floor((n-1)/2),C(n-k-1,k)b^k*a^(n-2k-1)}. - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (4,3).
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FORMULA
| a(n) = 4 a(n-1) + 3 a(n-2).
a(n) = (A086901(n+2) - A086901(n+1))/6. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Feb 01 2004
a(n)=sum{k=0..floor((n-1)/2), C(n-k-1, k)3^k*4^(n-2k-1)} - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005
((2+sqrt7)^n-(2-sqrt7)^n)/sqrt28. Offset 1. a(3)=19 [From Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 01 2010: (Start)
Limit(a(n+k)/a(k), k=infinity) = A108851(n)+A015530(n)*sqrt(7)
Limit(A108851(n)/A015530(n), n=infinity) = sqrt(7)
(End)
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MATHEMATICA
| a[n_]:=(MatrixPower[{{1, 6}, {1, 3}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, -1, 40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 20 2010]
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PROG
| (Other) sage: [lucas_number1(n, 4, -3) for n in xrange(0, 23)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
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CROSSREFS
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 01 2010: (Start)
Appears in A179596, A126473 and A179597.
(End)
Sequence in context: A017962 A192526 A084155 * A181880 A010907 A087449
Adjacent sequences: A015527 A015528 A015529 * A015531 A015532 A015533
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KEYWORD
| nonn,easy
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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