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A015442
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Generalized Fibonacci numbers: a(n) = a(n-1) + 7 a(n-2), a(0)=0, a(1)=1.
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10
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0, 1, 1, 8, 15, 71, 176, 673, 1905, 6616, 19951, 66263, 205920, 669761, 2111201, 6799528, 21577935, 69174631, 220220176, 704442593, 2245983825, 7177081976, 22898968751, 73138542583, 233431323840, 745401121921, 2379420388801
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| One obtains A015523 through a Binomial Transform, and A197189 by shifting one place left (starting 1,1,8 with offset 0) followed by a Binomial Transform. - R. J. Mathar, Oct 11 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=2, 8*a(n-1) equals the number of 8-colored compositions of n with all parts >=2, such that no adjacent parts have the same color.-Milan Janjic, Nov 26 2011
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LINKS
| Joerg Arndt, Fxtbook
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FORMULA
| O.g.f.: x/(1-x-7x^2). - R. J. Mathar, May 06 2008
a(n) = ( ((1+sqrt(29))/2)^(n+1) - ((1-sqrt(29))/2)^(n+1) )/sqrt(29).
Also a(n) = 8*a(n-2)+7*a(n-3) with characteristic polynomial x^3-8*x-7. - Roger L. Bagula, May 30 2007.
a(n+1)=Sum_{k, 0<=k<=n} A109466(n,k)*(-7)^(n-k). - Philippe DELEHAM, Oct 26 2008
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MATHEMATICA
| (* Mathematica programs from R. L. Bagula *)
(* recursion *) A15442[0] := 0; A15442[1] := 1; A15442[ 2] := 1; A15442[n_] := A15442[n] = 8*A15442[n - 2] + 7*A15442[n - 3]; Table[A15442[n], {n, 0, 25}]
(* matrix representation *) mat15442 = {{0, 1, 0}, {0, 0, 1}, {7, 8, 0}}; w15442[ 0] = {0, 1, 1}; w15442[n_] := w15442[n] = mat15442.w15442[n - 1]; Table[w15442[n][[1]], {n, 0, 25}]
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PROG
| (Other) sage: [lucas_number1(n, 1, -7) for n in xrange(0, 27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
| Cf. A015440, A015441.
Sequence in context: A161342 A048732 A193490 * A177199 A177165 A189003
Adjacent sequences: A015439 A015440 A015441 * A015443 A015444 A015445
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KEYWORD
| nonn,easy
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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