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A015443
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Generalized Fibonacci numbers: a(n) = a(n-1) + 8*a(n-2).
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24
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1, 1, 9, 17, 89, 225, 937, 2737, 10233, 32129, 113993, 371025, 1282969, 4251169, 14514921, 48524273, 164643641, 552837825, 1869986953, 6292689553, 21252585177, 71594101601, 241614783017, 814367595825, 2747285859961
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Construct a graph as follows: form the graph whose adjacency matrix is the tensor product of that of P_3 and [1,1;1,1], then add a loop at each of the extremity nodes. a(n-1) counts walks of length n between adjacent nodes. - Paul Barry, Nov 12 2004
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, 9*a(n-2) equals the number of 9-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
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, Fxtbook, p.318
Index to sequences with linear recurrences with constant coefficients, signature (1,8)
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FORMULA
| a(n) = (((1+sqrt(33))/2)^(n+1) - ((1-sqrt(33))/2)^(n+1))/sqrt(33).
a(n) = sum(k=0..n, A109466(n,k)*(-8)^(n-k) ). [From Philippe DELEHAM, Oct 26 2008]
G.f. 1/(1-x-8*x^2). - R. J. Mathar, Apr 07 2011
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MATHEMATICA
| Join[{a=1, b=1}, Table[c=b+8*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 16 2011*)
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PROG
| (Sage) [lucas_number1(n, 1, -8) for n in xrange(1, 27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
(MAGMA) [ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+8*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011
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CROSSREFS
| Cf. A015442, A015441.
Cf. A100302, A100303.
Sequence in context: A166705 A116526 A197396 * A121442 A049440 A177200
Adjacent sequences: A015440 A015441 A015442 * A015444 A015445 A015446
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KEYWORD
| nonn,easy
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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