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A015445
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Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).
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15
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1, 1, 10, 19, 109, 280, 1261, 3781, 15130, 49159, 185329, 627760, 2295721, 7945561, 28607050, 100117099, 357580549, 1258634440, 4476859381, 15804569341, 56096303770, 198337427839, 703204161769, 2488241012320
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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, 10*a(n-2) equals the number of 10-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
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FORMULA
| a(n) = (((1+sqrt(37))/2)^(n+1) - ((1-sqrt(37))/2)^(n+1))/sqrt(37).
a(n) = sum{k=0..floor(n/2), binomial(n-k, k)*9^k } - Paul Barry, Jul 20 2004
a(n) = sum{k=0..n, binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*3^(n-k)/2}; - Paul Barry, Aug 28 2005
a(n) = sum(k=0..n, A109466(n,k)*(-9)^(n-k) ). [From Philippe DELEHAM, Oct 26 2008]
a(n) = (1/37)*(1/2+(1/2)*sqrt(37))^n*sqrt(37)-(1/37)*(1/2-(1/2)*sqrt(37))^n*sqrt(37). [From Paolo P. Lava, Oct 01 2008] [May produce sequence with a different offset.]
G.f.: (-9*x-1)/(9*x^2+x-1) a(n)=(-703*(1/2-sqrt(37)/2)^n + 199*sqrt(37)*(1/2-sqrt(37)/2)^n-333*(1/2+sqrt(37)/2)^n + 171*sqrt(37)*(1/2+sqrt(37)/2)^n)/(74*(5*sqrt(37)-14)) [From Alexander R. Povolotsky, Oct 13 2010]
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PROG
| (Sage) [lucas_number1(n, 1, -9) for n in xrange(1, 25)] # [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)+9*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011
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CROSSREFS
| Cf. A015443, A015442, A026595.
Sequence in context: A131495 A060630 A070199 * A177203 A177167 A073222
Adjacent sequences: A015442 A015443 A015444 * A015446 A015447 A015448
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KEYWORD
| nonn,easy
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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EXTENSIONS
| Edited by N. J. A. Sloane, Oct 11 2010
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