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A015455
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a(n) = 9*a(n-1) + a(n-2) for n>1; a(0) = a(1) = 1.
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3
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1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226, 29726047448083, 270797521509973, 2466903741037840, 22472931190850533
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Generalized Fibonacci numbers.
As R. K. Guy suggested on the SeqFan list, the sequence could be extended "to the left side" by ..., 10, 1, 1, -8, 73, -665, 6058, -55187, 502741, -4579856, 41721445, ... by using the recurrence equation to get a(n-2) = a(n) - 9 a(n-1). The sequence 1,-8,73,... would have g.f. (1+x)/(1+9x-x^2).
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REFERENCES
| R. K. Guy, "A further family of sequences", SeqFan mailing list (www.seqfan.eu), Jun 13 2008
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index to sequences with linear recurrences with constant coefficients, signature (9,1).
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FORMULA
| G.f. = (1 - 8x)/(1 - 9x - x^2). - M. F. Hasler, Jun 14 2008
a(n)=Sum_{k, 0<=k<=n} 8^k*A055830(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2006
a(n) = round(1/2*(9/2 - 1/2*sqrt(85))^n + 7/170*sqrt(85)*(9/2 - 1/2*sqrt(85))^n - 7/170*sqrt(85)*(9/2 + 1/2*sqrt(85))^n + 1/2*(9/2 + 1/2*sqrt(85))^n). - Alexander R. Povolotsky (pevnev(AT)juno.com), Jun 13 2008
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PROG
| (PARI) a(n) = polcoeff((1-(O(x^n)+8)*x)/(1-9*x-x^2), n) \\ - M. F. Hasler, Jun 14 2008
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CROSSREFS
| Sequence in context: A079928 A002452 A096261 * A110410 A051789 A015467
Adjacent sequences: A015452 A015453 A015454 * A015456 A015457 A015458
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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 M. F. Hasler (www.univ-ag.fr/~mhasler), Jun 14 2008
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