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A015455 a(n) = 9*a(n-1) + a(n-2) for n>1; a(0) = a(1) = 1. 4
1, 1, 10, 91, 829, 7552, 68797, 626725, 5709322, 52010623, 473804929, 4316254984, 39320099785, 358197153049, 3263094477226, 29726047448083, 270797521509973, 2466903741037840, 22472931190850533 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

For n>=1, row sums of triangle for numbers 9^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 13 2012

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,9} containing no subwords ii, (i=0,1,...,8). - Milan Janjic, Jan 31 2015

REFERENCES

R. K. Guy, "A further family of sequences", SeqFan mailing list (www.seqfan.eu), Jun 13 2008

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (9,1).

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 Deléham, 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, Jun 13 2008

For n>=2, a(n)=F_n(9)+F_(n+1)(9), where F_n(x) is Fibonacci polynomial (cf.A049310): F_n(x)=sum{i=0,...,floor((n-1)/2)}C(n-i-1,i)x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012

PROG

(PARI) a(n) = polcoeff((1-(O(x^n)+8)*x)/(1-9*x-x^2), n) \\ M. F. Hasler, Jun 14 2008

(MAGMA) [n le 2 select 1 else 9*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 01 2015

CROSSREFS

Row m=9 of A135597.

Sequence in context: A231412 A002452 A096261 * A110410 A051789 A267833

Adjacent sequences:  A015452 A015453 A015454 * A015456 A015457 A015458

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

EXTENSIONS

Edited by M. F. Hasler, Jun 14 2008

STATUS

approved

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Last modified October 17 10:50 EDT 2017. Contains 293469 sequences.