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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 - 8*x)/(1 - 9*x - 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 MATHEMATICA CoefficientList[Series[(1 - 8*x)/(1 - 9*x - x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{9, 1}, {1, 1}, 50] (* G. C. Greubel, Dec 19 2017 *) 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 EXTENSIONS Edited by M. F. Hasler, Jun 14 2008 STATUS approved

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Last modified December 10 12:33 EST 2018. Contains 318047 sequences. (Running on oeis4.)