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%I #60 Dec 30 2023 23:40:38
%S 1,1,9,73,593,4817,39129,317849,2581921,20973217,170367657,1383914473,
%T 11241683441,91317382001,741780739449,6025563297593,48946287120193,
%U 397595860259137,3229713169193289,26235301213805449,213112122879636881
%N Generalized Fibonacci numbers.
%C a(n)/a(n-1) tends to (8 + 2*sqrt(17))/2 = exp ArcSinh 4 = A176458. - _Gary W. Adamson_, Dec 26 2007
%C For n>=1, row sums of triangle for numbers 8^k*C(m,k) with duplicated diagonals. - _Vladimir Shevelev_, Apr 13 2012
%C For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,8} containing no subwords ii, (i=0,1,...,7). - _Milan Janjic_, Jan 31 2015
%C a(n+1) is the number of nonary sequences of length n such that no two consecutive terms have distance 5. - _David Nacin_, May 31 2017
%H Vincenzo Librandi, <a href="/A015454/b015454.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,1)
%F a(n) = 8*a(n-1) + a(n-2).
%F a(n) = Sum_{k=0..n} 7^k*A055830(n,k). - _Philippe Deléham_, Oct 18 2006
%F G.f.: (1-7*x)/(1-8*x-x^2). - _Philippe Deléham_, Nov 20 2008
%F For n>=2, a(n) = F_n(8)+F_(n+1)(8), 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
%F a(n) = A041025(n) -7*A041025(n-1). - _R. J. Mathar_, Jul 06 2012
%t LinearRecurrence[{8, 1}, {1, 1}, 30] (* _Vincenzo Librandi_, Nov 08 2012 *)
%t CoefficientList[Series[(1-7*x)/(1-8*x-x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 19 2017 *)
%o (Magma) [n le 2 select 1 else 8*Self(n-1) + Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 08 2012
%o (PARI) x='x+O('x^30); Vec((1-7*x)/(1-8*x-x^2)) \\ _G. C. Greubel_, Dec 19 2017
%Y Row m=8 of A135597.
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
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