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A079262 Octanacci numbers: a(0)=a(1)=...=a(6)=0, a(7)=1; for n >= 8, a(n) = Sum_{i=1..8} a(n-i).

%I A079262 #92 Jun 03 2025 03:27:52
%S A079262 0,0,0,0,0,0,0,1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080,
%T A079262 16128,32192,64256,128257,256005,510994,1019960,2035872,4063664,
%U A079262 8111200,16190208,32316160,64504063,128752121,256993248,512966536,1023897200,2043730736
%N A079262 Octanacci numbers: a(0)=a(1)=...=a(6)=0, a(7)=1; for n >= 8, a(n) = Sum_{i=1..8} a(n-i).
%C A079262 a(n+7) is the number of compositions of n into parts <= 8. - _Joerg Arndt_, Sep 24 2020
%H A079262 T. D. Noe, <a href="/A079262/b079262.txt">Table of n, a(n) for n=0..207</a>
%H A079262 Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A079262 Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, <a href="https://doi.org/10.3390/math8091487">An Alternating Sum of Fibonacci and Lucas Numbers of Order k</a>, Mathematics (2020) Vol. 9, 1487.
%H A079262 Taras Goy and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Shattuck/shattuck20.html">Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
%H A079262 Tian-Xiao He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/He/he13.html">Impulse Response Sequences and Construction of Number Sequence Identities</a>, J. Int. Seq. 16 (2013) #13.8.2.
%H A079262 F. T. Howard and Curtis Cooper, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/49-3/HowardCooper.pdf">Some identities for r-Fibonacci numbers</a>, Fibonacci Quart. 49 (2011), no. 3, 231-243.
%H A079262 Omar Khadir, László Németh, and László Szalay, <a href="https://doi.org/10.1007/s00025-024-02284-3">Tiling of dominoes with ranked colors</a>, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
%H A079262 László Németh and László Szalay, <a href="https://arxiv.org/abs/2408.12196">Explicit solution of system of two higher-order recurrences</a>, arXiv:2408.12196 [math.NT], 2024. See p. 10.
%H A079262 Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
%H A079262 Fred J. Rispoli, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-3/paper43-3-5.pdf">Fibonacci Polytopes and Their Applications</a>, Fib. Q., 43,3 (2005), 227-233.
%H A079262 Kai Wang, <a href="https://www.researchgate.net/publication/344295426_IDENTITIES_FOR_GENERALIZED_ENNEANACCI_NUMBERS">Identities for generalized enneanacci numbers</a>, Generalized Fibonacci Sequences (2020).
%H A079262 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1,1,1).
%F A079262 G.f.: x^7/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8). - _Emeric Deutsch_, Apr 16 2005
%F A079262 a(1)..a(9) = 1, 1, 2, 4, 8, 16, 32, 64, 128. a(10) and following are given by 63*2^(n-8)+(1/2+sqrt(5/4))^(n-6)/sqrt(5)-(1/2-sqrt(5/4))^(n-6)/sqrt(5). Offset 10. a(10)=255. - Al Hakanson (hawkuu(AT)gmail.com), Feb 14 2009
%F A079262 Another form of the g.f.: f(z) = (z^7 - z^8)/(1 - 2*z + z^9), then a(n) = Sum_{i=0..floor((n-7)/9)} (-1)^i*binomial(n-7-8*i,i)*2^(n-7-9*i) - Sum_{i=0..floor((n-8)/9)} (-1)^i*binomial(n-8-8*i,i)*2^(n-8-9*i) with Sum_{i=m..n} alpha(i) = 0 for m>n. - _Richard Choulet_, Feb 22 2010
%F A079262 Sum_{k=0..7*n} a(k+b)*A171890(n,k) = a(8*n+b), b>=0.
%F A079262 a(n) = 2*a(n-1) - a(n-9). - _Vincenzo Librandi_, Dec 20 2010
%e A079262 a(16) = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255.
%p A079262 for j from 0 to 6 do a[j]:=0 od: a[7]:=1: for n from 8 to 45 do a[n]:=sum(a[n-i],i=1..8) od:seq(a[n],n=0..45); # _Emeric Deutsch_, Apr 16 2005
%t A079262 LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 1}, 50] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)
%t A079262 With[{nn=8},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* _Harvey P. Dale_, Aug 17 2013 *)
%Y A079262 Cf. A066178, A001592, A001591, A001630, A000073, A000045.
%Y A079262 Row 8 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y A079262 Cf. A253706, A253705. Primes and indices of primes in this sequence.
%K A079262 easy,nonn
%O A079262 0,10
%A A079262 _Michael Joseph Halm_, Feb 04 2003
%E A079262 Corrected by Joao B. Oliveira (oliveira(AT)inf.pucrs.br), Nov 25 2004