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A079262
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Octanacci numbers: a(0)=a(1)=...=a(6)=0, a(7)=1; for n >= 8, a(n) = Sum_{i=1..8} a(n-i).
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30
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0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872, 4063664, 8111200, 16190208, 32316160, 64504063, 128752121, 256993248, 512966536, 1023897200, 2043730736
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OFFSET
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0,10
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..207
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
T.-X. He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013) #13.8.2
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Fred J. Rispoli, Fibonacci Polytopes and Their Applications, Fib. Q., 43,3 (2005), 227-233.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1).
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FORMULA
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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
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
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
sum_{k=0..7*n} a(k+b)*A171890(n,k) = a(8*n+b), b>=0.
For a(0)=a(1)=..=a(6)=0, a(7)=a(8)=1, a(n) = 2*a(n-1) - a(n-9). - Vincenzo Librandi, Dec 20 2010
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EXAMPLE
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a(16) = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255.
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MAPLE
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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
for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-7-8*i, i)*2^(n-7-9*i), i=0..floor((n-7)/9))-sum((-1)^i*binomial(n-8-8*i, i)*2^(n-8-9*i), i=0..floor((n-8)/9)):od:seq(k(n), n=0..50); a:=taylor((z^7-z^8)/(1-2*z+z^9), z=0, 51); for p from 0 to 50 do j(p):=coeff(a, z, p):od :seq(j(p), p=0..50); # Richard Choulet, Feb 22 2010
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MATHEMATICA
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a=0; b=0; c=0; d=0; e=0; f=0; g=0; h=1; lst={a, b, c, d, e, f, g, h}; Do[k=a+b+c+d+e+f+g+h; AppendTo[lst, k]; a=b; b=c; c=d; d=e; e=f; f=g; g=h; h=k, {n, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 30 2008 *)
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 *)
With[{nn=8}, LinearRecurrence[Table[1, {nn}], Join[Table[0, {nn-1}], {1}], 50]] (* Harvey P. Dale, Aug 17 2013 *)
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CROSSREFS
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Cf. A066178, A001592, A001591, A001630, A000073, A000045.
Row 8 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Cf. A253706, A253705. Primes and indices of primes in this sequence.
Sequence in context: A145114 A172317 A234589 * A194631 A251746 A251760
Adjacent sequences: A079259 A079260 A079261 * A079263 A079264 A079265
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KEYWORD
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easy,nonn
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AUTHOR
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Michael Joseph Halm, Feb 04 2003
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EXTENSIONS
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Corrected by Joao B. Oliveira (oliveira(AT)inf.pucrs.br), Nov 25 2004
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STATUS
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approved
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