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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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9
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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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REFERENCES
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F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, http://www.math-cs.ucmo.edu/~curtisc/articles/howardcooper/genfib4.pdf.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..207
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
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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(1-9)=1,1,2,4,8,16,32,64,128. a(10 & following)=63*2^(n-8)+(.5+sqrt1.25)^(n-6)/sqrt5-(.5-sqrt1.25)^(n-6)/sqrt5. Offset 10. a(10)=255. [From Al Hakanson (hawkuu(AT)gmail.com), Feb 14 2009]
Another form of tjhe g.f.: f(z)=(z^7-z^8)/(1-2*z+z^9), then a(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)) with sum(alpha(i),i=m..n))=0 for m>n. [From Richard Choulet, Feb 22 2010]
sum_{k=0..7*n} A079262(k+b)*A171890(n,k) = A079262(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).[From 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); (Deutsch)
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); [From 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 [From 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).
Sequence in context: A008860 A145114 A172317 * A194631 A087079 A230579
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
More terms from Emeric Deutsch, Apr 16 2005
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STATUS
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approved
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