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A122189
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Heptanacci numbers: each term is the sum of the preceding 7 terms, with a(0),...,a(6) = 0,0,0,0,0,0,1.
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5
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0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600, 971364608, 1934923521
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OFFSET
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1,9
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COMMENTS
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See A066178 (essentially the same sequence) for more about the heptanacci numbers and other generalizations of the Fibonacci numbers (A000045).
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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.
B. E. Merkel, Probabilities of Consecutive Events in Coin Flipping, Master's Thesis, Univ. Cincinatti, May 11 2011; http://etd.ohiolink.edu/view.cgi/Merkel%20Benjamin%20E.pdf?ucin1307442290
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LINKS
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Table of n, a(n) for n=1..39.
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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). - R. J. Mathar, Feb 13 2009.
Another form of the g.f.: f(z)=(z^6-z^7)/(1-2*z+z^8), then a(n)=sum((-1)^i*binomial(n-6-7*i,i)*2^(n-6-8*i),i=0..floor((n-6)/8))-sum((-1)^i*binomial(n-7-7*i,i)*2^(n-7-8*i),i=0..floor((n-7)/8)) with sum(alpha(i),i=m..n)=0 for m>n. [From Richard Choulet, Feb 22 2010]
sum_{k=0..6*n} A122189(k+b)*A063265(n,k) = A122189(7*n+b), b>=0.
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MAPLE
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for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-6-7*i, i)*2^(n-6-8*i), i=0..floor((n-6)/8))-sum((-1)^i*binomial(n-7-7*i, i)*2^(n-7-8*i), i=0..floor((n-7)/8)):od:seq(k(n), n=0..50); a:=taylor((z^6-z^7)/(1-2*z+z^8), 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=1; lst={a, b, c, d, e, f, g}; Do[h=a+b+c+d+e+f+g; AppendTo[lst, h]; a=b; b=c; c=d; d=e; e=f; f=g; g=h, {n, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Sep 30 2008]
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 1}, 50] (* From Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
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CROSSREFS
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Cf. A000045 (k=2, Fibonacci numbers), A000073 (k=3, tribonacci) A000078 (k=4, tetranacci) A001591 (k=5, pentanacci) A001592 (k=6, hexanacci), A122189 (k=7, heptanacci).
Cf. A066178, A000322.
Sequence in context: A172316 A062258 A066178 * A194630 A133024 A060376
Adjacent sequences: A122186 A122187 A122188 * A122190 A122191 A122192
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KEYWORD
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nonn
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AUTHOR
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Roger Bagula and Gary W. Adamson, Oct 18 2006
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EXTENSIONS
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Edited by N. J. A. Sloane, Nov 20 2007
Removed wrong Binet-type formula R. J. Mathar, Feb 13 2009
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STATUS
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approved
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