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A122189 Heptanacci numbers: each term is the sum of the preceding 7 terms, with a(0),...,a(6) = 0,0,0,0,0,0,1. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

See A066178 (essentially the same sequence) for more about the heptanacci numbers and other generalizations of the Fibonacci numbers (A000045).

REFERENCES

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

LINKS

Table of n, a(n) for n=1..39.

FORMULA

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.

MAPLE

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]

MATHEMATICA

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] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

CROSSREFS

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: A062258 A239560 A066178 * A194630 A133024 A243085

Adjacent sequences:  A122186 A122187 A122188 * A122190 A122191 A122192

KEYWORD

nonn

AUTHOR

Roger Bagula and Gary W. Adamson, Oct 18 2006

EXTENSIONS

Edited by N. J. A. Sloane, Nov 20 2007

Removed wrong Binet-type formula R. J. Mathar, Feb 13 2009

STATUS

approved

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Last modified November 26 11:14 EST 2014. Contains 250056 sequences.