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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. 11
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

0,9

COMMENTS

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

LINKS

Robert Price, Table of n, a(n) for n = 0..1000

F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers.

B. E. Merkel, Probabilities of Consecutive Events in Coin Flipping, Master's Thesis, Univ. Cincinatti, May 11 2011.

Index to sequences with linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

FORMULA

G.f.: x^6/(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. - Richard Choulet, Feb 22 2010

sum_{k=0..6*n} a(k+b)*A063265(n,k) = a(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); # 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 (* 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 *)

a={0, 0, 0, 0, 0, 0, 1} For[n=7, n≤100, n++, sum=Plus@@a; Print[sum]; a=RotateLeft[a]; a[[7]]=sum] (* Robert Price, Dec 04 2014 *)

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, A248700.

Sequence in context: A062258 A239560 A066178 * A194630 A251672 A251747

Adjacent sequences:  A122186 A122187 A122188 * A122190 A122191 A122192

KEYWORD

nonn,changed

AUTHOR

Roger L. Bagula and Gary W. Adamson, Oct 18 2006

EXTENSIONS

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

Wrong Binet-type formula removed by R. J. Mathar, Feb 13 2009

STATUS

approved

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Last modified December 18 06:09 EST 2014. Contains 252079 sequences.