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%I #75 Nov 30 2023 10:35:40
%S 0,0,0,0,0,0,1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936,15808,
%T 31489,62725,124946,248888,495776,987568,1967200,3918592,7805695,
%U 15548665,30972384,61695880,122895984,244804400,487641600,971364608,1934923521
%N Heptanacci numbers: each term is the sum of the preceding 7 terms, with a(0),...,a(6) = 0,0,0,0,0,0,1.
%C See A066178 (essentially the same sequence) for more about the heptanacci numbers and other generalizations of the Fibonacci numbers (A000045).
%H Robert Price, <a href="/A122189/b122189.txt">Table of n, a(n) for n = 0..1000</a>
%H Tomás Aguilar-Fraga, Jennifer Elder, Rebecca E. Garcia, Kimberly P. Hadaway, Pamela E. Harris, Kimberly J. Harry, Imhotep B. Hogan, Jakeyl Johnson, Jan Kretschmann, Kobe Lawson-Chavanu, J. Carlos Martínez Mori, Casandra D. Monroe, Daniel Quiñonez, Dirk Tolson III, and Dwight Anderson Williams II, <a href="https://arxiv.org/abs/2311.14055">Interval and L-interval Rational Parking Functions</a>, arXiv:2311.14055 [math.CO], 2023. See p. 14.
%H Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H Taras Goy and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Shattuck/shattuck20.html">Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
%H T.-X. He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/He/he13.html">Impulse Response Sequences and Construction of Number Sequence Identities</a>, J. Int. Seq. 16 (2013) #13.8.2.
%H F. T. Howard and Curtis Cooper, <a href="http://www.fq.math.ca/Papers1/49-3/HowardCooper.pdf">Some identities for r-Fibonacci numbers</a>, Fibonacci Quart. 49 (2011), no. 3, 231-243.
%H B. E. Merkel, <a href="http://rave.ohiolink.edu/etdc/view?acc_num=ucin1307442290">Probabilities of Consecutive Events in Coin Flipping</a>, Master's Thesis, Univ. Cincinatti, May 11 2011.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1,1).
%F G.f.: x^6/(1-x-x^2-x^3-x^4-x^5-x^6-x^7). - _R. J. Mathar_, Feb 13 2009
%F G.f.: Sum_{n >= 0} x^(n+5) * [ Product_{k = 1..n} (k + k*x + k*x^2 + k*x^3 + k*x^4 + k*x^5 + x^6)/(1 + k*x + k*x^2 + k*x^3 + k*x^4 + k*x^5 + k*x^6) ]. - _Peter Bala_, Jan 04 2015
%F Another form of the g.f.: f(z) = (z^6-z^7)/(1-2*z+z^8), then a(n) = Sum_{i=0..floor((n-6)/8)} (-1)^i*binomial(n-6-7*i,i)*2^(n-6-8*i) - Sum_{i=0..floor((n-7)/8)} (-1)^i*binomial(n-7-7*i,i)*2^(n-7-8*i) with Sum_{i=m..n} alpha(i) = 0 for m>n. - _Richard Choulet_, Feb 22 2010
%F Sum_{k=0..6*n} a(k+b)*A063265(n,k) = a(7*n+b), b>=0.
%F a(n) = 2*a(n-1) - a(n-8). - _Joerg Arndt_, Sep 24 2020
%p 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
%t LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 1}, 50] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)
%t 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 *)
%o (PARI) a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,1,1,1,1,1,1]^n*[0;0;0;0;0;0;1])[1,1] \\ _Charles R Greathouse IV_, Jun 20 2015
%Y 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).
%Y Cf. A066178, A000322, A248700.
%K nonn,easy
%O 0,9
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 18 2006
%E Edited by _N. J. A. Sloane_, Nov 20 2007
%E Wrong Binet-type formula removed by _R. J. Mathar_, Feb 13 2009
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