OFFSET
0,9
COMMENTS
LINKS
Robert Price, Table of n, a(n) for n = 0..1000
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, Interval and L-interval Rational Parking Functions, arXiv:2311.14055 [math.CO], 2023. See p. 14.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
Tian-Xiao He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013) #13.8.2.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
Benjamin E. Merkel, Probabilities of Consecutive Events in Coin Flipping, Master's Thesis, Univ. Cincinatti, May 11 2011.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
Index entries for 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
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
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
Sum_{k=0..6*n} a(k+b)*A063265(n,k) = a(7*n+b), b>=0.
a(n) = 2*a(n-1) - a(n-8). - Joerg Arndt, Sep 24 2020
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy,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