OFFSET
0,8
COMMENTS
a(n+5) is the number of ways of throwing n with an unstated number of standard dice and so the row sum of A061676; for example a(9)=8 is the number of ways of throwing a total of 4: 4, 3+1, 2+2, 1+3, 2+1+1, 1+2+1, 1+1+2 and 1+1+1+1; if order did not distinguish partitions (i.e. the dice were indistinguishable) then this would produce A001402 instead. - Henry Bottomley, Apr 01 2002
Number of permutations (p(i)) [of the numbers 1 to n, presumably? - N. J. A. Sloane, Jan 22 2021] satisfying -k<=p(i)-i<=r, i=1..n-5, with k=1, r=5. - Vladimir Baltic, Jan 17 2005
a(n+5) is the number of compositions of n with no part greater than 6. - Vladimir Baltic, Jan 17 2005
Equivalently, for n>=0: a(n+6) is the number of binary strings with length n where at most 5 ones are consecutive, see fxtbook link below. - Joerg Arndt, Apr 08 2011
REFERENCES
Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..3361 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), pp. 307-309
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135.
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.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 13
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.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Hexanacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1).
FORMULA
G.f.: x^5/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6). - Simon Plouffe in his 1992 dissertation
G.f.: Sum_{n >= 0} x^(n+5) * [ Product_{k = 1..n} (k + k*x + k*x^2 + k*x^3 + k*x^4 + x^5)/(1 + k*x + k*x^2 + k*x^3 + k*x^4 + k*x^5) ]. - Peter Bala, Jan 04 2015
Another form of the g.f.: f(z) = (z^5-z^6)/(1-2*z+z^7); then a(n) = Sum_((-1)^i*binomial(n-5-6*i,i)*2^(n-5-7*i), i=0..floor((n-5)/7))-Sum_((-1)^i*binomial(n-6-6*i,i)*2^(n-6-7*i), i=0..floor((n-6)/7)) with Sum_(alpha(i), i=m..n) = 0 for m>n. - Richard Choulet, Feb 22 2010
Sum_{k=0..5*n} a(k+b)*A063260(n,k) = a(6*n+b), b>=0.
a(n) = 2*a(n-1)-a(n-7). - Vincenzo Librandi, Dec 19 2010
lim n-> oo a(n)/a(n-1) = A118427. - R. J. Mathar, Mar 11 2024
MATHEMATICA
CoefficientList[Series[x^5/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6), {x, 0, 50}], x]
a[0] = a[1] = a[2] = a[3] = a[4] = 0; a[5] = a[6] = 1; a[n_] := a[n] = 2 a[n - 1] - a[n - 7]; Array[a, 36]
LinearRecurrence[{1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
PROG
(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; 1, 1, 1, 1, 1, 1]^n*[0; 0; 0; 0; 0; 1])[1, 1] \\ Charles R Greathouse IV, Apr 08 2016
(PARI) a(n)= my(x='x, p=polrecip(1 - x - x^2 - x^3 - x^4 - x^5 - x^6)); polcoef(lift(Mod(x, p)^n), 5);
vector(31, n, a(n-1)) \\ Joerg Arndt, May 16 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Nov 16 2000
STATUS
approved