A001592 Hexanacci numbers: a(n+1) = a(n)+...+a(n-5) with a(0)=...=a(4)=0, a(5)=1.

0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968, 920319, 1825529, 3621088, 7182728, 14247536, 28261168, 56058368, 111196417, 220567305, 437513522, 867844316, 1721441096, 3414621024

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Row 6 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).

Programs

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 *)

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

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

Extensions

More terms from Robert G. Wilson v, Nov 16 2000