%I M1128 N0431 #149 Sep 29 2024 03:01:20

%S 0,0,0,0,0,1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617,15109,

%T 29970,59448,117920,233904,463968,920319,1825529,3621088,7182728,

%U 14247536,28261168,56058368,111196417,220567305,437513522,867844316,1721441096,3414621024

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

%C 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

%C 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

%C a(n+5) is the number of compositions of n with no part greater than 6. - _Vladimir Baltic_, Jan 17 2005

%C 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

%D Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Indranil Ghosh, <a href="/A001592/b001592.txt">Table of n, a(n) for n = 0..3361</a> (terms 0..200 from T. D. Noe)

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp. 307-309

%H Vladimir Baltic, <a href="http://pefmath.etf.rs/vol4num1/AADM-Vol4-No1-119-135.pdf">On the number of certain types of strongly restricted permutations</a>, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135.

%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 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H I. Flores, <a href="http://www.fq.math.ca/Scanned/5-3/flores.pdf">k-Generalized Fibonacci numbers</a>, Fib. Quart., 5 (1967), 258-266.

%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 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=13">Encyclopedia of Combinatorial Structures 13</a>

%H Omar Khadir, László Németh, and László Szalay, <a href="https://doi.org/10.1007/s00025-024-02284-3">Tiling of dominoes with ranked colors</a>, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.

%H Sergey Kirgizov, <a href="https://arxiv.org/abs/2201.00782">Q-bonacci words and numbers</a>, arXiv:2201.00782 [math.CO], 2022.

%H László Németh and László Szalay, <a href="https://arxiv.org/abs/2408.12196">Explicit solution of system of two higher-order recurrences</a>, arXiv:2408.12196 [math.NT], 2024. See p. 10.

%H Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexanacciNumber.html">Hexanacci Number</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1,1).

%F G.f.: x^5/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6). - _Simon Plouffe_ in his 1992 dissertation

%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 + x^5)/(1 + k*x + k*x^2 + k*x^3 + k*x^4 + k*x^5) ]. - _Peter Bala_, Jan 04 2015

%F 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

%F Sum_{k=0..5*n} a(k+b)*A063260(n,k) = a(6*n+b), b>=0.

%F a(n) = 2*a(n-1)-a(n-7). - _Vincenzo Librandi_, Dec 19 2010

%F lim n-> oo a(n)/a(n-1) = A118427. - _R. J. Mathar_, Mar 11 2024

%t CoefficientList[Series[x^5/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6), {x, 0, 50}], x]

%t 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]

%t LinearRecurrence[{1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 1}, 50] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)

%o (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

%o (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);

%o vector(31,n,a(n-1)) \\ _Joerg Arndt_, May 16 2021

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

%K nonn,easy

%O 0,8

%A _N. J. A. Sloane_

%E More terms from _Robert G. Wilson v_, Nov 16 2000