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 A001592 Hexanacci numbers: a(n+1) = a(n)+...+a(n-5) with a(0)=...=a(4)=0, a(5)=1. (Formerly M1128 N0431) 25

%I M1128 N0431

%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 satisfying -k<=p(i)-i<=r, i=1..n-5, with k=1, r=5. - _Vladimir Baltic_, Jan 17 2005

%C a(n)=number of compositions of n-5 with no part greater than 6. Example: a(12)=63 because we have 63 compositions of 7: 7=1+1+1+1+1+1+1=2+1+1+1+1+1=...=2+2+1+1+1=...=2+2+2+1=...=3+1+1+1+1=... =3+2+1+1=...=3+2+2=...=3+3+1=...=4+1+1+1=...=4+2+1=...=4+3=3+4=5+1+1 =1+5+1=1+1+5=5+2=2+5=6+1=1+6 - _Vladimir Baltic_, Jan 17 2005

%C For n>=0: a(n+5) 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 I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.

%D F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, http://www.math-cs.ucmo.edu/~curtisc/articles/howardcooper/genfib4.pdf.

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

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

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=13">Encyclopedia of Combinatorial Structures 13</a>

%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="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

%H _Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 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>

%F x^5/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6)

%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. [From _Richard Choulet_, Feb 22 2010]

%F sum_{k=0..5*n} A001592(k+b)*A063260(n,k) = A001592(6*n+b), b>=0.

%F a(n) = 2*a(n-1)-a(n-7) with initial values 0, 0, 0, 0, 0, 1, 1. [From Vincenzo Librandi, Dec 19 2010]

%p A001592:=-1/(-1+z+z**2+z**3+z**4+z**5+z**6); [_Simon Plouffe_ in his 1992 dissertation.]

%p for n from 0 to 50 do k(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)):od:seq(k(n),n=0..50); a:=taylor((z^5-z^6)/(1-2*z+z^7),z=0,51);for p from 0 to 50 do j(p):=coeff(a,z,p):od :seq(j(p),p=0..50); [From _Richard Choulet_, Feb 22 2010]

%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] (* From Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

%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

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