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A001592
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Hexanacci numbers: a(n+1) = a(n)+...+a(n-5) with a(0)=...=a(4)=0, a(5)=1.
(Formerly M1128 N0431)
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25
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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
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OFFSET
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0,8
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COMMENTS
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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 satisfying -k<=p(i)-i<=r, i=1..n-5, with k=1, r=5. - Vladimir Baltic, Jan 17 2005
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
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]
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REFERENCES
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I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, http://www.math-cs.ucmo.edu/~curtisc/articles/howardcooper/genfib4.pdf.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Joerg Arndt, 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
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 13
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 S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Hexanacci Number
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FORMULA
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x^5/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6)
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]
sum_{k=0..5*n} A001592(k+b)*A063260(n,k) = A001592(6*n+b), b>=0.
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]
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MAPLE
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A001592:=-1/(-1+z+z**2+z**3+z**4+z**5+z**6); [Simon Plouffe in his 1992 dissertation.]
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]
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MATHEMATICA
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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] (* From Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
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CROSSREFS
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Row 6 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Sequence in context: A062259 A001949 A210031 * A194629 A217832 A140134
Adjacent sequences: A001589 A001590 A001591 * A001593 A001594 A001595
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Robert G. Wilson v, Nov 16 2000
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STATUS
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approved
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