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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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36
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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 (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
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REFERENCES
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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).
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LINKS
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FORMULA
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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.
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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] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
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PROG
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(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);
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CROSSREFS
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Row 6 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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