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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)
32
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

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 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]

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

T. D. Noe, Table of n, a(n) for n=0..200

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

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.

F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers.

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 séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Eric Weisstein's World of Mathematics, Hexanacci Number

Index to sequences with linear recurrences with constant coefficients, signature (1,1,1,1,1,1).

FORMULA

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. [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. [Vincenzo Librandi, Dec 19 2010]

MAPLE

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); # Richard Choulet, Feb 22 2010

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 *)

CROSSREFS

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

Sequence in context: A210031 A239558 A239559 * A194629 A251710 A217832

Adjacent sequences:  A001589 A001590 A001591 * A001593 A001594 A001595

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

STATUS

approved

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Last modified December 19 02:47 EST 2014. Contains 252175 sequences.