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A001591
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Pentanacci numbers: a(n) = a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5), a(0)=a(1)=a(2)=a(3)=0, a(4)=1.
(Formerly M1122 N0429)
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44
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0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351, 678355061, 1333610936, 2621810068
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OFFSET
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0,7
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COMMENTS
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Number of permutations satisfying -k<=p(i)-i<=r, i=1..n-4, with k=1, r=4. - Vladimir Baltic, Jan 17 2005
a(n)=number of compositions of n-4 with no part greater than 5. Example: a(12)=61 because we have 61 compositions of 8: 8=1+1+1+1+1+1+1+1=2+1+1+1+1+1+1=...=2+2+1+1+1+1=...=2+2+2+1+1=...=2+2+2+2 =3+1+1+1+1+1=...=3+2+1+1+1=...=3+2+2+1=...=3+3+1+1=...=3+3+2=... =4+1+1+1+1=...=4+2+1+1=...=4+2+2=...=4+3+1=...=5+1+1+1=...=5+2+1=...=5+3=3+5 - Vladimir Baltic, Jan 17 2005
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REFERENCES
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I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
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 12
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, Pentanacci Number
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
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x^4/(1 - x - x^2 - x^3 - x^4 - x^5)
G.f.: 1/(1-z-z^2-z^3-z^4-z^5) . (Simon Plouffe) [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 17 2009]
Another form of the g.f.: f(z)=(z^4-z^5)/(1-2*z+z^6); then a(n)=sum((-1)^i*binomial(n-4-5*i,i)*2^(n-4-6*i),i=0..floor((n-4)/6))-sum((-1)^i*binomial(n-5-5*i,i)*2^(n-5-6*i),i=0..floor((n-5)/6)) with convention sum(alpha(i),i=m..n)=0 for m>n. [From Richard Choulet (richard.choulet(AT)orange.fr), Feb 22 2010]
a(n)=sum(sum(binomial(k,r)*sum(binomial(r,m)*sum(binomial(m,j)*binomial(j,n-m-k-j-r),j,0,m),m,0,r),r,0,k),k,1,n), n>0 [From Vladimir Kruchinin, Aug 30 2010]
sum_{k=0..4*n} A001591(k+b)*A035343(n,k) = A001591(5*n+b), b>=0.
a(n) = 2*a(n-1)-a(n-6) with initial values 0, 0, 0, 0, 1, 1. [From Vincenzo Librandi, Dec 19 2010]
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MAPLE
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A001591:=-1/(-1+z+z**2+z**3+z**4+z**5); [Conjectured by Simon Plouffe in his 1992 dissertation.]
g:=1/(1-z-z^2-z^3-z^4-z^5): gser:=series(g, z=0, 49): seq((coeff(gser, z, n)), n=-4..32); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 17 2009]
a:=taylor((z^4-z^5)/(1-2*z+z^6), z=0, 51); for p from 0 to 50 do j(p):=coeff(a, z, p):od :seq(j(p), p=0..50); for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-4-5*i, i)*2^(n-4-6*i), i=0..floor((n-4)/6))-sum((-1)^i*binomial(n-5-5*i, i)*2^(n-5-6*i), i=0..floor((n-5)/6)):od:seq(k(n), n=0..50); [From Richard Choulet (richard.choulet(AT)orange.fr), Feb 22 2010]
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MATHEMATICA
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CoefficientList[Series[x^4/(1 - x - x^2 - x^3 - x^4 - x^5), {x, 0, 50}], x]
a[0] = a[1] = a[2] = a[3] = 0; a[4] = a[5] = 1; a[n_] := a[n] = 2 a[n - 1] - a[n - 6]; Array[a, 37, 0]
LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 0, 0, 1}, 50] (* From Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
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PROG
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(Maxima) a(n):=sum(sum(binomial(k, r)*sum(binomial(r, m)*sum(binomial(m, j)*binomial(j, n-m-k-j-r), j, 0, m), m, 0, r), r, 0, k), k, 1, n); [From Vladimir Kruchinin, Aug 30 2010]
(PARI) a=vector(100); a[4]=a[5]=1; for(n=6, #a, a[n]=a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]); a \\ Charles R Greathouse IV, Jul 15 2011
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CROSSREFS
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Row 5 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Sequence in context: A189076 A192656 A128761 * A194628 A003240 A018487
Adjacent sequences: A001588 A001589 A001590 * A001592 A001593 A001594
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KEYWORD
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nonn,easy,changed
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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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