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A001591 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)
47
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

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

The pentanomial (A035343(n)) transform of a(n) is a(5n+4), n>=0. - Bob Selcoe, Jun 10 2014

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.

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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 12

Kruchinin, Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565

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, Pentanacci Number

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

FORMULA

G.f. x^4/(1 - x - x^2 - x^3 - x^4 - x^5). - Simon Plouffe in his 1992 dissertation.

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. - Richard Choulet, 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. - 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. - Vincenzo Librandi, Dec 19 2010

EXAMPLE

n=2: a(14) = (1*1 + 2*1 + 3*2 + 4*4 + 5*8 + 4*16 + 3*31 + 2*61 + 1*120) = 464. - Bob Selcoe, Jun 10 2014

G.f. = x^4 + x^5 + 2*x^6 + 4*x^7 + 8*x^8 + 16*x^9 + 31*x^10 + 120*x^11 + ...

MAPLE

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); # Zerinvary Lajos, 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); # Richard Choulet, Feb 22 2010

MATHEMATICA

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

a[n_] := a[n] = Sum[Sum[Binomial[k, r]*Sum[Binomial[r, m]*Sum[Binomial[m, j]*Binomial[j, n - 4 - m - k - j - r], {j, 0, m}], {m, 0, r}], {r, 0, k}], {k, 1, n - 4}]; a[4] = 1; Table[a[n], {n, 0, 37}] (* After Kruchinin's formula. - L. Edson Jeffery, Jul 18 2014 *)

PROG

(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); /* 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

(Maxima) a(n):=mod(floor(10^((n-4)*(n+1))*10^(5*(n+1))*(10^(n+1)-1)/(10^(6*(n+1))-2*10^(5*(n+1))+1)), 10^n); /* Tani Akinari, Apr 10 2014 */

CROSSREFS

Row 5 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).

Cf. A035343 (pentanomial coefficients).

Sequence in context: A192656 A128761 A239557 * A194628 A003240 A018487

Adjacent sequences:  A001588 A001589 A001590 * A001592 A001593 A001594

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Apr 30 1991

EXTENSIONS

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

STATUS

approved

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Last modified October 21 14:25 EDT 2014. Contains 248377 sequences.