Search: id:A001591
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%I A001591 M1122 N0429
%S A001591 0,0,0,0,1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930,13624,26784,
%T A001591 52656,103519,203513,400096,786568,1546352,3040048,5976577,11749641,
%U A001591 23099186,45411804,89277256,175514464,345052351,678355061,1333610936
%N A001591 Pentanacci numbers: a(n+1)=a(n)+...+a(n-4).
%C A001591 Number of permutations satisfying -k<=p(i)-i<=r, i=1..n-4, with k=1, 
               r=4. - Vladimir Baltic (baltic(AT)matf.bg.ac.yu), Jan 17 2005
%C A001591 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 (baltic(AT)matf.bg.ac.yu), Jan 17 2005
%D A001591 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001591 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001591 S. 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.
%D A001591 I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
%D A001591 V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal 
               triangles, Fib. Quart., 7 (1969), 341-358, 393.
%D A001591 Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas 
               n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 
               05.4.4.
%H A001591 T. D. Noe, <a href="b001591.txt">Table of n, a(n) for n=0..200</a>
%H A001591 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A001591 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001591 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001591 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001591 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=12">
               Encyclopedia of Combinatorial Structures 12</a>
%H A001591 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%H A001591 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PentanacciNumber.html">Pentanacci Number</a>
%F A001591 x^4/(1 - x - x^2 - x^3 - x^4 - x^5)
%F A001591 G.f.: 1/(1-z-z^2-z^3-z^4-z^5) . (S.Plouffe) [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 17 2009]
%p A001591 A001591:=-1/(-1+z+z**2+z**3+z**4+z**5); [Conjectured by S. Plouffe in 
               his 1992 dissertation.]
%p A001591 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]
%t A001591 CoefficientList[Series[x^4/(1 - x - x^2 - x^3 - x^4 - x^5), {x, 0, 50}], 
               x]
%Y A001591 Row 5 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y A001591 Sequence in context: A006775 A104993 A128761 this_sequence A003240 A018487 
               A010747
%Y A001591 Adjacent sequences: A001588 A001589 A001590 this_sequence A001592 A001593 
               A001594
%K A001591 nonn
%O A001591 0,7
%A A001591 N. J. A. Sloane (njas(AT)research.att.com).
%E A001591 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 16 2000

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