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A013979 Expansion of 1/(1 - x^2 - x^3 - x^4) = 1/((1 + x)*(1 - x - x^3)). 10
1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 17, 24, 36, 52, 77, 112, 165, 241, 354, 518, 760, 1113, 1632, 2391, 3505, 5136, 7528, 11032, 16169, 23696, 34729, 50897, 74594, 109322, 160220, 234813, 344136, 504355, 739169, 1083304, 1587660, 2326828, 3410133, 4997792, 7324621 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

For n>0, number of compositions (ordered partitions) of n into 2's, 3's and 4's. - Len Smiley, May 08 2001

Diagonal sums of trinomial triangle A071675 (Riordan array (1, x(1+x+x^2))). - Paul Barry, Feb 15 2005

For n>1, a(n) is number of compositions of n-2 into parts 1 and 2 with no 3 consecutive 1's. For example: a(7) = 5 because we have: 2+2+1, 2+1+2, 1+2+2, 1+2+1+1, 1+1+2+1. - Geoffrey Critzer, Mar 15 2014

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

C. K. Fan, A Hecke algebra quotient and some combinatorial applications, J. Algebraic Combin. 5 (1996), no. 3, 175-189.

C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f^0_n.]

R. Mullen, On Determining Paint by Numbers Puzzles with Nonunique Solutions, JIS 12 (2009) 09.6.5

Index entries for linear recurrences with constant coefficients, signature (0,1,1,1).

FORMULA

a(n) = sum{k=0..floor(n/2), sum{i=0..floor(n/2), C(k, 2i+3k-n)C(2i+3k-n, i)}}. - Paul Barry, Feb 15 2005

a(n) = a(n-4) + a(n-3) + a(n-2). - Jon E. Schoenfield, Aug 07 2006

a(n)+a(n+1) = A000930(n+1). - R. J. Mathar, Mar 14 2011

a(n) = (1/3)*(A000930(n) + A097333(n-2) + (-1)^n), n>1. - Ralf Stephan, Aug 15 2013

a(n) = (-1)^n * A077889(-4-n) = A107458(n+4) for all n in Z. - Michael Somos, Jun 20 2015

EXAMPLE

G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 5*x^7 + 8*x^8 + 11*x^9 + ...

MATHEMATICA

a=b=c=0; d=1; lst={d}; Do[AppendTo[lst, e=a+b+c]; a=b; b=c; c=d; d=e, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, May 28 2010 *)

a[ n_] := If[ n < 0, SeriesCoefficient[ x^4 / (1 + x + x^2 - x^4), {x, 0, -n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3 - x^4), {x, 0, n}]]; (* Michael Somos, Jun 20 2015 *)

PROG

(Haskell)

a013979 n = a013979_list !! n

a013979_list = 1 : 0 : 1 : 1 : zipWith (+) a013979_list

   (zipWith (+) (tail a013979_list) (drop 2 a013979_list))

-- Reinhard Zumkeller, Mar 23 2012

CROSSREFS

Cf. A060945 (Ordered partitions into 1's, 2's and 4's), A107458.

First differences of A023435.

Cf. A001634.

Sequence in context: A240734 A328460 A238478 * A107458 A274142 A006206

Adjacent sequences:  A013976 A013977 A013978 * A013980 A013981 A013982

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 19 22:04 EST 2020. Contains 332060 sequences. (Running on oeis4.)