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A025767 Expansion of 1/((1-x)*(1-x^3)*(1-x^4)). 3
1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 17, 18, 20, 22, 24, 26, 28, 30, 33, 35, 37, 40, 43, 45, 48, 51, 54, 57, 60, 63, 67, 70, 73, 77, 81, 84, 88, 92, 96, 100, 104, 108, 113, 117, 121, 126, 131, 135, 140, 145, 150, 155, 160, 165, 171, 176, 181, 187, 193, 198 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Apply the Riordan array (1/(1-x^4),x) to floor((n+3)/3). - Paul Barry, Jan 20 2006

Partitions of n into parts 1, 3, and 4. - David Neil McGrath, Aug 30 2014

Also, a(n-4) is equal to the number of partitions mu of n of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is odd or vice-versa (see below example). - John M. Campbell, Jan 29 2016

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

FORMULA

G.f.: 1/((1-x)*(1-x^3)*(1-x^4)).

a(n) = floor(n^2/24+n/3+1).

a(n) = sum{k=0..floor(n/4), floor((n-4k+3)/3)}. - Paul Barry, Jan 20 2006

Euler transform of length 4 sequence [ 1, 0, 1, 1]. - Michael Somos, Nov 09 2007

a(-8 - n) = a(n). - Michael Somos, Nov 09 2007

a(n) = n^2/24 + n/3 + 83/144 + (-1)^n/16 + A061347(n+1)/9 + A056594(n)/4. - R. J. Mathar, Mar 31 2011

a(n) = a(n-1)+a(n-3)-a(n-5)-a(n-7)+a(n-8). - David Neil McGrath, Aug 30 2014

EXAMPLE

The a(4)=3 partitions of 4 into parts 1, 3, and 4 are (4), (3,1), and (1,1,1,1). - David Neil McGrath, Aug 30 2014

From John M. Campbell, Jan 29 2016: (Start)

Letting n=12, there are a(n-4)=a(8)=6 partitions mu of n=12 of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is odd or vice-versa:

(10,1,1) |- n

(8,3,1) |- n

(7,3,2) |- n

(6,5,1) |- n

(6,3,3) |- n

(5,5,2) |- n

(End)

MAPLE

A056594 := proc(n) op(1+(n mod 4), [1, 0, -1, 0]) ; end proc:

A061347 := proc(n) op(1+(n mod 3), [-2, 1, 1]) ; end proc:

A025767 := proc(n) n^2/24+n/3+83/144+(-1)^n/16 +A061347(n+1)/9 +A056594(n)/4 ; end proc: # R. J. Mathar, Mar 31 2011

MATHEMATICA

Table[Floor[n^2/24 + n/3 + 1], {n, 0, 60}] (* Vincenzo Librandi, Aug 31 2014 *)

PROG

(PARI) a(n)=if(n<0, 0, (n^2+8*n)\24+1)

(PARI) {a(n) = round( ((n + 4)^2 - 1) / 24 )} /* Michael Somos, Nov 09 2007 */

(PARI) Vec(1/((1-x)*(1-x^3)*(1-x^4)) + O(x^80)) \\ Michel Marcus, Jan 29 2016

(MAGMA) [Floor(n^2/24 + n/3 + 1): n in [0..70]]; // Vincenzo Librandi, Aug 31 2014

CROSSREFS

A008621(n) = A002265(n+4) = a(n) - a(n-3).

Sequence in context: A011883 A034886 A011882 * A091848 A017886 A029038

Adjacent sequences:  A025764 A025765 A025766 * A025768 A025769 A025770

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 3 12:40 EST 2016. Contains 278735 sequences.