A025767 Expansion of 1/((1-x)*(1-x^3)*(1-x^4)).

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

Offset

0,4

Comments

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

Number of 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

With four 0's prepended and offset 0, a(n) is the number of partitions of n into four parts whose 2nd and 3rd largest parts are equal. - Wesley Ivan Hurt, Jan 05 2021

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-4*k+3)/3)}. - Paul Barry, Jan 20 2006

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

a(n) = a(-8 - n) for all n in Z. - 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

a(n) = Sum_{k=1..floor((n+4)/4)} Sum_{j=k..floor((n+4-k)/3)} Sum_{i=j..floor((n+4-j-k)/2)} [j = i], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 17 2021

a(n)-a(n-1) = A008679(n). - R. J. Mathar, Jun 23 2021

a(n)-a(n-4) = A008620(n). - R. J. Mathar, Jun 23 2021

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