A053799 Number of basis partitions of n+9 with Durfee square size 3.

1, 2, 4, 8, 12, 18, 26, 34, 44, 56, 68, 82, 98, 114, 132, 152, 172, 194, 218, 242, 268, 296, 324, 354, 386, 418, 452, 488, 524, 562, 602, 642, 684, 728, 772, 818, 866, 914, 964, 1016, 1068, 1122, 1178, 1234, 1292, 1352, 1412, 1474, 1538, 1602, 1668, 1736

Offset: 0

Author

James Sellers, Mar 27 2000

Keywords

• easy
• nonn

Comments

a(n) is the number of solutions in integers (x,y,z) of |x| + 2|y| + 3|z| = |n|. - Michael Somos, Jul 17 2018

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 12*x^4 + 18*x^5 + 26*x^6 + 34*x^7 + ... - _Michael Somos_, Jul 17 2018
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• M. D. Hirschhorn, Basis partitions and Rogers-Ramanujan partitions, Discrete Math. 205 (1999), 241-243.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Programs

Mathematica

LinearRecurrence[{2,-1,1,-2,1},{1,2,4,8,12,18},60] (* Harvey P. Dale, Aug 25 2015 *)
a[ n_] := 2 Quotient[ n^2, 3] + 2 - Boole[n == 0]; (* Michael Somos, Jul 17 2018 *)
a[ n_] := SeriesCoefficient[ (1 + x^2) (1 + x^3) / ((1 - x)^3 (1 + x + x^2)), {x, 0, Abs@n}]; (* Michael Somos, Jul 17 2018 *)
a[ n_] := Length @ FindInstance[ Abs[x] + 2 Abs[y] + 3 Abs[z] == Abs[n], {x, y, z}, Integers, 10^9]; (* Michael Somos, Jul 17 2018 *)

PARI

{a(n) = n^2 \ 3 * 2 + 2 - (n==0)}; /* Michael Somos, Jul 17 2018 */

Formula

For n>0, a(n) = 2*(1+floor(n^2/3)) = 2*A087483(n-1) = 2*(1+A000212(n)). - Max Alekseyev, Dec 05 2013

G.f.: (1+x)*(1+x^2)*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) = (1+x)*(1+x^2)*(1-x+x^2)/((1-x)^3*(1+x+x^2)).

a(n) = A000982(n)+A008749(n). - John Mason, Jan 08 2015

From Michael Somos, Jul 17 2018: (Start)

Euler transform of length 6 sequence [2, 1, 2, -1, 0, -1].

a(n+1) - 2*a(n) + a(n-1) = 1 + (-1)^n if |n|>1.

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