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

0,2

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

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).

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)

Example

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

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 *)

Prog

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

Crossrefs

Sequence in context: A293495 A375978 A330130 * A343949 A284122 A212585

Adjacent sequences: A053796 A053797 A053798 * A053800 A053801 A053802

Keyword

easy,nonn

Author

James A. Sellers, Mar 27 2000

Status

approved