A053799 - OEIS

%I #30 Jul 18 2018 09:43:03

%S 1,2,4,8,12,18,26,34,44,56,68,82,98,114,132,152,172,194,218,242,268,

%T 296,324,354,386,418,452,488,524,562,602,642,684,728,772,818,866,914,

%U 964,1016,1068,1122,1178,1234,1292,1352,1412,1474,1538,1602,1668,1736

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

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

%H Seiichi Manyama, <a href="/A053799/b053799.txt">Table of n, a(n) for n = 0..10000</a>

%H M. D. Hirschhorn, <a href="https://doi.org/10.1016/S0012-365X(99)00030-8">Basis partitions and Rogers-Ramanujan partitions</a>, Discrete Math. 205 (1999), 241-243.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

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

%F 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)).

%F a(n) = A000982(n)+A008749(n). - _John Mason_, Jan 08 2015

%F From _Michael Somos_, Jul 17 2018: (Start)

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

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

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

%e 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

%t LinearRecurrence[{2,-1,1,-2,1},{1,2,4,8,12,18},60] (* _Harvey P. Dale_, Aug 25 2015 *)

%t a[ n_] := 2 Quotient[ n^2, 3] + 2 - Boole[n == 0]; (* _Michael Somos_, Jul 17 2018 *)

%t 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 *)

%t 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 *)

%o (PARI) {a(n) = n^2 \ 3 * 2 + 2 - (n==0)}; /* _Michael Somos_, Jul 17 2018 */

%K easy,nonn

%O 0,2

%A _James A. Sellers_, Mar 27 2000