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A053799 Number of basis partitions of n+9 with Durfee square size 3. 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 (list; graph; refs; listen; history; text; internal format)
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: A080476 A256885 A293495 * A284122 A212585 A085891

Adjacent sequences:  A053796 A053797 A053798 * A053800 A053801 A053802

KEYWORD

easy,nonn

AUTHOR

James A. Sellers, Mar 27 2000

STATUS

approved

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Last modified July 18 11:25 EDT 2019. Contains 325138 sequences. (Running on oeis4.)