A307018 Total number of parts of size 3 in the partitions of n into parts of size 2 and 3.

0, 0, 0, 1, 0, 1, 2, 1, 2, 4, 2, 4, 6, 4, 6, 9, 6, 9, 12, 9, 12, 16, 12, 16, 20, 16, 20, 25, 20, 25, 30, 25, 30, 36, 30, 36, 42, 36, 42, 49, 42, 49, 56, 49, 56, 64, 56, 64, 72, 64, 72, 81, 72, 81, 90, 81, 90, 100, 90, 100, 110, 100, 110, 121, 110, 121, 132

Offset

0,7

Links

Table of n, a(n) for n=0..66.

Index entries for linear recurrences with constant coefficients, signature (0,1,2,0,-2,-1,0,1).

Formula

a(n+2) = A321202(n) - A114209(n+1).

a(3n+1) = A002620(n+2).

a(3n+2) = A002620(n+1).

a(3n+3) = A002620(n+2).

G.f.: x^3/((1+x)*(1+x+x^2)^2*(1-x)^3). - Alois P. Heinz, Mar 19 2019

a(n) = a(n-2) + 2*a(n-3) - 2*a(n-5) - a(n-6) + a(n-8). - G. C. Greubel, Apr 03 2019

a(n) = (6*n*(2 + n) + 8*(4 + 3*n)*cos(2*n*Pi/3) - 8*sqrt(3)*n*sin(2*n*Pi/3) - 5 - 27*(-1)^n)/216. - Stefano Spezia, Apr 21 2022

Mathematica

LinearRecurrence[{0, 1, 2, 0, -2, -1, 0, 1}, {0, 0, 0, 1, 0, 1, 2, 1}, 80] (* G. C. Greubel, Apr 03 2019 *)
Table[(6n(2+n)-5-27(-1)^n+8(4+3n)Cos[2n Pi/3]-8Sqrt[3]n Sin[2n Pi/3])/216, {n, 0, 66}] (* Stefano Spezia, Apr 21 2022 *)

Prog

(PARI) my(x='x+O('x^80)); concat([0, 0, 0], Vec(x^3/((1-x^2)*(1-x^3)^2))) \\ G. C. Greubel, Apr 03 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 80); [0, 0, 0] cat Coefficients(R!( x^3/((1-x^2)*(1-x^3)^2) )); // G. C. Greubel, Apr 03 2019
(Sage) (x^3/((1-x^2)*(1-x^3)^2)).series(x, 80).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019
(GAP) a:=[0, 0, 0, 1, 0, 1, 2, 1];; for n in [9..80] do a[n]:=a[n-2]+2*a[n-3] -2*a[n-5]-a[n-6]+a[n-8]; od; a; # G. C. Greubel, Apr 03 2019

Crossrefs

Cf. A000041, A002620, A103221, A114209, A321202.

Sequence in context: A234359 A099500 A120253 * A274624 A263050 A060547

Adjacent sequences: A307015 A307016 A307017 * A307019 A307020 A307021

Keyword

nonn,easy

Author

Andrew Ivashenko, Mar 19 2019

Extensions

More terms from Alois P. Heinz, Mar 19 2019

Status

approved