A165190 G.f.: 1/((1-x^4)*(1-x^5)).

1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 5, 5, 4, 4, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6

Offset: 0

Author

Alford Arnold, Sep 24 2009

Keywords

• nonn
• easy

Comments

A121262 convolved with A079998. The two sequences have very simple generating functions and can be mapped to the numeric partitions 4=4 and 5=5 respectively.

Number of partitions of n into parts 4 and 5. - Joerg Arndt, Aug 28 2015

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
• Index to Molien series
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1,0,0,0,-1).

Crossrefs

Cf. A057077, A079998, A117444, A121262.

Programs

Magma

[Floor((n+4)/4) - Floor((n+4)/5) : n in [0..100]]; // Wesley Ivan Hurt, Aug 27 2015

Maple

A165190:=n->floor((n+4)/4) - floor((n+4)/5): seq(A165190(n), n=0..100); # Wesley Ivan Hurt, Aug 27 2015

Mathematica

CoefficientList[Series[1/((1-x^4)(1-x^5)),{x,0,110}],x] (* or *) LinearRecurrence[{0,0,0,1,1,0,0,0,-1},{1,0,0,0,1,1,0,0,1},110] (* Harvey P. Dale, Aug 16 2012 *)
Table[Floor[(n + 4)/4] - Floor[(n + 4)/5], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 27 2015 *)

Formula

1 followed by the Euler transform of the finite sequence [0,0,0,1,1].

G.f.: 1/((1-x)^2*(1+x)*(1+x^2)*(1+x+x^2+x^3+x^4)). [R. J. Mathar, Oct 07 2009]

a(n) = A117444(n+2)/5 + n/20 + 9/40 + (-1)^n/8 + A057077(n)/4. [R. J. Mathar, Oct 07 2009]

a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(5)=1, a(6)=0, a(7)=0, a(8)=1, a(n) = a(n-4)+a(n-5)-a(n-9), n>8. - Harvey P. Dale, Aug 16 2012

a(n) = floor((n+4)/4) - floor((n+4)/5). - Wesley Ivan Hurt, Aug 27 2015

a(n)+a(n-2) = A008616(n). - R. J. Mathar, Jun 23 2021

Extensions

Removed duplicate of comment in A165188; Euler transform formula corrected - R. J. Mathar, Oct 07 2009