A163985 Sum of all isolated parts of all partitions of n.

0, 1, 2, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113

Offset: 0

Author

Omar E. Pol, Aug 14 2009

Keywords

• easy
• nonn
• less

Comments

Note that for n >= 3 the isolated parts of all partitions of n are n and n-1.

Examples

			For n=4, the five partitions of 4 are {(4);(2,2);(3,1);(2,1,1);(1,1,1,1)}. Since 1 and 2 are repeated parts and 3 and 4 are not repeated parts (or isolated parts) then a(4) = 3 + 4 = 7.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Omar E. Pol, Illustration of the shell model of partitions (2D view).
• Omar E. Pol, Illustration of the shell model of partitions (3D view).
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A000041, A005408, A140139, A163986.

Programs

Magma

[0,1,2] cat [2*n-1: n in [3..60]]; // Vincenzo Librandi, Dec 23 2015

Mathematica

Join[{0, 1, 2}, Table[2 n - 1, {n, 3, 60}]] (* Vincenzo Librandi, Dec 23 2015 *)

PARI

a(n) = if (n<3, n, 2*n-1); \\ Michel Marcus, Dec 23 2015

Formula

a(n) = n for n<3, a(n) = 2*n-1 for n>=3.

a(n) = A140139(n), n>=1.

a(n) = A130773(n-1), n >=2. - R. J. Mathar, Jan 25 2023

From Stefano Spezia, Apr 21 2025: (Start)

G.f.: x*(1 + 2*x^2 - x^3)/(1 - x)^2.

E.g.f.: 1 - x^2/2 - exp(x)*(1 - 2*x). (End)