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

0,3

Comments

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

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).

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)

Example

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.

Mathematica

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

Prog

(Magma) [0, 1, 2] cat [2*n-1: n in [3..60]]; // Vincenzo Librandi, Dec 23 2015
(PARI) a(n) = if (n<3, n, 2*n-1); \\ Michel Marcus, Dec 23 2015

Crossrefs

Cf. A000041, A005408, A140139, A163986.

Sequence in context: A247421 A007069 A182463 * A130773 A140139 A184737

Adjacent sequences: A163982 A163983 A163984 * A163986 A163987 A163988

Keyword

easy,nonn,less

Author

Omar E. Pol, Aug 14 2009

Status

approved