A082562 a(n) = number of values of m such that m can be expressed as the sum of distinct odd numbers with largest odd number in the sum = 2n+1.

1, 2, 4, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323, 360, 399, 440, 483, 528, 575, 624, 675, 728, 783, 840, 899, 960, 1023, 1088, 1155, 1224, 1295, 1368, 1443, 1520, 1599, 1680, 1763, 1848, 1935, 2024, 2115, 2208, 2303, 2400, 2499

Offset: 0

Author

Naohiro Nomoto, May 05 2003

Keywords

• easy
• nonn

Comments

Beginning with the third term, the first differences are the odd positive integers. - John W. Layman, Feb 28 2012

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A082548, A082547.

Programs

Mathematica

Join[{1, 2, 4}, LinearRecurrence[{3, -3, 1}, {8, 15, 24}, 80]] (* and *) Join[{1, 2, 4}, Table[n^2 - 1, {n, 3, 80}]] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

PARI

Vec((1-x+x^2+x^3+x^4-x^5)/(1-x)^3 + O(x^100)) \\ Colin Barker, Feb 15 2016

Formula

For n>2, a(n) = n^2-1. The values of m are all values from 2n+1 to (n+1)^2 except 2n+3 and n^2+2n-1. - David Wasserman, Sep 16 2004

From Colin Barker, Feb 15 2016: (Start)

a(n) = n^2-1 for n>2.

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5.

G.f.: (1-x+x^2+x^3+x^4-x^5) / (1-x)^3.

(End)

Extensions

More terms from David Wasserman, Sep 16 2004