A092530 a(0) = 0; for n > 0, a(n) = T(n) + k where T(n) is the n-th triangular number (A000217) and k (see A026741) is the smallest positive number such that a(n) is divisible by n.

0, 2, 4, 9, 12, 20, 24, 35, 40, 54, 60, 77, 84, 104, 112, 135, 144, 170, 180, 209, 220, 252, 264, 299, 312, 350, 364, 405, 420, 464, 480, 527, 544, 594, 612, 665, 684, 740, 760, 819, 840, 902, 924, 989, 1012, 1080, 1104, 1175, 1200, 1274, 1300, 1377, 1404, 1484

Offset: 0

Author

N. J. A. Sloane, Apr 08 2004

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• A. W. Vyawahare and K. M. Purohit, Near pseudo Smarandache function, Smarandache Notions, 14 (2004), 42-61.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Equals A000217 + A026741.

Programs

Maple

seq(n*(1+ceil(n/2)), n=0..53); # Zerinvary Lajos and Klaus Brockhaus, Apr 10 2007

Mathematica

{0}~Join~Array[Block[{k = 1}, While[GCD[#1, #2 + k] < #1, k++]; #2 + k] & @@ {#, (#^2 + #)/2} &, 53] (* or *)
CoefficientList[Series[x (2 + 2 x + x^2 - x^3)/((1 - x)^3*(1 + x)^2), {x, 0, 53}], x] (* Michael De Vlieger, Feb 03 2019 *)
LinearRecurrence[{1,2,-2,-1,1},{0,2,4,9,12},60] (* Harvey P. Dale, Sep 21 2024 *)

PARI

for(n=0,53,print1(n*(1+ceil(n/2)),",")); \\ Klaus Brockhaus, Apr 10 2007

PARI

concat(0, Vec(x*(2 + 2*x + x^2 - x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Feb 03 2019

Formula

a(0) = 0, a(2n) = a(2n-1) + n, a(2n-1) = a(2n-2) + 3n-1. - Amarnath Murthy, Jul 04 2004

From Colin Barker, Feb 03 2019: (Start)

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

a(n) = (n*(2 + n)) / 2 for n even.

a(n) = (n*(3 + n)) / 2 for n odd.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

(End)