A193711 Partial sums of the union of squares and triangular numbers.

1, 4, 8, 14, 23, 33, 48, 64, 85, 110, 138, 174, 219, 268, 323, 387, 453, 531, 612, 703, 803, 908, 1028, 1149, 1285, 1429, 1582, 1751, 1922, 2112, 2308, 2518, 2743, 2974, 3227, 3483, 3759, 4048, 4348, 4672, 4997, 5348, 5709, 6087, 6487, 6893, 7328, 7769, 8234

Offset: 1

Author

Reinhard Zumkeller, Aug 03 2011

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A000290, A000292, A000330, A005214.

Programs

Haskell

a193711 n = a193711_list !! (n-1)
a193711_list = scanl1 (+) a005214_list :: [Integer]

Mathematica

Module[{upto=500,sq,tr},sq=Range[Floor[Sqrt[upto]]]^2;tr=Accumulate[ Range[ Floor[ (Sqrt[1+8*upto]-1)/2]]];Accumulate[Union[Join[tr,sq]]]] (* Harvey P. Dale, Jun 24 2018 *)

Formula

a(n) = Sum_{k=1..n} A005214(k).

a(n) ~ c * n^3, where c = 1 - 2*sqrt(2)/3 = 0.0571909... . - Amiram Eldar, Apr 04 2025

Extensions

Program and b-file corrected by Reinhard Zumkeller, Aug 04 2011