A157130 - OEIS

A157130

Partial sums of A128201.

1, 4, 8, 13, 20, 29, 40, 53, 68, 84, 101, 120, 141, 164, 189, 216, 245, 276, 309, 344, 380, 417, 456, 497, 540, 585, 632, 681, 732, 785, 840, 897, 956, 1017, 1080, 1144, 1209, 1276, 1345, 1416, 1489, 1564, 1641, 1720, 1801, 1884, 1969, 2056, 2145, 2236, 2329

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..51.

FORMULA

a(n) = (n-r)^2+(4*r^3+6*r^2+2*r)/3 where r = floor((sqrt(1+8*n)-1)/4). - Simplified by Gerald Hillier, Apr 14 2015

EXAMPLE

First three terms of A128201 are 1, 3, 4, hence a(3) = 1+3+4 = 8.

MAPLE

a:= n-> (r-> (n-r)^2+(4*r^3+6*r^2+2*r)/3)(floor((sqrt(1+8*n)-1)/4)):

seq(a(n), n=1..51); # Alois P. Heinz, Nov 11 2025

PROG

(Magma) [(n-r)^2+(4*r^3+6*r^2+2*r)/3 where r is Floor((Sqrt(1+8*n)-1)/4): n in [1..51]];

(PARI) {for(n=1, 51, r=floor((sqrt(1+8*n)-1)/4); print1((n-r)^2+(4*r^3+6*r^2+2*r)/3, ", "))}

(Python)

from math import isqrt

def A157130(n): return (r:=isqrt((n<<3)+1)-1>>2)*(r+2)*((r<<2)|1)//3+n*(n-(r<<1)) # Chai Wah Wu, Nov 11 2025

CROSSREFS

Cf. A128201 (union of A000290 and A005408), A000290 (squares), A005408 (odd numbers).

Sequence in context: A387355 A071994 A023661 * A172050 A376854 A312219

Adjacent sequences: A157127 A157128 A157129 * A157131 A157132 A157133

KEYWORD

nonn

AUTHOR

Gerald Hillier, Feb 23 2009

EXTENSIONS

Edited and extended by Klaus Brockhaus, Feb 24 2009

STATUS

approved