A264118 Numbers that can be represented as S1+S2, as T1+T2, and as S3+T3, where S1, S2, S3 are positive squares, and T1, T2, T3 are positive triangular numbers.

2, 25, 29, 37, 61, 65, 72, 100, 101, 106, 130, 136, 137, 157, 169, 200, 202, 205, 226, 232, 234, 241, 274, 277, 289, 340, 353, 389, 400, 409, 416, 421, 436, 442, 445, 466, 490, 514, 520, 522, 562, 577, 596, 610, 625, 637, 640, 661, 666, 676, 697, 724, 757, 765, 785, 794, 820

Offset

1,1

Comments

Intersection of A000404, A051533, A133254.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(1) = 2 = 1+1.
25 is a term because 25 = 16 + 9 = 10 + 15 = 4 + 21, and 16, 9, 4 are squares, 10, 15, 21 are triangular numbers.

Maple

N:= 1000: # for terms <= N
SS:= {seq(seq(i^2+j^2, i=1..min(j, floor(sqrt(N-j^2)))), j=1..floor(sqrt(N)))}:
TS:= {seq(seq(i*(i+1)/2 + j^2, j = 1..floor(sqrt(N-i*(i+1)/2))), i=1..floor((sqrt(1+8*N)-1)/2))}:
TT:= {seq(seq(i*(i+1)/2 + j*(j+1)/2, i = 1 .. min(j, floor((sqrt(1+8*(N-j*(j+1)/2))-1)/2))), j=1..floor((sqrt(1+8*N)-1)/2))}:
sort(convert(SS intersect TS intersect TT, list)); # Robert Israel, Mar 19 2026

Crossrefs

Cf. A000217, A000290, A000404, A051533, A133254, A264101.

Sequence in context: A041723 A067443 A163497 * A054292 A248546 A369670

Adjacent sequences: A264115 A264116 A264117 * A264119 A264120 A264121

Keyword

nonn

Author

Alex Ratushnyak, Nov 03 2015

Status

approved