A182427 Triangular numbers that can be represented as a sum of a nonzero square number and a nonzero triangular number.

10, 15, 28, 45, 55, 91, 136, 190, 210, 231, 253, 325, 378, 406, 435, 496, 561, 595, 666, 703, 741, 820, 861, 903, 946, 990, 1081, 1128, 1176, 1225, 1378, 1431, 1540, 1596, 1711, 1770, 1830, 1891, 2080, 2145, 2211, 2278, 2346, 2415, 2485, 2556, 2701, 2926, 3160, 3321

Offset

1,1

Comments

Theorem (by Ivan N. Ianakiev): There are infinitely many such numbers. Proof: Any triangular number of the form A000217(n^2) for n>1 is such a number, as A000217(n^2) = A000217(n^2-1) + A000290(n), for n>=1. Observation: Other numbers not of the form A000217(n^2), for example 15 and 28, are also in A182427. - Ivan N. Ianakiev, May 30 2012

For any integer k>1, all triangular numbers with indices of the form 3*k-2 (A060544) are terms as (3*k-2)*(3*k-1)/2 = (2*k-1)^2 + (k-1)*k/2. - Ivan N. Ianakiev, Nov 25 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Example

10, 15, 28 are in the sequence because 10 = 2^2 + 3*4/2 = 3^2 + 1*2/2, 15 = 3^2 + 3*4/2, 28 = 5^2 + 2*3/2.

Prog

(PARI) isok(t) = {for (k=1, sqrtint(t), my(tt = t - k^2); if ((tt) && ispolygonal(tt, 3), return (1)); ); }
lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); if (isok(t), print1(t, ", ")); ); } \\ Michel Marcus, Nov 25 2015

Crossrefs

Cf. A000217, A000290, A037270.

Sequence in context: A246447 A254362 A373488 * A171956 A271743 A060535

Adjacent sequences: A182424 A182425 A182426 * A182428 A182429 A182430

Keyword

nonn

Author

Ivan N. Ianakiev, Apr 28 2012

Status

approved