

A182427


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


3



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
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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^21) + 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*k2 (A060544) are terms as (3*k2)*(3*k1)/2 = (2*k1)^2 + (k1)*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: A051990 A246447 A254362 * A171956 A271743 A060535
Adjacent sequences: A182424 A182425 A182426 * A182428 A182429 A182430


KEYWORD

nonn


AUTHOR

Ivan N. Ianakiev, Apr 28 2012


STATUS

approved



