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A182427 Triangular numbers that can be represented as a sum of a nonzero square number and a nonzero triangular number. 3

%I #31 Dec 15 2015 06:42:35

%S 10,15,28,45,55,91,136,190,210,231,253,325,378,406,435,496,561,595,

%T 666,703,741,820,861,903,946,990,1081,1128,1176,1225,1378,1431,1540,

%U 1596,1711,1770,1830,1891,2080,2145,2211,2278,2346,2415,2485,2556,2701,2926,3160,3321

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

%C 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

%C 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

%H Alois P. Heinz, <a href="/A182427/b182427.txt">Table of n, a(n) for n = 1..1000</a>

%e 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.

%o (PARI) isok(t) = {for (k=1, sqrtint(t), my(tt = t - k^2); if ((tt) && ispolygonal(tt, 3), return (1)););}

%o lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); if (isok(t), print1(t, ", ")););} \\ _Michel Marcus_, Nov 25 2015

%Y Cf. A000217, A000290, A037270.

%K nonn

%O 1,1

%A _Ivan N. Ianakiev_, Apr 28 2012

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