A218273 Square triangular numbers that can be expressed as sums of a positive square number and a positive triangular number. Intersection of A182427 and A214937.

1225, 1413721, 48024900, 1631432881, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600, 85170343853180456676, 2893284510173841030625, 98286503002057414584576, 3338847817559778254844961, 113422539294030403250144100

Offset: 1

Author

Ivan N. Ianakiev, Oct 25 2012

Keywords

• nonn
• more

Comments

Theorem (I. N. Ianakiev): There are infinitely many such numbers. Proof: Any A001110(2n+1), for n>0, is such a number as A001110(2n+1) = (2a+1)^2+(4a^2+4a)(4a^2+4a+1)(1/2), where a = (A002315(n)-1)(1/2). Note: other numbers, not of the form A001110(2n+1), e.g. A001110(6), are also in the sequence (see the example below).

Every term is divisible by its digital root (A010888). - Ivan N. Ianakiev, Oct 17 2013

Examples

			a(3) = A001110(6) = 48024900 = 6918^2 + [576*577*(1/2)].
		

Crossrefs

Cf. A000217, A000290, A001110, A182427, A214937.

Extensions

a(8)-a(13) from Donovan Johnson, Nov 02 2012