A214937 -id:A214937 - cp's OEIS frontend

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

Ivan N. Ianakiev, Oct 25 2012

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

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

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

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

A215069 Natural numbers that when squared can be expressed as sums of a positive square number and a positive triangular number.

Original entry on oeis.org

2, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85

Offset: 1

Ivan N. Ianakiev, Aug 02 2012

nonn

There are infinitely many terms (see A214937 for a proof)

			2 and 7 are in the sequence because 2^2=1^2+2*3/2 and 7^2=2^2+9*10/2

Ivan N. Ianakiev, Table of n, a(n) for n = 1..1000

Cf. A000217, A000290, A214937

a(n) = sqrt(A214937(n))

cp's OEIS Frontend

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.

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