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

4, 16, 25, 49, 64, 81, 100, 121, 169, 196, 256, 289, 361, 400, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2401, 2500, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481

Offset

1,1

Comments

Theorem (I. N. Ianakiev): There are infinitely many such numbers. Proof: There are infinitely many square triangular numbers (A001110) and every (2t+1)-th of them is odd because A001110(0)=0, A001110(1)=1 and A001110(n)=34*a(n-1)-a(n-2)+2, for n>=2. Any sqrt(A001110(2t+1)) is odd (i. e. is in A005408) and can be written as p^2-q^2 because A005408(n)=A000290(n+1)-A000290(n). The unique values of p and q (p>q>0) for each sqrt(A001110(2t+1)) generate (when t>0) a unique Pythagorean triple with a unique hypotenuse (a=p^2-q^2, b=2pq, c=p^2+q^2). Therefore, there are infinitely many such hypotenuses squared.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

4 and 49 are in the sequence because 2^2=1^2+2*3/2 and 7^2=2^2+9*10/2

Crossrefs

Cf. A000217, A000290, A182427.

Sequence in context: A343726 A202303 A350835 * A235001 A087055 A135556

Adjacent sequences: A214934 A214935 A214936 * A214938 A214939 A214940

Keyword

nonn

Author

Ivan N. Ianakiev, Jul 30 2012

Status

approved