

A214937


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


3



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
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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(n1)a(n2)+2, for n>=2. Any sqrt(A001110(2t+1)) is odd (i. e. is in A005408) and can be written as p^2q^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^2q^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: A269157 A343726 A202303 * A235001 A087055 A135556
Adjacent sequences: A214934 A214935 A214936 * A214938 A214939 A214940


KEYWORD

nonn


AUTHOR

Ivan N. Ianakiev, Jul 30 2012


STATUS

approved



