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A214937 Square numbers that can be expressed as sums of a positive square number and a positive triangular number. 3

%I

%S 4,16,25,49,64,81,100,121,169,196,256,289,361,400,484,529,576,625,676,

%T 729,784,841,900,961,1024,1156,1225,1369,1444,1521,1600,1681,1764,

%U 1849,1936,2025,2116,2209,2401,2500,2704,2809,2916,3025,3136,3249,3364,3481

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

%C 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.

%H Charles R Greathouse IV, <a href="/A214937/b214937.txt">Table of n, a(n) for n = 1..10000</a>

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

%Y Cf. A000217, A000290, A182427.

%K nonn

%O 1,1

%A _Ivan N. Ianakiev_, Jul 30 2012

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Last modified August 8 23:02 EDT 2020. Contains 336300 sequences. (Running on oeis4.)