A224419 Numbers n such that triangular(n) + triangular(2*n) is a square.

0, 1, 25, 216, 1849, 36481, 311904, 2666689, 52606009, 449765784, 3845364121, 75857828929, 648561949056, 5545012396225, 109386936710041, 935225880773400, 7995904029992761, 157735886878050625, 1348595071513294176, 11530088066237165569, 227455039491212291641, 1944673157896289428824, 16626378995609962758169, 327990009210441246496129

Offset: 1

Author

Alex Ratushnyak, Apr 18 2013

Keywords

• nonn

Comments

8 of the first 10 terms are of the form x^y. The two exceptions are a(7) = 311904 = 2^5 * 3^3 * 19^2 and a(10) = 449765784 = 2^3 * 3^5 * 13^2 * 37^2.

The corresponding squares are given by A075873(2*n-1)^2. E.g., triangular(a(10)) + triangular(2*a(10)) = 711142146^2 = A075873(19)^2.

Locations of squares in A147875, equivalent to solving the Diophantine equation n*(5*n+3)=2*s^2. - R. J. Mathar, Apr 19 2013

Links

• Max Alekseyev, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1442,-1442,0,-1,1).

Crossrefs

Cf. A000217, A075873.

Cf. A220186 (numbers n such that triangular(2*n) - triangular(n) is a square).

Programs

Mathematica

LinearRecurrence[{1,0,1442,-1442,0,-1,1},{0,1,25,216,1849,36481,311904},30]  (* Harvey P. Dale, Jan 23 2015 *)

Python

import math
for i in range(1<<30):
        s = i*(i+1)/2 + i*(2*i+1)
        t = int(math.sqrt(s))
        if s == t*t:  print(i)

Formula

a(n) = (A228209(2*n-1) - 3) / 10. - Max Alekseyev, Sep 04 2013

G.f.: x^2*(x+1)*(x^4 + 23*x^3 + 168*x^2 + 23*x + 1) / (x^6 - 1442*x^3 + 1) / (1-x). - Max Alekseyev, Sep 04 2013

Extensions

Terms a(11) onward from Max Alekseyev, Sep 04 2013