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A224419
Numbers n such that triangular(n) + triangular(2*n) is a square.
4
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
1,3
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
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
MATHEMATICA
LinearRecurrence[{1, 0, 1442, -1442, 0, -1, 1}, {0, 1, 25, 216, 1849, 36481, 311904}, 30] (* Harvey P. Dale, Jan 23 2015 *)
PROG
(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)
CROSSREFS
Cf. A220186 (numbers n such that triangular(2*n) - triangular(n) is a square).
Sequence in context: A324794 A348201 A264493 * A372950 A297864 A065939
KEYWORD
nonn
AUTHOR
Alex Ratushnyak, Apr 18 2013
EXTENSIONS
Terms a(11) onward from Max Alekseyev, Sep 04 2013
STATUS
approved