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A230363 Factorials representable as b^2 + triangular(c).

%I A230363 #21 Dec 28 2024 19:34:05
%S A230363 1,1,2,6,24,120,362880,3628800,39916800,479001600,6227020800,
%T A230363 1307674368000,121645100408832000
%N A230363 Factorials representable as b^2 + triangular(c).
%C A230363 Numbers n such that n! is representable as a sum of a square and a triangular number: 0, 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 15, 19, ... .
%C A230363 1! = 1/2*1*(1+1), 3! = 1/2*3*(3+1) and 5! = 1/2*15*(15+1)/2 are triangular terms of the sequence. Next such term, if it exists is greater than 10000!. - _Farideh Firoozbakht_, Oct 18 2013
%F A230363 A014133 INTERSECT A000142. - _R. J. Mathar_, Oct 11 2014
%e A230363 13! = 66708^2+1/2*59616(59616+1) = 78693^2+1/2*8298(8298+1), so 13! = 6227020800 is in the sequence. What is the next term of the sequence which has more than one representation of the form b^2 + triangular(c)? - _Farideh Firoozbakht_, Oct 18 2013
%o A230363 (Python)
%o A230363 import math
%o A230363 f=1
%o A230363 for n in range(1, 1000000):
%o A230363     f *= n
%o A230363     t = b = 0
%o A230363     while t<=f:
%o A230363         x = f-t
%o A230363         a = int(math.sqrt(x))
%o A230363         if a*a==x:
%o A230363             print(f, end=", ")
%o A230363             break
%o A230363         b += 1
%o A230363         t = b*(b+1)//2
%Y A230363 Cf. A000142, A000290, A000217, A014133.
%K A230363 nonn,more
%O A230363 1,3
%A A230363 _Alex Ratushnyak_, Oct 17 2013
%E A230363 Initial 1 added by _Farideh Firoozbakht_, Oct 18 2013