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A230364
Triangular numbers representable as b! + c^2.
0
1, 3, 6, 10, 15, 28, 55, 66, 105, 120, 136, 171, 231, 325, 406, 465, 561, 820, 1081, 1770, 2016, 2145, 2211, 3160, 3321, 5778, 7750, 11026, 13041, 13695, 15400, 17020, 23220, 34716, 41616, 55945, 60031, 70876, 75078, 100576, 106953, 126756, 196251, 260281, 263175, 374545
OFFSET
1,2
MATHEMATICA
nf = 9; tb = Table[b!, {b, nf}]; nn = Ceiling[Sqrt[tb[[-1]]]]; ts = Range[0, nn]^2; tri = Table[n (n + 1)/2, {n, (Sqrt[1 + 8 nn^2] - 1)/2}]; u = Union[Select[Flatten[Outer[Plus, tb, ts]], # <= nn^2 &]]; Intersection[tri, u] (* T. D. Noe, Oct 18 2013 *)
PROG
(Python)
import math
factorials = [1] * 1024
f = 1
for n in range(2, 1025):
f *= n
factorials[n-1] = f
for n in range(1L<<30):
t = n*(n+1)/2
for a in factorials:
r = t - a
if r<0: break
b = int(math.sqrt(r))
if b*b==r:
print str(t)+', ',
break
CROSSREFS
KEYWORD
nonn
AUTHOR
Alex Ratushnyak, Oct 17 2013
STATUS
approved