A230364 Triangular numbers representable as b! + c^2.

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

Links

Table of n, a(n) for n=1..46.

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(1<<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(t, end=', ')
            break

Crossrefs

Cf. A000217, A000142, A000290.

Sequence in context: A365696 A346734 A366132 * A214282 A130200 A202269

Adjacent sequences: A230361 A230362 A230363 * A230365 A230366 A230367

Keyword

nonn

Author

Alex Ratushnyak, Oct 17 2013

Status

approved