A014133 Sum of a square and a triangular number.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 35, 36, 37, 39, 40, 42, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 64, 65, 66, 67, 70, 71, 72, 74, 75, 77, 78, 79, 80, 81, 82, 84, 85, 87, 91, 92

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1001 terms from Harvey P. Dale)

Formula

{k: A101428(k) > 0}. - R. J. Mathar, Apr 28 2020

k is a term iff each prime factor congruent to 5 or 7 (mod 8) of 8*k+1 comes in even degree. - Max Alekseyev, Mar 30 2026

Maple

isA014133 := proc(n)
    local c, t ;
    for c from 0 do
        t := c*(c+1)/2 ;
        if t > n then
            return false;
        end if;
        if issqr(n-t) then
            return true;
        end if;
    end do:
end proc:
for n from 0 to 100 do
    if isA014133(n) then
        printf("%d, ", n) ;
    end if;
end do: # R. J. Mathar, Oct 11 2014
# Alternative:
q:= n-> ormap(x-> issqr(8*x+1), [n-i^2$i=0..isqrt(n)]):
select(q, [$0..100])[];  # Alois P. Heinz, Jan 14 2026

Mathematica

With[{nn=20}, Select[Union[Flatten[Outer[Plus, Range[0, nn]^2, Accumulate[ Range[ 0, nn]]]]], #<=(nn(nn+1))/2&]] (* Harvey P. Dale, Dec 29 2019 *)
fQ[n_] := Block[{s = t = 0, lmt = 1 + Floor[(Sqrt[ 8n +1] -1)/2]}, While[t < lmt, If[ Mod[ Sqrt[n -t (t +1)/2], 1] == 0, s = 1; t = lmt]; t++]; s == 1]; Select[Range[0, 92], fQ] (* Robert G. Wilson v, Jan 14 2026 *)

Crossrefs

Cf. A000217, A000290, A101428, A014134 (complement).

Sequence in context: A214879 A187226 A026470 * A070115 A073571 A328242

Adjacent sequences: A014130 A014131 A014132 * A014134 A014135 A014136

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Offset changed to 1 by Alois P. Heinz, Jan 14 2026

Status

approved