A379697 For n >= 3, a(n) is the least k >= 0 such that (k + 1)*(2*n + k) / 2 is a triangular number (A000217).

0, 2, 5, 0, 1, 20, 4, 0, 2, 6, 8, 2, 0, 10, 119, 3, 4, 20, 0, 1, 5, 14, 2, 5, 1, 0, 32, 6, 7, 464, 20, 2, 8, 0, 24, 8, 2, 4, 65, 9, 10, 47, 0, 3, 11, 30, 17, 2, 3, 1, 59, 12, 0, 2, 21, 4, 14, 38, 40, 14, 4, 42, 101, 0, 16, 74, 2, 5, 17, 46, 48, 17, 5, 1, 11, 0, 19, 125, 10, 6, 20, 54, 1, 20, 6, 44, 272, 21, 0

Offset: 3

Author

Ctibor O. Zizka, Dec 30 2024

Keywords

• nonn
• look

Comments

Also for n >= 3, a(n) is the least k >= 0 such that the Sum_{i = 0..k} (n + i) is a triangular number (A000217). For n = 0, 1 the Sum is a triangular number for all n. For n = 2, there is no solution.

Examples

			n = 4: the least k >= 0 such that (k + 1)*(8 + k)/2 is a triangular number is k = 2, thus a(4) = 2.
n = 6: the least k >= 0 such that (k + 1)*(12 + k)/2 is a triangular number is k = 0, thus a(6) = 0.
		

Links

• Robert Israel, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A000217, A074377, A379676.

Programs

Maple

f:= proc(n) local t, alpha, beta;
      t:= n^2-n;
      alpha:= convert(select(type, numtheory:-divisors(t),odd),list);
      beta:= map(s -> (s+t/s - 1)/2 - n, alpha);
      min(select(`>=`,beta,0))
end proc:
map(f, [$3..100]); # Robert Israel, Jan 30 2025

Mathematica

a[n_] := Module[{k = 0}, While[! IntegerQ[Sqrt[4*(k + 1)*(2*n + k) + 1]], k++]; k]; Array[a, 100, 3] (* Amiram Eldar, Dec 30 2024 *)

PARI

a(n) = my(k=0); while (!ispolygonal((k + 1)*(2*n + k)/2, 3), k++); k; \\ Michel Marcus, Dec 30 2024

Formula

a(n) = 0 for n from A000217.

a(n) = 1 for n from A074377 AND n is not a triangular number.