A379676 For n >= 0, a(n) is the least k >= 2 such that (n + 1)*(2*k + n) / 2 is a triangular number (A000217).

3, 7, 4, 15, 7, 5, 10, 31, 13, 6, 16, 12, 19, 10, 7, 63, 25, 11, 28, 22, 8, 15, 34, 21, 37, 16, 40, 9, 43, 20, 46, 127, 14, 21, 18, 10, 55, 25, 15, 19, 61, 26, 64, 45, 11, 30, 70, 44, 73, 31, 21, 55, 79, 35, 12, 70, 22, 36, 88, 18, 91, 40, 34, 255, 31, 13, 100, 19, 28, 24, 106, 92, 109, 46, 29, 78, 25, 14, 118, 91, 121, 51, 124, 63, 42, 55, 35, 39, 133, 43, 15

Offset

0,1

Comments

Also for n >= 0, a(n) is the least k >= 2 such that the Sum_{i = 0..n} (k + i) is a triangular number (A000217). For k = 0, 1 the Sum is a triangular number for all n. The sequences A076114 and A076116 are for square sum and cube sum.

Links

Table of n, a(n) for n=0..90.

Formula

For i >= 0, a(2^i - 1) = 2^(i + 2) - 1, max. values of a(n).

For i >= 0, a(i*(i + 3)/2) = i + 3, min. values of a(n).

For i >= 1, i is not from A083390, a(2*i) = (3*i + 1).

Example

n = 4: the least k >= 2 such that (4 + 1)*(2*k + 4)/2 = 5*k + 10 is a triangular number is k = 7, thus a(4) = 7.
n = 5: the least k >= 2 such that (5 + 1)*(2*k + 5)/2 = 6*k + 15 is a triangular number is k = 5, thus a(5) = 5.

Mathematica

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

Prog

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

Crossrefs

Cf. A000217, A076114, A076116, A083390.

Sequence in context: A112305 A231396 A231463 * A218616 A323173 A324184

Adjacent sequences: A379673 A379674 A379675 * A379677 A379678 A379679

Keyword

nonn

Author

Ctibor O. Zizka, Dec 29 2024

Status

approved