A385254 Distinct terms in A386369.

0, 1, 2, 6, 18, 26, 70, 82, 222, 642, 1902, 5682, 17022, 51042, 59778, 59958, 77774, 107258, 268870, 285010, 361086, 930666, 1084314, 1498134, 3813282, 5053994, 13240150, 14183598, 15487758, 15579122, 18418666, 18622506, 23809998, 58728474, 64572254, 65013058

Offset

1,3

Comments

a(1) = 0. a(n) is the smallest positive integer > a(n-1) such that Sum_{m = 1..n-1} (a(m+1)-a(m))*a(m) is a perfect square.

Links

David A. Corneth, Table of n, a(n) for n = 1..1018

David A. Corneth, PARI program

Example

The first 6 terms of A386369 are 0, 1, 2, 2, 2, 2 which has partial sum 9. We have A386369(7) = 6. To find a(4) we look for the next term in A386369 that is larger than 6 i.e. solve 6*(k-6) + 9 = s^2 for some k. Rewrite gives 6*(k-6) = s^2 - 9 = (s-3)(s + 3). So we have 4 cases:
   1 | s - 3, 6 | s + 3
   2 | s - 3, 3 | s + 3
   3 | s - 3, 2 | s + 3
   6 | s - 3, 1 | s + 3
Solving for smallest t > 6 gives s = 9. So 6*(k-6) = 9^2 - 9 = 72 and so k = 18.

Mathematica

Module[{s = 0, a = 0}, Table[If[IntegerQ[Sqrt[s += a]], a = k-1, Nothing], {k, 10^5}]]

Prog

(PARI) \\ See Corneth link

Crossrefs

Cf. A386369.

Sequence in context: A032649 A237989 A185382 * A257065 A066286 A324541

Adjacent sequences: A385251 A385252 A385253 * A385255 A385256 A385257

Keyword

nonn

Author

David A. Corneth and Paolo Xausa, Jul 29 2025

Extensions

More terms from Michael De Vlieger, Jul 29 2025

Status

approved