A373026 a(n) is the least positive integer k such that 3*n^2 + 2*n - k is a square.

1, 7, 8, 7, 4, 20, 17, 12, 5, 31, 24, 15, 4, 40, 29, 16, 1, 47, 32, 15, 69, 52, 33, 12, 76, 55, 32, 7, 81, 56, 29, 111, 84, 55, 24, 116, 85, 52, 17, 119, 84, 47, 8, 120, 81, 40, 160, 119, 76, 31, 161, 116, 69, 20, 160, 111, 60, 7, 157, 104, 49, 207, 152, 95, 36, 204, 145, 84, 21, 199, 136, 71

Offset

1,2

Comments

The scatterplot shows an interesting structure where terms are on descending hatches.

Terms on each hatch are quite well fitted by a polynomial of degree 2.

The parity of the term indices alternates from one hatch to the next and that of two consecutive terms alternates on the same hatch.

For terms on a given hatch, the differences of order 2 quickly become constant and equal to 2.

The fixed points begin 1, 16, 225, 3136, etc. They appear to be all squares and to come from A098301.

Links

Table of n, a(n) for n=1..72.

Example

a(1) = 1 because 3*1^2 + 2*1 = 5 and 5-1 is a square. So, 1 is a term.
a(2) = 7 because 3*2^2 + 2*2 = 16 and 16-1, 16-2, 16-3, 16-4, 16-5, 16-6 are not squares, but 16-7 is. So, 7 is a term.

Prog

(PARI) a(n) = my(m=3*n^2+2*n-1); m+1-sqrtint(m)^2; \\ Michel Marcus, May 20 2024

Crossrefs

Cf. A000005, A000290, A005408.

Cf. A373016, A373028.

Sequence in context: A220610 A021931 A100264 * A272877 A255685 A154192

Adjacent sequences: A373023 A373024 A373025 * A373027 A373028 A373029

Keyword

nonn,easy

Author

Claude H. R. Dequatre, May 20 2024

Status

approved