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

Author

Claude H. R. Dequatre, May 20 2024

Keywords

• nonn
• easy

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.

Examples

			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.
		

Crossrefs

Cf. A000005, A000290, A005408.

Cf. A373016, A373028.

Programs

PARI

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