A373028 a(n) is the least positive integer k such that 3*n^2 + 2*n + k^2 is a square.

2, 3, 4, 5, 6, 1, 8, 9, 10, 2, 12, 13, 14, 3, 16, 10, 18, 4, 20, 21, 2, 5, 24, 25, 10, 6, 28, 29, 30, 7, 32, 33, 34, 8, 36, 3, 38, 9, 20, 41, 42, 10, 44, 45, 46, 11, 48, 49, 50, 12, 4, 53, 54, 13, 56, 33, 58, 14, 60, 23, 62, 15, 64, 65, 66, 5, 68, 69, 70, 17, 72, 73, 74, 18, 76, 49, 78, 19, 80, 33, 6

Offset: 1

Author

Claude H. R. Dequatre, May 20 2024

Keywords

• nonn
• easy

Comments

Many runs of consecutive integers appear in the sequence. The first ones are [2,3,4,5,6], [8,9,10], [12,13,14], [20,21], [24,25], [28,29,30], ...

The scatterplot shows an interesting structure where distinct sets of terms can be seen on three straight lines. The greater the slope of the line, the higher the density of terms. The remaining terms are more randomly distributed between the three lines.

More detailed observations:

For the line of lowest slope, the term and term index parity alternates from one term to the next and if the term index is even, the term is odd and reciprocally. Terms of the same set and their indices are in arithmetic progression of respective common difference of 1 and 15.

For the line of medium slope, all term indices are even and the term parity alternates from one term to the next. Terms of the same set and their indices are in arithmetic progression of respective common difference 1 and 4.

For the line of greatest slope, if the term index is even, then the term is odd and reciprocally. All terms are equal to their index + 1.

There are no fixed points a(n) = n since 3*n^2+2*n + n^2 = (2*n)^2 + 2*n falls between (2*n)^2 and (2*n+1)^2, so not square.

Examples

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

Crossrefs

Cf. A129861, A373016, A373026.

Programs

PARI

a(n) = my(m=3*n^2+2*n, k=1); while (!issquare(m+k^2), k++); k; \\ Michel Marcus, May 20 2024