A373016 a(n) is the least positive integer k such that 3*n^2 + 2*n + k is a square.

4, 9, 3, 8, 15, 1, 8, 17, 28, 4, 15, 28, 43, 9, 24, 41, 60, 16, 35, 56, 4, 25, 48, 73, 11, 36, 63, 92, 20, 49, 80, 113, 31, 64, 99, 9, 44, 81, 120, 20, 59, 100, 143, 33, 76, 121, 3, 48, 95, 144, 16, 65, 116, 169, 31, 84, 139, 196, 48, 105, 164, 8, 67, 128, 191, 25, 88, 153, 220, 44, 111, 180

Offset: 1

Author

Claude H. R. Dequatre, May 20 2024

Keywords

• nonn
• easy

Comments

The scatterplot shows an interesting crosshatch structure where all terms are at the intersection of ascending and descending hatches.

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

For terms on ascending hatches, the parity of the term indices does not change on a given hatch but alternates from one hatch to the next and on the same hatch, the parity of two consecutive terms alternates.

For terms on descending hatches, the parity of the indices of two consecutive terms alternates on the same hatch and that of terms does not change on the same hatch but alternates from one hatch to the next.

All squares exclusively are in ascending order on the same ascending hatch at n = 6, 10, 14, 18, 22, ... but some squares can be also found at the intersection of other hatches.

The first differences of the indices of the terms located on ascending and descending hatches are respectively equal to 4 and 3. For terms that are on the ascending and descending hatches, the differences of order 2 quickly become constant and equal to 2 and 4, respectively.

The fixed points begin 3, 48, 675, 9408, etc. They are all divisible by 3 and their parity seems to alternate. It appears that they are the positive terms of A007654.

Examples

			a(1) = 4 because 3*1^2 + 2*1 = 5 and 5 + 1, 5 + 2, 5 + 3 are not squares, but 5 + 4 is. So, 4 is a term.
a(2) = 9 because 3*2^2 + 2*2 = 16 and 16 + 1, 16 + 2, 16 + 3, 16 + 4, 16 + 5, 16 + 6, 16 + 7, 16 + 8 are not squares, but 16 + 9 is. So, 9 is a term.
		

Crossrefs

Cf. A000005, A000290, A005408.

Cf. A045944, A080883.

Cf. A373026, A373028.

Sequences with similar scatterplot and pin plot graphs: A141130, A141131, A141134, A141135.

Programs

PARI

a(n)=my(t=3*n^2+2*n); (sqrtint(t)+1)^2-t \\ Charles R Greathouse IV, May 21 2024

Formula

a(n) is the smallest square greater than 3*n^2 + 2*n, minus 3*n^2 + 2*n. - Charles R Greathouse IV, May 21 2024

1 <= a(n) <= floor(sqrt(12)*n) + 3. I believe both bounds are tight infinitely often. - Charles R Greathouse IV, May 21 2024

a(n) = A080883(A045944(n)). - Michel Marcus, May 22 2024