A357281 The numbers of a square spiral with 1 in the center, lying at integer points of the right branch of the parabola y=n^2.

1, 9, 79, 355, 1077, 2581, 5299, 9759, 16585, 26497, 40311, 58939, 83389, 114765, 154267, 203191, 262929, 334969, 420895, 522387, 641221, 779269, 938499, 1120975, 1328857, 1564401, 1829959, 2127979, 2461005, 2831677, 3242731, 3696999, 4197409

Offset: 0

Author

Nicolay Avilov, Sep 22 2022

Keywords

• nonn
• easy

Comments

On a square spiral with 1 in the center is a parabola y=n^2. The coordinate system is defined by the On and Oy axes, which intersect at the center of the spiral. The points of the parabola with integer coordinates located on the right branch of the parabola y=n^2 determine the terms of the sequence. In the attached figure, the terms are highlighted in red cells.

Examples

			a(0) = 1;
a(4) = 4*4^4 + 3*4^2 + 4 + 1 = 1024 + 48 + 5 = 1077.
		

Links

• Nicolay Avilov, Figure illustrating the terms a(0) - a(3)
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A033951 (x axes), A054552 (y axes).

Cf. A000583, A056108.

Programs

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{1,9,79,355,1077},40] (* Harvey P. Dale, Oct 15 2023 *)

Formula

a(n) = 4*n^4 + 3*n^2 + n + 1.

a(n) = 4*A000583(n) + A056108(n).

G.f.: (1 + 4*x + 44*x^2 + 40*x^3 + 7*x^4)/(1 - x)^5. - Stefano Spezia, Sep 24 2022