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

0,2

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.

Links

Table of n, a(n) for n=0..32.

Nicolay Avilov, Figure illustrating the terms a(0) - a(3)

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

Example

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

Mathematica

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

Crossrefs

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

Cf. A000583, A056108.

Sequence in context: A279683 A123918 A044577 * A172203 A293721 A198857

Adjacent sequences: A357278 A357279 A357280 * A357282 A357283 A357284

Keyword

nonn,easy

Author

Nicolay Avilov, Sep 22 2022

Status

approved