A357281 - OEIS

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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

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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