A320281 Terms that are on the positive x-axis of the square spiral built with 2*k, 2*k+1, 2*k+1 for k >= 0.

0, 1, 7, 18, 35, 57, 84, 117, 155, 198, 247, 301, 360, 425, 495, 570, 651, 737, 828, 925, 1027, 1134, 1247, 1365, 1488, 1617, 1751, 1890, 2035, 2185, 2340, 2501, 2667, 2838, 3015, 3197, 3384, 3577, 3775, 3978

Offset: 0

Author

Paul Curtz, Oct 09 2018

Keywords

• nonn
• easy

Comments

Resulting spiral:

28--29--29--30--31--31--32

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27 13--14--15--15--16--17

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27 13 4---5---5---6 17

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26 12 3 0---1 7 18

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25 11 3---2---1 7 19

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25 11--10---9---9---8 19

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24--23--23--22--21--21--20

.

a(n) mod 9 is of period 27. a(n) mod 10 is of period 30.

The NE diagonal starting at 1 is A301696. - Klaus Purath, May 15 2021

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A000969.

Programs

Mathematica

LinearRecurrence[{2,-1,1,-2,1},{0,1,7,18,35},100] (* Paolo Xausa, Nov 13 2023 *)

PARI

concat(0, Vec(x*(1 + 5*x + 5*x^2 + 5*x^3) / ((1 - x)^3*(1 + x + x^2)) + O(x^50))) \\ Colin Barker, Oct 09 2018

Formula

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), a(0)=0, a(1)=1, a(2)=7, a(3)=18, a(4)=35.

a(n+2) - 2*a(n-1) + a(n) = period 3: repeat [5, 5, 6].

a(-n) = 0, 5, 15, 30, 51, 77, 108, 145, ... is the sequence of the terms on the positive y-axis.

G.f.: x*(1 + 5*x + 5*x^2 + 5*x^3) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Oct 09 2018