A320259 Terms that are on the y-axis of the square spiral built with 2*k, 2*k+1, 2*k+1 for k >= 0.

0, 2, 5, 9, 15, 22, 30, 40, 51, 63, 77, 92, 108, 126, 145, 165, 187, 210, 234, 260, 287, 315, 345, 376, 408, 442, 477, 513, 551, 590, 630, 672, 715, 759, 805, 852, 900, 950, 1001, 1053, 1107, 1162, 1218, 1276, 1335

Offset: 0

Author

Paul Curtz, Oct 08 2018

Keywords

• nonn

Comments

a(n) mod 9 is of period 27.

The 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

Examples

			G.f. = 2*x + 5*x^2 + 9*x^3 + 15*x^4 + 22*x^5 + 30*x^6 + ... - _Michael Somos_, Nov 13 2018
		

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, A004396, A004523, A004767, A004772 (first differences), A211480, A002264, A143978.

Programs

GAP

a:=[0,2,5,9,15];; for n in [6..50] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-2*a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Oct 08 2018

Maple

seq(coeff(series(x^2*(2+x+x^2)/((1-x)^3*(1+x+x^2)),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Oct 08 2018

Mathematica

LinearRecurrence[{2,-1,1,-2,1}, {0, 2, 5, 9, 15}, 50] (* or *)
CoefficientList[Series[x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 09 2018 *)
a[ n_] := Quotient[(n + 1) (2 n + 1), 3]; (* Michael Somos, Nov 13 2018 *)

PARI

concat(0, Vec(x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Oct 08 2018

PARI

{a(n) = (n + 1) * (2*n + 1) \ 3}; /* Michael Somos, Nov 13 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)=2, a(2)=5, a(3)=9, a(4)=15.

a(n) = a(n-1) + A004772(n+1), a(0)=0, n>0.

a(n+15) = a(n-15) + 10*A004767(n).

a(-n-1) = ({0} U A000969(n)) = 0, 1, 3, 7, ... = b(n), the full x-axis terms.

a(-n-1) + a(n) = 0, 3, 8, 16, ... = A211480(n+1).

a(n) = b(n) + A004523(n+1).

G.f.: x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Oct 08 2018

a(n) = A143978(n) + A002264(n+2).

a(n) = A000969(-2-n) for all n in Z. - Michael Somos, Nov 13 2018