A329404 Interleave 2*n*(3*n-1), (2*n+1)*(6*n+1) for n >= 0.

0, 1, 4, 21, 20, 65, 48, 133, 88, 225, 140, 341, 204, 481, 280, 645, 368, 833, 468, 1045, 580, 1281, 704, 1541, 840, 1825, 988, 2133, 1148, 2465, 1320, 2821, 1504, 3201, 1700, 3605, 1908, 4033, 2128, 4485, 2360, 4961

Offset: 0

Author

Paul Curtz, Nov 13 2019

Keywords

• nonn
• easy

Comments

a(n) + a(n+3) = 21, 21, 69, 69, 153, 153, ...

Hexagonal spiral for A026741:

.

33--17--35--18

/

16 8--17---9--19

/ / \

31 15 5---3---7 10

/ / / \ \

15 7 2 0===1===4==21==>

\ \ \ / / /

29 13 3---1 9 11

\ \ / /

14 6--11---5 23

\ /

27--13--25--12

.

a(n) is the horizontal sequence from 0.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A005449, A014641, A026741, A033579, A165355.

Programs

Mathematica

LinearRecurrence[{0,3,0,-3,0,1},{0,1,4,21,20,65},100] (* Paolo Xausa, Nov 13 2023 *)

PARI

concat(0, Vec(x*(1 + 4*x + 18*x^2 + 8*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^3) + O(x^45))) \\ Colin Barker, Nov 13 2019

Formula

a(n) = n * A165355(n-1).

From Colin Barker, Nov 13 2019: (Start)

G.f.: x*(1 + 4*x + 18*x^2 + 8*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^3).

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5.

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

From Amiram Eldar, Dec 27 2024: (Start)

Sum_{n>=1} 1/a(n) = Pi/(8*sqrt(3)) + 9*log(3)/8.

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi/(8*sqrt(3)) - 3*log(3)/8. (End)