A330451 a(n) = a(n-3) + 20*n - 30 for n > 2, with a(0)=0, a(1)=3, a(2)=13.

0, 3, 13, 30, 53, 83, 120, 163, 213, 270, 333, 403, 480, 563, 653, 750, 853, 963, 1080, 1203, 1333, 1470, 1613, 1763, 1920, 2083, 2253, 2430, 2613, 2803, 3000, 3203, 3413, 3630, 3853, 4083, 4320, 4563, 4813, 5070

Offset: 0

Author

Paul Curtz, Mar 01 2020

Keywords

• nonn
• easy

Comments

Main N-S vertical in the pentagonal spiral for A002264:

16

16 10 10

16 9 5 5 10

15 9 4 1 2 5 11

15 9 4 1 0 0 2 6 11

15 8 4 1 0 2 6 11

14 8 3 3 3 6 12

14 8 7 7 7 12

14 13 13 13 12

The main S-N vertical is A194275.

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. A000290, A002264, abs(A084103(n+3)), A194275, A244636.

Cf. A049347.

Programs

Mathematica

Table[2/9(-1+15n^2+Cos[2n*Pi/3]),{n,0,39}] (* Stefano Spezia, Mar 02 2020 *)

PARI

concat(0, Vec(x*(1 + x)*(3 + 4*x + 3*x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Mar 02 2020

Python

def A330451(n): return 10*n**2//3 # Chai Wah Wu, Aug 12 2025

Formula

G.f.: x*(1 + x)*(3 + 4*x + 3*x^2) / ((1 - x)^3*(1 + x + x^2)). - Colin Barker, Mar 02 2020

a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).

a(n) = (2/9)*(-1 + 15*n^2 + cos(2*n*Pi/3)). - Stefano Spezia, Mar 02 2020

a(3*n) = 30*n^2.

a(n) = floor(10*n^2/3). - Chai Wah Wu, Aug 12 2025