A322465 Numbers on the 0-9-10-line in a spiral on an equilateral triangular lattice.

0, 9, 10, 31, 32, 65, 66, 111, 112, 169, 170, 239, 240, 321, 322, 415, 416, 521, 522, 639, 640, 769, 770, 911, 912, 1065, 1066, 1231, 1232, 1409, 1410, 1599, 1600, 1801, 1802, 2015, 2016, 2241, 2242, 2479, 2480, 2729, 2730, 2991, 2992, 3265, 3266, 3551, 3552

Offset: 0

Author

Hans G. Oberlack, Dec 09 2018

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 9, 10, ... in the triangle spiral.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Hans G. Oberlack, Triangle spiral line 0-9-10
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Bisection (even part) gives A202804.

Programs

Maple

a:= n-> `if`(n::even, n*((3/2)*n+2), (n+1)*((3/2)*(n+1)+2)-1): seq(a(n), n=0..50); # Muniru A Asiru, Dec 20 2018

PARI

concat(0, Vec(x*(9 + x + 3*x^2 - x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Dec 18 2018

Formula

For even n: a(n) = n*((3/2)*n+2).

For odd n: a(n) = a(n+1)-1 = (n+1)*((3/2)*(n+1)+2)-1.

From Colin Barker, Dec 18 2018: (Start)

G.f.: x*(9 + x + 3*x^2 - x^3) / ((1 - x)^3*(1 + x)^2).

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

(End)