A063436 Write 1, 2, 3, 4, ... counterclockwise in a hexagonal spiral around 0 starting left down, then a(n) is the sequence found by reading from 0 in the vertical upward direction.
0, 15, 54, 117, 204, 315, 450, 609, 792, 999, 1230, 1485, 1764, 2067, 2394, 2745, 3120, 3519, 3942, 4389, 4860, 5355, 5874, 6417, 6984, 7575, 8190, 8829, 9492, 10179, 10890, 11625, 12384, 13167, 13974, 14805, 15660, 16539, 17442, 18369, 19320, 20295, 21294, 22317
Offset: 0
Examples
The spiral begins:
.
16--15--14
/ \
17 5---4 13
/ / \ \
18 6 0 3 12
/ / / / /
19 7 1---2 11 26
\ \ / /
20 8---9--10 25
\ /
21--22--23--24
Links
- Harry J. Smith, Table of n, a(n) for n = 0..1000
- Leo Tavares, Illustration: Stellar Triangles.
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
-
Mathematica
a[n_] := 3*n*(4*n + 1); Array[a, 40, 0] (* Amiram Eldar, Mar 27 2022 *)
-
PARI
{ for (n=0, 1000, write("b063436.txt", n, " ", n*(12*n + 3)) ) } \\ Harry J. Smith, Aug 21 2009
Formula
a(n) = 3*n*(4*n+1) = 3*A007742(n).
a(n) = 24*n + a(n-1) - 9 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
From Colin Barker, Jul 07 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 3*x*(5 + 3*x)/(1-x)^3. (End)
From Amiram Eldar, Mar 27 2022: (Start)
Sum_{n>=1} 1/a(n) = 4/3 - Pi/6 - log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(2)) + log(2)/3 + sqrt(2)*log(sqrt(2)+1)/3 - 4/3. (End)
E.g.f.: 3*x*(5 + 4*x)*exp(x). - Elmo R. Oliveira, Oct 31 2024
Comments