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

Author

Floor van Lamoen, Jul 21 2001

Keywords

• nonn
• easy

Comments

Related to parity of Beatty sequences for exp(-(1/2)/n). Let f(k,n)=-sum(i=1,n,sum(j=1,i,(-1)^floor(j*exp(-(1/2)/n)))), then a(n)=Max{f(k,n) : 1<=k<=4*a(n)-2} and for 0<=i<=4*a(n)-3, f(i,n)=f(4*a(n)-2-i,n). - Benoit Cloitre, May 26 2004

Or, sum of multiples of 2 and 3 from 0 to 6*n. - Zak Seidov, Aug 06 2016

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).

Crossrefs

Cf. A000567, A003154, A007742, A062783, A272399.

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)

a(n) = A272399(n+1) - A003154(n+1). - Leo Tavares, Mar 24 2022

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