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

0,2

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

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).

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

Example

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

Mathematica

a[n_] := 3*n*(4*n + 1); Array[a, 40, 0] (* Amiram Eldar, Mar 27 2022 *)

Prog

(PARI) { for (n=0, 1000, write("b063436.txt", n, " ", n*(12*n + 3)) ) } \\ Harry J. Smith, Aug 21 2009

Crossrefs

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

Sequence in context: A219384 A198955 A341563 * A010004 A172073 A059145

Adjacent sequences: A063433 A063434 A063435 * A063437 A063438 A063439

Keyword

nonn,easy

Author

Floor van Lamoen, Jul 21 2001

Status

approved