A289999 Sierpinski cuboctahedral numbers: a(n) = 16*4^n - 12*2^n + 9.

13, 49, 217, 937, 3913, 16009, 64777, 260617, 1045513, 4188169, 16764937, 67084297, 268386313, 1073643529, 4294770697, 17179475977, 68718690313, 274876334089, 1099508482057, 4398040219657, 17592173461513, 70368719011849, 281474926379017, 1125899806179337, 4503599426043913, 18014398106828809

Offset: 0

Author

Steven Beard, Sep 03 2017

Keywords

• nonn
• easy

Comments

Sierpinski cuboctahedron constructed by joining eight Sierpinski tetrahedra of sequence 4, 10, 34, 130, 514, 2050, 8194... (4^n*2)+2 (the double of A052539). This sequence is also Sierpinski recursion for the octahemioctahedron A274974.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Wikipedia, Sierpinski tetrahedron.
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Cf. A005902, A290396, A274974, A281699, A067771, A279511.

Programs

Mathematica

CoefficientList[Series[(13 - 42 x + 56 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* Michael De Vlieger, Sep 03 2017 *)
Table[16*4^n-12*2^n+9,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{13,49,217},30] (* Harvey P. Dale, Dec 31 2018 *)

PARI

Vec((13 - 42*x + 56*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Sep 03 2017

PARI

a(n) = 16*4^n - 12*2^n + 9 \\ Charles R Greathouse IV, Nov 03 2017

Formula

a(n) = -3*2^(n + 2) + 2^(2n + 4) + 9.

From Colin Barker, Sep 03 2017: (Start)

G.f.: (13 - 42*x + 56*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).

a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.

(End)