A329774 a(n) = n+1 for n <= 2; otherwise a(n) = 3*a(n-3)+1.

1, 2, 3, 4, 7, 10, 13, 22, 31, 40, 67, 94, 121, 202, 283, 364, 607, 850, 1093, 1822, 2551, 3280, 5467, 7654, 9841, 16402, 22963, 29524, 49207, 68890, 88573, 147622, 206671, 265720, 442867, 620014, 797161, 1328602, 1860043, 2391484, 3985807

Offset: 0

Author

N. J. A. Sloane, Nov 27 2019

Keywords

• nonn
• easy

Comments

Robert Fathauer observed that if the "warp and woof" construction used by Jim Conant in his recursive dissection of a square (see A328078) is applied to a triangle, one obtains the Sierpinski gasket.

The present sequence gives the number of regions after the n-th generation of this dissection of a triangle.

References

• Robert Fathauer, Email to N. J. A. Sloane, Oct 14 2019.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Illustration of initial terms.
• Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3).

Crossrefs

A mixture of A003462, A060816, and A237930. Cf. A328078.

Programs

Maple

f:=proc(n) option remember;
if n<=2 then n+1 else 3*f(n-3)+1; fi; end;
[seq(f(n),n=0..50)];

PARI

Vec((1 + x + x^2 - 2*x^3) / ((1 - x)*(1 - 3*x^3)) + O(x^40)) \\ Colin Barker, Nov 27 2019

Formula

From Colin Barker, Nov 27 2019: (Start)

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

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

(End)