A166132 a(n) = 1 + (4*9^n - 9*4^n) / 5.

1, 37, 469, 4789, 45397, 417781, 3796885, 34319413, 309464533, 2787540085, 25097297941, 225913430197, 2033371866709, 18300950780149, 164710972940437, 1482408420140341, 13341714435968725, 120075584542541173

Offset: 1

Author

Jonathan Vos Post, Oct 06 2010

Keywords

• nonn
• easy

Comments

Variable N_{LNR}(n) of the number of grid points triangulating snowflakes (Neuberger et al.).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500
• John M. Neuberger, Nandor Sieben, James W. Swift, Computing Eigenfunctions on the Koch Snowflake: A New Grid and Symmetry, arXiv:1010.0775 [math.DS], 2010.
• Index entries for linear recurrences with constant coefficients, signature (14,-49,36).

Programs

Maple

A := proc(n) 1+(4*9^n-9*4^n)/5 ; end proc: seq(A(n),n=1..60) ;

Mathematica

LinearRecurrence[{14, -49, 36}, {1, 37, 469}, 50] (* G. C. Greubel, Apr 26 2016 *)
Table[1 + (4*9^n - 9*4^n)/5, {n, 24}] (* or *)
Rest@ CoefficientList[Series[-x (1 + 23 x)/((x - 1) (4 x - 1) (9 x - 1)), {x, 0, 24}], x] (* Michael De Vlieger, Apr 27 2016 *)

Formula

a(n) = 14*a(n-1) - 49*a(n-2) + 36*a(n-3).

G.f.: -x*(1+23*x) / ((x-1)*(4*x-1)*(9*x-1)).

a(n) = A002451(n-1) + 23*A002451(n-2).

E.g.f.: (1/5)*(5*exp(x) + 4*exp(9*x) - 9*exp(4*x)). - G. C. Greubel, Apr 26 2016