OFFSET
0,2
COMMENTS
Inspired by A061777, let us assign the label "1" to an origin hexagon; at n-th generation add a hexagon at each expandable vertex, i.e. a vertex such that the new added generations will not overlap to the existing ones, but overlapping among new generations are allowed. The non-overlapping squares will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. The hexagons count is A003215. See llustration. For n >= 1, (a(n) - a(n-1))/6 is A000225
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
FORMULA
a(0) = 1, for n >= 1, a(n) = 6*A000225(n) + a(n-1).
From Colin Barker, Sep 21 2014: (Start)
a(n) = (-11+3*2^(2+n)-6*n).
a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3).
G.f.: -(x+1)*(2*x+1) / ((x-1)^2*(2*x-1)).
(End)
MATHEMATICA
LinearRecurrence[{4, -5, 2}, {1, 7, 25}, 50] (* Paolo Xausa, Aug 22 2024 *)
PROG
(PARI) a(n) = if (n<1, 1, 6*(2^n-1)+a(n-1))
for (n=0, 50, print1(a(n), ", "))
(PARI) Vec(-(x+1)*(2*x+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 21 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Sep 21 2014
EXTENSIONS
More terms from Colin Barker, Sep 21 2014
STATUS
approved