OFFSET
0,2
COMMENTS
Inspired by A061777, let us assign the label "1" to an origin pentagon; at the n-th generation add a pentagon at each expandable vertex, i.e., a vertex such that the new added generations will not overlap existing ones, but overlapping among new generations is allowed. Each nonoverlapping pentagon will have the same label value as its predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. The pentagon count is A005891. See illustration. [Edited for grammar/style by Peter Munn, Jan 14 2023]
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 (2,1,-4,2).
FORMULA
a(0) = 1, for n >= 1, a(n) = 5*A027383(n-1) + a(n-1). [Offset corrected by Peter Munn, Apr 20 2023]
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+2*a(n-4). G.f.: -(2*x^3+3*x^2+4*x+1) / ((x-1)^2*(2*x^2-1)). - Colin Barker, Sep 21 2014
MATHEMATICA
LinearRecurrence[{2, 1, -4, 2}, {1, 6, 16, 36}, 50] (* Paolo Xausa, Aug 21 2024 *)
PROG
(PARI)
{
b=0; a=1; print1(1, ", ");
for (n=0, 50,
b=b+2^floor(n/2);
a=a+5*b;
print1(a, ", ")
)
}
(PARI)
Vec(-(2*x^3+3*x^2+4*x+1)/((x-1)^2*(2*x^2-1)) + 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