A247619 Start with a single pentagon; at n-th generation add a pentagon at each expandable vertex; a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

1, 6, 16, 36, 66, 116, 186, 296, 446, 676, 986, 1456, 2086, 3036, 4306, 6216, 8766, 12596, 17706, 25376, 35606, 50956, 71426, 102136, 143086, 204516, 286426, 409296, 573126, 818876, 1146546, 1638056, 2293406, 3276436, 4587146, 6553216, 9174646, 13106796

Offset: 0

Author

Kival Ngaokrajang, Sep 21 2014

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A005891, A027383, A061777, A247618, A247620.

See A358632 for a related concept.

Programs

Mathematica

LinearRecurrence[{2, 1, -4, 2}, {1, 6, 16, 36}, 50] (* Paolo Xausa, Aug 21 2024 *)

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

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

Extensions

More terms from Colin Barker, Sep 21 2014