A247620 Start with a single hexagon; at n-th generation add a hexagon at each expandable vertex; a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

1, 7, 25, 67, 157, 343, 721, 1483, 3013, 6079, 12217, 24499, 49069, 98215, 196513, 393115, 786325, 1572751, 3145609, 6291331, 12582781, 25165687, 50331505, 100663147, 201326437, 402653023, 805306201, 1610612563, 3221225293, 6442450759, 12884901697

Offset: 0

Author

Kival Ngaokrajang, Sep 21 2014

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000225, A061777, A003215, A247618, A247619.

Programs

Mathematica

A247620[n_] := 3*2^(n+2) - 6*n - 11; Array[A247620, 50, 0] (* or *)
LinearRecurrence[{4, -5, 2}, {1, 7, 25}, 50] (* Paolo Xausa, Aug 22 2024 *)

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

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)

Extensions

More terms from Colin Barker, Sep 21 2014