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

0,2

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

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 *)

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

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

Sequence in context: A299262 A001296 A000970 * A240156 A155245 A155291

Adjacent sequences: A247617 A247618 A247619 * A247621 A247622 A247623

Keyword

nonn,easy

Author

Kival Ngaokrajang, Sep 21 2014

Extensions

More terms from Colin Barker, Sep 21 2014

Status

approved