A247618 Start with a single square; at n-th generation add a square at each expandable vertex; a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

1, 5, 17, 45, 105, 229, 481, 989, 2009, 4053, 8145, 16333, 32713, 65477, 131009, 262077, 524217, 1048501, 2097073, 4194221, 8388521, 16777125, 33554337, 67108765, 134217625, 268435349, 536870801, 1073741709, 2147483529, 4294967173, 8589934465, 17179869053, 34359738233

Offset

0,2

Comments

Inspired by A061777, let us assign label "1" to an origin square; at n-th generation add a square 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 squares count is A001844. See illustration. For n >= 1, (a(n) - a(n-1))/4 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) = 4*A000225(n) + a(n-1).

From Colin Barker, Sep 21 2014: (Start)

a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3).

a(n) = -7+2^(3+n)-4*n.

G.f.: -(2*x^2+x+1) / ((x-1)^2*(2*x-1)). (End)

Mathematica

A247618[n_] := 2^(n + 3) - 4*n - 7; Array[A247618, 35, 0] (* Paolo Xausa, May 07 2026 *)

Prog

(PARI)
a(n) = if (n<1, 1, 4*(2^n-1)+a(n-1))
for (n=0, 50, print1(a(n), ", "))
(PARI)
Vec(-(2*x^2+x+1) / ((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 21 2014

Crossrefs

Cf. A000225, A061777, A001844, A247619, A247620.

Sequence in context: A133252 A396298 A299335 * A269962 A048612 A320554

Adjacent sequences: A247615 A247616 A247617 * A247619 A247620 A247621

Keyword

nonn,easy

Author

Kival Ngaokrajang, Sep 20 2014

Extensions

More terms from Paolo Xausa, May 07 2026

Status

approved