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

Offset: 0

Author

Kival Ngaokrajang, Sep 20 2014

Keywords

• nonn
• easy

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

• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

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

Programs

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

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)