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A211012
Total area of all squares and rectangles after 2^n stages in the toothpick structure of A139250, assuming the toothpicks have length 2.
7
0, 0, 8, 48, 224, 960, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224, 17179607040, 68718952448, 274876858368, 1099509530624, 4398042316800, 17592177655808, 70368727400448, 281474943156224
OFFSET
0,3
COMMENTS
All internal regions in the toothpick structure are squares and rectangles. The area of every internal region is a power of 2.
Similar to A271061. - Robert Price, Mar 30 2016
For n=3,5,..., also the number of minimum vertex colorings in the n-sunlet graph. - Eric W. Weisstein, Mar 03 2024
LINKS
Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
Eric Weisstein's World of Mathematics, Sunlet Graph.
FORMULA
a(n) = 2^n * (2^n-2) = A000079(n)*(A000079(n) - 2) = A159786(2^n) = 8*A006516(n-1), n>=1.
G.f.: 8*x^2 / ((1-2*x)*(1-4*x)). - Colin Barker, Mar 30 2016
a(n) = 6*a(n-1)-8*a(n-2) for n>2. - Colin Barker, Mar 30 2016
EXAMPLE
For n = 3 the area of all squares and rectangles in the toothpick structure after 2^3 stages equals the area of a rectangle of size 8X6, so a(3) = 8*6 = 48.
PROG
(PARI) concat(vector(2), Vec(8*x^2/((1-2*x)*(1-4*x)) + O(x^50))) \\ Colin Barker, Mar 30 2016
CROSSREFS
Row sums of triangle A211017, n>=1.
Sequence in context: A261975 A087914 A271061 * A081084 A230931 A073390
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Sep 21 2012
STATUS
approved