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A367700
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Number of degree 2 vertices in the n-Menger sponge graph.
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5
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12, 72, 744, 11256, 201960, 3871416, 76138536, 1512609912, 30171384168, 602782587960, 12050495247528, 240968665611768, 4819043435788776, 96378229818994104, 1927543485550004520, 38550700825394191224, 771012665426135994984, 15420242499878035355448, 308404763528431125030312
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OFFSET
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1,1
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COMMENTS
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The level 0 Menger sponge graph is a single vertex. The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.
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LINKS
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FORMULA
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a(n) = (1/17)*20^n + (2/5)*8^n + (216/85)*3^n.
a(n) = 20*a(n-1) - (3/5)*8^n - (72/5)*3^n.
G.f.: 12*x*(1 - 25*x + 120*x^2)/((1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 27 2023
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EXAMPLE
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The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices. Thus a(1) = 12.
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MATHEMATICA
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LinearRecurrence[{31, -244, 480}, {12, 72, 744}, 25] (* Paolo Xausa, Nov 28 2023 *)
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PROG
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(Python)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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