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A336479
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For any number n with k binary digits, a(n) is the number of tilings T of a size k staircase polyomino (as described in A335547) such that the sizes of the polyominoes at the base of T correspond to the lengths of runs of consecutive equal digits in the binary representation of n.
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3
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1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 3, 2, 3, 1, 1, 1, 2, 8, 5, 11, 18, 8, 5, 3, 5, 11, 7, 3, 5, 1, 1, 1, 2, 13, 8, 26, 42, 18, 11, 26, 42, 94, 58, 29, 47, 13, 8, 5, 8, 29, 18, 36, 58, 26, 16, 7, 11, 26, 16, 5, 8, 1, 1, 1, 2, 21, 13, 60, 97, 42, 26, 87, 141, 317
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OFFSET
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0,6
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COMMENTS
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a(0) = 1 corresponds to the empty polyomino.
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LINKS
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FORMULA
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A335547(n) = Sum_{k = 2^(n-1)..2^n-1} a(k).
a(2^k-1) = 1 for any k >= 0.
a(2^k) = 1 for any k >= 0.
a(3*2^k) = A000045(k+1) for any k >= 0.
a(7*2^k) = A123392(k) for any k >= 0.
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EXAMPLE
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For n = 13, the binary representation of 13 is "1101", so we count the tilings of a size 4 staircase polyomino whose base has the following shape:
.....
. .
. .....
. .
+---+ .....
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| 1 1 | 0 | 1 |
+-------+---+---+
there are 3 such tilings:
+---+ +---+ +---+
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+---+---+ + +---+ +---+---+
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+---+---+---+ +---+---+---+ +---+ +---+
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| +---+---+---+ | +---+---+---+ | +---+---+---+
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+-------+---+---+, +-------+---+---+, +-------+---+---+
so a(13) = 3.
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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