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A337120
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Factor complexity (number of subwords of length n) of the regular paperfolding sequence (A014577), and all generalized paperfolding sequences.
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4
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1, 2, 4, 8, 12, 18, 23, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228
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OFFSET
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0,2
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LINKS
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FORMULA
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a(1..6) = 2,4,8,12,18,23, then a(n) = 4*n for n>=7. [Allouche]
G.f.: (1 + x^2)*(1 + 2*x^3 - x^6) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>8.
(End)
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EXAMPLE
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For n=4, all length 4 subwords except 0000, 0101, 1010, 1111 occur, so a(4) = 16-4 = 12. (These words do not occur because odd terms in a paperfolding sequence alternate, so a subword wxyz must have w!=y or x!=z.)
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MATHEMATICA
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LinearRecurrence[{2, -1}, {1, 2, 4, 8, 12, 18, 23, 28, 32}, 100] (* Paolo Xausa, Feb 29 2024 *)
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PROG
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(PARI) Vec((1 + x^2)*(1 + 2*x^3 - x^6) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Sep 08 2020
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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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