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A246075
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Paradigm shift sequence for a (-3,5) production scheme with replacement.
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12
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 32, 36, 40, 44, 48, 52, 56, 64, 72, 80, 88, 96, 104, 112, 128, 144, 160, 176, 192, 208, 224, 256, 288, 320, 352, 384, 416, 448, 512, 576, 640, 704, 768, 832, 896, 1024, 1152, 1280, 1408, 1536, 1664, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 3584, 4096
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OFFSET
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1,2
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COMMENTS
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This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple incremental action, bundle existing output as an integrated product (which requires p=-3 steps), or implement the current bundled action (which requires q=5 steps). The first use of a novel bundle erases (or makes obsolete) all prior actions. How large an output can be generated in n time steps?"
1. A production scheme with replacement R(p,q) eliminates existing output followinging a bundling action, while an additive scheme A(p,q) retains the output. The schemes correspond according to A(p,q)=R(p-q,q), with the replacement scheme serving as the default presentation.
2. This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with production schemes R(1,1) and R(2,1) with replacement, respectively.
3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 2.
4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
5. For large n, the sequence is recursively defined.
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LINKS
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Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2).
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FORMULA
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a(n) = (qd+r) * d^(C-R) * (d+1)^R, where r = (n-Cp) mod q, Q = floor( (R-Cp)/q ), R = Q mod (C+1), and d = floor ( Q/(C+1) ).
a(n) = 2*a(n-7) for all n >= 14.
G.f.: x*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)^2 / (1 -2*x^7). - Colin Barker, Nov 18 2016
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MATHEMATICA
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Join[Range[6], LinearRecurrence[PadLeft[{2}, 7], Range[7, 13], 65]] (* Jean-François Alcover, Sep 25 2017 *)
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PROG
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(PARI) Vec(x*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)^2 / (1 -2*x^7) + O(x^100)) \\ Colin Barker, Nov 18 2016
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CROSSREFS
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Paradigm shift sequences for implementation size p=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.
Paradigm shift sequences for p<0: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577.
Sequence in context: A035062 A032964 A033066 * A246078 A246085 A017907
Adjacent sequences: A246072 A246073 A246074 * A246076 A246077 A246078
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KEYWORD
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nonn,easy
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AUTHOR
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Jonathan T. Rowell, Aug 13 2014
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STATUS
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approved
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