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A246077
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Paradigm shift sequence for (-1,-3) production scheme with replacement.
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11
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 64, 72, 84, 96, 112, 128, 144, 168, 192, 224, 256, 288, 336, 384, 448, 512, 576, 672, 768, 896, 1024, 1152, 1344, 1536, 1792, 2048, 2304, 2688, 3072, 3584, 4096, 4608, 5376, 6144, 7168, 8192, 9216
(list;
graph;
refs;
listen;
history;
text;
internal format)
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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=-1 steps), or implement the current bundled action (which requires q=3 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 following 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.
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LINKS
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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-5) for all n >= 16.
G.f.: x*(1 +x^2) * (1 +2*x +2*x^2 +2*x^3 +3*x^4 +2*x^5 +x^8-x^10 +x^12) / (1 -2*x^5). - Colin Barker, Nov 19 2016
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PROG
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(PARI) Vec(x*(1 +x^2) * (1 +2*x +2*x^2 +2*x^3 +3*x^4 +2*x^5 +x^8-x^10 +x^12) / (1 -2*x^5) + O(x^100)) \\ Colin Barker, Nov 19 2016
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CROSSREFS
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Paradigm shifts with q<0: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577.
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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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