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A246101
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Paradigm shift sequence for (5,3) production scheme with replacement.
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9
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 27, 30, 33, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 80, 85, 90, 99, 108, 120, 132, 144, 160, 176, 192, 208, 224, 240, 260, 280, 300, 325, 350, 375, 400, 432, 480, 528, 576, 640, 704, 768, 832, 896, 960, 1040, 1120, 1200, 1300, 1400, 1500, 1625, 1750, 1920
(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 = 5 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 = 4.
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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Jonathan T. Rowell, Table of n, a(n) for n = 1..150
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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) ).
Recursive: a(n) = 4*a(n-17) for all n >= 75.
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CROSSREFS
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Paradigm shift sequences with q=3: A029747, A029750, A246077, A246081, A246085, A246089, A246093, A246097, A246101.
Paradigm shift sequences with p=5: A193457, A246100, A246101, A246102, A246103.
Sequence in context: A275164 A076499 A092629 * A241673 A258009 A108192
Adjacent sequences: A246098 A246099 A246100 * A246102 A246103 A246104
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KEYWORD
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nonn
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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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