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A246099
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Paradigm shift sequence for (4,5) production scheme with replacement.
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13
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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, 22, 23, 24, 25, 26, 27, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 108, 117, 126, 135, 144, 153, 162, 171, 180, 192, 204, 216, 228, 240, 256, 272, 288, 304, 320, 336, 352, 378, 405, 432, 459, 486, 513, 540, 576, 612, 648, 684, 720, 768, 816, 864, 912, 960, 1024, 1088, 1152, 1216, 1296, 1377, 1458, 1539, 1620, 1728, 1836, 1944, 2052, 2160, 2304, 2448, 2592, 2736, 2880, 3072, 3264, 3456, 3648, 3888, 4131, 4374, 4617, 4864, 5184, 5508, 5832, 6156, 6480, 6912, 7344, 7776, 8208, 8640, 9216, 9792, 10368, 10944, 11664, 12393, 13122, 13851, 14592, 15552, 16524, 17496, 18468, 19456, 20736, 22032, 23328
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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=4 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 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 = 3.
4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
5. The optimal action string that generates this integer sequence (x=a^k PV^m ...) has inversions in the minimum number of bundling procedures P. These inversions are persistent as they are present when the recursive formula for the output first holds.
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LINKS
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Table of n, a(n) for n=1..142.
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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) = 3*a(n-19) for all n >= 164.
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CROSSREFS
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Paradigm shift sequences for q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.
Paradigm shift sequences for q=4: A193456, A246096, A246097, A246098, A246099.
Sequence in context: A080683 A174228 A197640 * A280865 A124231 A154470
Adjacent sequences: A246096 A246097 A246098 * A246100 A246101 A246102
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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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