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A246099 Paradigm shift sequence for (4,5) production scheme with replacement. 13


%S 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,

%T 27,28,30,33,36,39,42,45,48,51,54,57,60,64,68,72,76,80,84,88,92,96,

%U 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

%N Paradigm shift sequence for (4,5) production scheme with replacement.

%C 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?"

%C 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.

%C 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.

%C 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 3.

%C 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

%C 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.

%F 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) ).

%F Recursive: a(n) = 3*a(n-19) for all n >= 164.

%Y Paradigm shift sequences for q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.

%Y Paradigm shift sequences for q=4: A193456, A246096, A246097, A246098, A246099.

%K nonn

%O 1,2

%A _Jonathan T. Rowell_, Aug 13 2014

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