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A246098 Paradigm shift sequence for (4,4) production scheme with replacement.

%I A246098 #6 Aug 15 2014 23:08:16
%S A246098 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,27,30,
%T A246098 33,36,39,42,45,48,52,56,60,64,68,72,76,80,85,90,99,108,117,126,135,
%U A246098 144,156,168,180,192,208,224,240,256,272,288,304,324,351,378,405,432,468,504,540,576,624,672,720,768,832,896,960,1024,1088,1152,1216,1296,1404,1512,1620,1728,1872,2016,2160,2304,2496,2688,2880,3072,3328,3584,3840,4096,4352,4608,4864,5184
%N A246098 Paradigm shift sequence for (4,4) production scheme with replacement.
%C A246098 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=4 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 A246098 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 A246098 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 A246098 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 4.
%C A246098 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%F A246098 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 A246098 Recursive:  a(n) = 4*a(n-20) for all n >= 64.
%Y A246098 Paradigm shift sequences with q=4: A029750, A103969, A246074, A246078, A246082, A246086, A246090, A246094, A246098, A246102.
%Y A246098 Paradigm shift sequence with p=4: A193456, A246096, A246097, A246098, A246099.
%K A246098 nonn
%O A246098 1,2
%A A246098 _Jonathan T. Rowell_, Aug 13 2014