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%I #20 Sep 25 2017 08:55:15
%S 1,2,3,4,5,6,7,8,9,10,11,12,14,16,18,20,22,24,28,32,36,40,44,48,56,64,
%T 72,80,88,96,112,128,144,160,176,192,224,256,288,320,352,384,448,512,
%U 576,640,704,768,896,1024,1152,1280,1408,1536,1792,2048,2304,2560,2816,3072,3584,4096,4608,5120,5632,6144,7168,8192,9216
%N Paradigm Shift Sequence for a (-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 followinging 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 = 2.
%C 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%C 5. The paradigm shift sequence for the R(-4,5) scheme is also the solution to the R(-2,4) scheme.
%H Colin Barker, <a href="/A246074/b246074.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2).
%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 a(n) = 2*a(n-6) for all n >= 12.
%F G.f.: x*(1+x)^2 * (1-x+x^2)^2 * (1+x+x^2)^2 / (1-2*x^6). - _Colin Barker_, Nov 18 2016
%t Join[{1, 2, 3, 4, 5}, LinearRecurrence[{0, 0, 0, 0, 0, 2}, {6, 7, 8, 9, 10, 11}, 64]] (* _Jean-François Alcover_, Sep 25 2017 *)
%o (PARI) Vec(x*(1+x)^2 * (1-x+x^2)^2 * (1+x+x^2)^2 / (1-2*x^6) + O(x^100)) \\ _Colin Barker_, Nov 18 2016
%Y Paradigm shift sequences with implementation step q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.
%Y Paradigm shift sequences with negative bundling steps: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577.
%K nonn,easy
%O 1,2
%A _Jonathan T. Rowell_, Aug 13 2014
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