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A246077 Paradigm shift sequence for (-1,-3) production scheme with replacement.

%I A246077 #12 Nov 19 2016 03:07:11
%S A246077 1,2,3,4,5,6,7,8,9,10,12,14,16,18,21,24,28,32,36,42,48,56,64,72,84,96,
%T A246077 112,128,144,168,192,224,256,288,336,384,448,512,576,672,768,896,1024,
%U A246077 1152,1344,1536,1792,2048,2304,2688,3072,3584,4096,4608,5376,6144,7168,8192,9216
%N A246077 Paradigm shift sequence for (-1,-3) production scheme with replacement.
%C A246077 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=-1 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?"
%C A246077 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 A246077 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 A246077 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 2.
%C A246077 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%H A246077 Colin Barker, <a href="/A246077/b246077.txt">Table of n, a(n) for n = 1..1000</a>
%H A246077 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,2).
%F A246077 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 A246077 a(n) = 2*a(n-5) for all n >= 16.
%F A246077 G.f.: x*(1 +x^2) * (1 +2*x +2*x^2 +2*x^3 +3*x^4 +2*x^5 +x^8-x^10 +x^12) / (1 -2*x^5). - _Colin Barker_, Nov 19 2016
%o A246077 (PARI) Vec(x*(1 +x^2) * (1 +2*x +2*x^2 +2*x^3 +3*x^4 +2*x^5 +x^8-x^10 +x^12) / (1 -2*x^5) + O(x^100)) \\ _Colin Barker_, Nov 19 2016
%Y A246077 Paradigm shifts with q=3: A029747, A029750, A246077, A246081, A246085, A246089, A246093, A246097, A246101.
%Y A246077 Paradigm shifts with q<0: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577.
%K A246077 nonn,easy
%O A246077 1,2
%A A246077 _Jonathan T. Rowell_, Aug 13 2014