%I #8 Nov 19 2016 05:51:54
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,15,18,21,24,28,32,36,40,45,50,55,63,72,
%T 84,96,112,128,144,160,180,200,225,252,288,336,384,448,512,576,640,
%U 720,800,900,1008,1152,1344,1536,1792,2048,2304,2560,2880,3200,3600,4032,4608,5376,6144,7168,8192,9216,10240,11520,12800,14400,16128,18432,21504,24576,28672
%N Paradigm shift sequence for (3,2) 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=3 steps), or implement the current bundled action (which requires q=2 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 = 4.
%C 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%H Colin Barker, <a href="/A246092/b246092.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,4).
%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) = 4*a(n-11) for all n >= 36.
%F G.f.: x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +11*x^10 +8*x^11 +5*x^12 +3*x^13 +2*x^14 +x^15 +x^21 +2*x^22 +3*x^23 +3*x^24 +5*x^34) / (1 -4*x^11). - _Colin Barker_, Nov 19 2016
%o (PARI) Vec(x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +11*x^10 +8*x^11 +5*x^12 +3*x^13 +2*x^14 +x^15 +x^21 +2*x^22 +3*x^23 +3*x^24 +5*x^34) / (1 -4*x^11) + O(x^100)) \\ _Colin Barker_, Nov 19 2016
%Y Paradigm shift sequences with q=2: A029744, A029747, A246080, A246084, A246088, A246092, A246096, A246100.
%Y Paradigm shift sequences with p=3: A193455, A246092, A246093, A246094, A246095.
%K nonn,easy
%O 1,2
%A _Jonathan T. Rowell_, Aug 13 2014