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A246088 Paradigm shift sequence for (2,2) production scheme with replacement.

%I A246088 #9 Nov 19 2016 06:26:11
%S A246088 1,2,3,4,5,6,7,8,9,10,11,12,15,18,21,24,28,32,36,40,45,54,63,72,84,96,
%T A246088 112,128,144,162,189,216,252,288,336,384,448,512,576,648,756,864,1008,
%U A246088 1152,1344,1536,1792,2048,2304,2592,3024,3456,4032,4608,5376,6144,7168,8192,9216,10368,12096,13824,16128,18432,21504,24576,28672,32768,36864,41472
%N A246088 Paradigm shift sequence for (2,2) production scheme with replacement.
%C A246088 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=2 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 A246088 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 A246088 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 A246088 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 4.
%C A246088 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%H A246088 Colin Barker, <a href="/A246088/b246088.txt">Table of n, a(n) for n = 1..1000</a>
%H A246088 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,4).
%F A246088 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 A246088 a(n) = 4*a(n-10) for all n >= 32.
%F A246088 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 +7*x^10 +4*x^11 +3*x^12 +2*x^13 +x^14 +x^20 +6*x^21 +3*x^22 +2*x^29 +9*x^30) / ((1 -2*x^5) * (1 +2*x^5)). - _Colin Barker_, Nov 19 2016
%o A246088 (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 +7*x^10 +4*x^11 +3*x^12 +2*x^13 +x^14 +x^20 +6*x^21 +3*x^22 +2*x^29 +9*x^30) / ((1 -2*x^5) * (1 +2*x^5)) + O(x^100)) \\ _Colin Barker_, Nov 19 2016
%Y A246088 Paradigm shift sequences with q=2: A029744, A029747, A246080, A246084, A246088, A246092, A246096, A246100.
%Y A246088 Paradigm shift sequences with p=2: A193286, A246088, A246089, A246090, A246091.
%K A246088 nonn,easy
%O A246088 1,2
%A A246088 _Jonathan T. Rowell_, Aug 13 2014