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A246084 Paradigm shift sequence for (1,2) production scheme with replacement.

%I A246084 #9 Nov 19 2016 05:51:59
%S A246084 1,2,3,4,5,6,7,8,9,10,12,15,18,21,24,28,32,36,45,54,63,72,84,96,112,
%T A246084 135,162,189,216,252,288,336,405,486,567,648,756,864,1008,1215,1458,
%U A246084 1701,1944,2268,2592,3024,3645,4374,5103,5832,6804,7776,9072,10935,13122,15309,17496,20412,23328,27216,32805,39366,45927
%N A246084 Paradigm shift sequence for (1,2) production scheme with replacement.
%C A246084 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=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 A246084 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 A246084 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 A246084 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 3.
%C A246084 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%C A246084 5. For large n, the sequence is recursively defined.
%H A246084 Colin Barker, <a href="/A246084/b246084.txt">Table of n, a(n) for n = 1..1000</a>
%H A246084 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,3).
%F A246084 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 A246084 a(n) = 3*a(n-7) for all n >= 26.
%F A246084 G.f.: x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +5*x^7 +3*x^8 +x^9 +x^15 +2*x^16 +4*x^24) / (1 -3*x^7). - _Colin Barker_, Nov 19 2016
%o A246084 (PARI) Vec(x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +5*x^7 +3*x^8 +x^9 +x^15 +2*x^16 +4*x^24) / (1 -3*x^7) + O(x^100)) \\ _Colin Barker_, Nov 19 2016
%Y A246084 Paradigm shift sequences with q=2: A029744, A029747, A246080, A246084, A246088, A246092, A246096, A246100.
%Y A246084 Paradigm shift sequences with p=1: A178715, A246084, A246085, A246086, A246087.
%K A246084 nonn,easy
%O A246084 1,2
%A A246084 _Jonathan T. Rowell_, Aug 13 2014