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%I #15 Nov 18 2016 20:07:46
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,22,24,26,28,30,33,36,40,
%T 44,48,52,56,60,66,72,80,88,96,104,112,120,132,144,160,176,192,208,
%U 224,240,264,288,320,352,384,416,448,480,528,576,640,704,768,832,896,960,1056,1152,1280,1408,1536,1664,1792
%N Paradigm shift sequence for the (-2,5) 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=-2 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 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 = 2.
%C 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%H Colin Barker, <a href="/A246076/b246076.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,2).
%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-8) for all n >= 25.
%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 +7*x^8 +6*x^9 +5*x^10 +4*x^11 +3*x^12 +2*x^13 +x^14 +x^23) / (1 -2*x^8). - _Colin Barker_, Nov 18 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 +7*x^8 +6*x^9 +5*x^10 +4*x^11 +3*x^12 +2*x^13 +x^14 +x^23) / (1 -2*x^8) + O(x^100)) \\ _Colin Barker_, Nov 18 2016
%Y Paradigm shift sequences with q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.
%Y Paradigm shift sequences with p<0: 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