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A246083 Paradigm shift sequence for (0,5) production scheme with replacement. 13


%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,24,26,28,30,33,

%T 36,39,42,45,48,51,54,57,60,66,72,78,84,90,99,108,117,126,135,144,153,

%U 162,171,180,198,216,234,252,270,297,324,351,378,405,432,459,486,513,540,594,648,702,756,810,891,972,1053,1134,1215,1296,1377,1458

%N Paradigm shift sequence for (0,5) 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=0 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 = 3.

%C 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

%C 5. For large n, the sequence is recursively defined.

%H Colin Barker, <a href="/A246083/b246083.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,3).

%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) = 3*a(n-15) for all n >= 25.

%F G.f.: x*(1 +x +x^2 +x^3 +x^4)^2 * (1 +2*x^5 +3*x^10 +x^15) / (1 -3*x^15). - _Colin Barker_, Nov 18 2016

%o (PARI) Vec(x*(1 +x +x^2 +x^3 +x^4)^2 * (1 +2*x^5 +3*x^10 +x^15) / (1 -3*x^15) + O(x^100)) \\ _Colin Barker_, Nov 18 2016

%Y Paradigm shift sequences for q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.

%Y Paradigm shift sequences for p=0: A000792, A246080, A246081, A246082, A246083.

%K nonn,easy

%O 1,2

%A _Jonathan T. Rowell_, Aug 13 2014

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