Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #9 Sep 17 2014 15:59:41
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,24,27,30,33,36,
%T 40,44,48,52,56,60,65,70,75,80,85,90,99,108,120,132,144,160,176,192,
%U 208,224,240,260,280,300,325,350,375,400,432,480,528,576,640,704,768,832,896,960,1040,1120,1200,1300,1400,1500,1625,1750,1920
%N Paradigm shift sequence for (5,3) 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 = 5 steps), or implement the current bundled action (which requires q = 3 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 Jonathan T. Rowell, <a href="/A246101/b246101.txt">Table of n, a(n) for n = 1..150</a>
%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 Recursive: a(n) = 4*a(n-17) for all n >= 75.
%Y Paradigm shift sequences with q=3: A029747, A029750, A246077, A246081, A246085, A246089, A246093, A246097, A246101.
%Y Paradigm shift sequences with p=5: A193457, A246100, A246101, A246102, A246103.
%K nonn
%O 1,2
%A _Jonathan T. Rowell_, Aug 13 2014