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 A246099 Paradigm shift sequence for (4,5) production scheme with replacement. 13
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 108, 117, 126, 135, 144, 153, 162, 171, 180, 192, 204, 216, 228, 240, 256, 272, 288, 304, 320, 336, 352, 378, 405, 432, 459, 486, 513, 540, 576, 612, 648, 684, 720, 768, 816, 864, 912, 960, 1024, 1088, 1152, 1216, 1296, 1377, 1458, 1539, 1620, 1728, 1836, 1944, 2052, 2160, 2304, 2448, 2592, 2736, 2880, 3072, 3264, 3456, 3648, 3888, 4131, 4374, 4617, 4864, 5184, 5508, 5832, 6156, 6480, 6912, 7344, 7776, 8208, 8640, 9216, 9792, 10368, 10944, 11664, 12393, 13122, 13851, 14592, 15552, 16524, 17496, 18468, 19456, 20736, 22032, 23328 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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=4 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?" 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. 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. 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 3. 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d. 5. The optimal action string that generates this integer sequence (x=a^k PV^m ...) has inversions in the minimum number of bundling procedures P. These inversions are persistent as they are present when the recursive formula for the output first holds. LINKS FORMULA 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) ). Recursive:  a(n) = 3*a(n-19) for all n >= 164. CROSSREFS Paradigm shift sequences for q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103. Paradigm shift sequences for q=4: A193456, A246096, A246097, A246098, A246099. Sequence in context: A080683 A174228 A197640 * A280865 A124231 A154470 Adjacent sequences:  A246096 A246097 A246098 * A246100 A246101 A246102 KEYWORD nonn AUTHOR Jonathan T. Rowell, Aug 13 2014 STATUS approved

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Last modified April 9 04:05 EDT 2020. Contains 333343 sequences. (Running on oeis4.)