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 A246082 Paradigm shift sequence for (0,4) production scheme with replacement. 10
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 54, 60, 66, 72, 81, 90, 99, 108, 117, 126, 135, 144, 162, 180, 198, 216, 243, 270, 297, 324, 351, 378, 405, 432, 486, 540, 594, 648, 729, 810, 891, 972, 1053, 1134, 1215, 1296, 1458, 1620, 1782 (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=0 steps), or implement the current bundled action (which requires q=4 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. For large n, the sequence is recursively defined. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,3). 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) ). a(n) = 3*a(n-12) for all n >= 20. G.f.: x*(1+x)^2 * (1+x^2)^2 * (1+2*x^4+3*x^8+x^12) / (1-3*x^12). - Colin Barker, Nov 19 2016 PROG (PARI) Vec(x*(1+x)^2 * (1+x^2)^2 * (1+2*x^4+3*x^8+x^12) / (1-3*x^12) + O(x^100)) \\ Colin Barker, Nov 19 2016 CROSSREFS Paradigm shift sequences with q=4: A029750, A103969, A246074, A246078, A246082, A246086, A246090, A246094, A246098, A246102. Paradigm shift sequences with p=0: A000792, A246080, A246081, A246082, A246083. Sequence in context: A032966 A122937 A246076 * A060340 A241989 A246089 Adjacent sequences:  A246079 A246080 A246081 * A246083 A246084 A246085 KEYWORD nonn,easy AUTHOR Jonathan T. Rowell, Aug 13 2014 STATUS approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)