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 A246075 Paradigm shift sequence for a (-3,5) production scheme with replacement. 12
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 32, 36, 40, 44, 48, 52, 56, 64, 72, 80, 88, 96, 104, 112, 128, 144, 160, 176, 192, 208, 224, 256, 288, 320, 352, 384, 416, 448, 512, 576, 640, 704, 768, 832, 896, 1024, 1152, 1280, 1408, 1536, 1664, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 3584, 4096 (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=-3 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 followinging 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 = 2. 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 (0,0,0,0,0,0,2). 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) = 2*a(n-7) for all n >= 14. G.f.: x*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)^2 / (1 -2*x^7). - Colin Barker, Nov 18 2016 MATHEMATICA Join[Range, LinearRecurrence[PadLeft[{2}, 7], Range[7, 13], 65]] (* Jean-François Alcover, Sep 25 2017 *) PROG (PARI) Vec(x*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6)^2 / (1 -2*x^7) + O(x^100)) \\ Colin Barker, Nov 18 2016 CROSSREFS Paradigm shift sequences for implementation size p=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103. Paradigm shift sequences for p<0: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577. Sequence in context: A035062 A032964 A033066 * A246078 A246085 A017907 Adjacent sequences:  A246072 A246073 A246074 * A246076 A246077 A246078 KEYWORD nonn,easy AUTHOR Jonathan T. Rowell, Aug 13 2014 STATUS approved

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Last modified April 12 18:58 EDT 2021. Contains 342932 sequences. (Running on oeis4.)