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 A246086 Paradigm shift sequence for (1,4) production scheme with replacement. 10
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 66, 72, 81, 90, 99, 108, 117, 126, 135, 144, 156, 168, 180, 198, 216, 243, 270, 297, 324, 351, 378, 405, 432, 468, 504, 540, 594, 648, 729, 810, 891, 972, 1053, 1134, 1215, 1296, 1404, 1512, 1620, 1782, 1944, 2187, 2430, 2673, 2916, 3159, 3402, 3645, 3888, 4212 (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=1 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. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,3). 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-13) for all n >= 32. G.f.: x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +11*x^10 +12*x^11 +13*x^12 +11*x^13 +9*x^14 +7*x^15 +5*x^16 +3*x^17 +2*x^18 +x^19 +x^29 +2*x^30) / (1-3*x^13). - Colin Barker, Nov 19 2016 PROG (PARI) Vec(x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +11*x^10 +12*x^11 +13*x^12 +11*x^13 +9*x^14 +7*x^15 +5*x^16 +3*x^17 +2*x^18 +x^19 +x^29 +2*x^30) / (1-3*x^13) + 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=1: A178715, A246084, A246085, A246086, A246087. Sequence in context: A080681 A272322 A246079 * A194417 A246093 A261924 Adjacent sequences:  A246083 A246084 A246085 * A246087 A246088 A246089 KEYWORD nonn,easy AUTHOR Jonathan T. Rowell, Aug 13 2014 STATUS approved

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Last modified July 15 00:20 EDT 2020. Contains 335762 sequences. (Running on oeis4.)