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A246079 Paradigm shift sequence for (-1,5) production scheme with replacement.

%I A246079 #15 Sep 25 2017 08:55:01
%S A246079 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,22,24,26,28,30,33,36,
%T A246079 39,42,45,48,52,56,60,66,72,78,84,90,99,108,117,126,135,144,156,168,
%U A246079 180,198,216,234,252,270,297,324,351,378,405,432,468,504,540,594,648,702,756,810,891,972,1053,1134,1215,1296,1404
%N A246079 Paradigm shift sequence for (-1,5) production scheme with replacement.
%C A246079 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=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?"
%C A246079 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 A246079 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 A246079 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 3.
%C A246079 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%H A246079 Colin Barker, <a href="/A246079/b246079.txt">Table of n, a(n) for n = 1..1000</a>
%H A246079 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,3).
%F A246079 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 A246079 a(n) = 3*a(n-14) for all n >= 33.
%F A246079 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 +14*x^13 +12*x^14 +10*x^15 +8*x^16 +6*x^17 +5*x^18 +4*x^19 +3*x^20 +2*x^21 +x^22 +x^30 +2*x^31) / (1 -3*x^14). - _Colin Barker_, Nov 18 2016
%t A246079 Join[Range[18], LinearRecurrence[PadLeft[{3}, 14], {20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 52, 56}, 55]] (* _Jean-François Alcover_, Sep 25 2017 *)
%o A246079 (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 +14*x^13 +12*x^14 +10*x^15 +8*x^16 +6*x^17 +5*x^18 +4*x^19 +3*x^20 +2*x^21 +x^22 +x^30 +2*x^31) / (1 -3*x^14) + O(x^100)) \\ _Colin Barker_, Nov 18 2016
%Y A246079 Paradigm shift sequences with q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.
%Y A246079 Paradigm shift sequences with p<0: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577.
%K A246079 nonn,easy
%O A246079 1,2
%A A246079 _Jonathan T. Rowell_, Aug 13 2014