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A246087 Paradigm shift sequence for (1,5) production scheme with replacement.

%I A246087 #13 Nov 18 2016 20:08:20
%S A246087 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,26,28,30,
%T A246087 33,36,39,42,45,48,51,54,57,60,64,68,72,78,84,90,99,108,117,126,135,
%U A246087 144,153,162,171,180,192,204,216,234,252,270,297,324,351,378,405,432,459,486,513,540,576,612,648,702,756,810,891,972,1053,1134,1215,1296
%N A246087 Paradigm shift sequence for (1,5) production scheme with replacement.
%C A246087 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 A246087 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 A246087 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 A246087 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 3.
%C A246087 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%H A246087 Colin Barker, <a href="/A246087/b246087.txt">Table of n, a(n) for n = 1..1000</a>
%H A246087 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3).
%F A246087 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 A246087 a(n) = 3*a(n-16) for all n >= 39.
%F A246087 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 +15*x^14 +16*x^15 +14*x^16 +12*x^17 +10*x^18 +8*x^19 +6*x^20 +4*x^21 +3*x^22 +2*x^23 +x^24 +x^36 +2*x^37) / (1 -3*x^16). - _Colin Barker_, Nov 18 2016
%t A246087 CoefficientList[Series[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 + 15 x^14 + 16 x^15 + 14 x^16 + 12 x^17 + 10 x^18 + 8 x^19 + 6 x^20 + 4 x^21 + 3 x^22 + 2 x^23 x^24 + x^36 + 2 x^37)/(1 - 3 x^16), {x, 0, 80}], x] (* _Michael De Vlieger_, Nov 18 2016 *)
%o A246087 (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 +15*x^14 +16*x^15 +14*x^16 +12*x^17 +10*x^18 +8*x^19 +6*x^20 +4*x^21 +3*x^22 +2*x^23 +x^24 +x^36 +2*x^37) / (1 -3*x^16) + O(x^100)) \\ _Colin Barker_, Nov 18 2016
%Y A246087 Paradigm shift sequences with q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.
%Y A246087 Paradigm shift sequences with p=1: A178715, A246084, A246085, A246086, A246087.
%K A246087 nonn,easy
%O A246087 1,2
%A A246087 _Jonathan T. Rowell_, Aug 13 2014