|
%I #18 Nov 18 2016 20:07:55
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,18,20,22,24,27,30,33,36,40,44,48,
%T 54,60,66,72,81,90,99,108,120,132,144,162,180,198,216,243,270,297,324,
%U 360,396,432,486,540,594,648,729,810,891,972,1080,1188,1296,1458,1620,1782,1944,2187,2430,2673,2916,3240,3564,3888,4374
%N Paradigm shift sequence for (-1,4) production scheme with replacement.
%C 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?"
%C 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 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 3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 3.
%C 4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
%C 5. For large n, the sequence is recursively defined.
%H Colin Barker, <a href="/A246078/b246078.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,3).
%F 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 a(n) = 3*a(n-11) for all n >= 26.
%F 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 +9*x^11 +7*x^12 +5*x^13 +4*x^14 +3*x^15 +2*x^16 +x^17 +x^23 +2*x^24) / (1 -3*x^11). - _Colin Barker_, Nov 18 2016
%t 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 + 9 x^11 + 7 x^12 + 5 x^13 + 4 x^14 + 3 x^15 + 2 x^16 + x^17 + x^23 + 2 x^24)/(1 - 3 x^11), {x, 0, 71}], x] (* _Michael De Vlieger_, Nov 18 2016 *)
%o (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 +9*x^11 +7*x^12 +5*x^13 +4*x^14 +3*x^15 +2*x^16 +x^17 +x^23 +2*x^24) / (1 -3*x^11) + O(x^100)) \\ _Colin Barker_, Nov 18 2016
%Y Paradigm shift sequences with q=4: A029750, A103969, A246074, A246078, A246082, A246086, A246090, A246094, A246098, A246102.
%Y Paradigm shift sequences with p<0: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577.
%K nonn,easy
%O 1,2
%A _Jonathan T. Rowell_, Aug 13 2014
|
