|
| |
|
|
A246074
|
|
Paradigm Shift Sequence for a (-4,5) production scheme with replacement.
|
|
18
|
|
|
|
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 28, 32, 36, 40, 44, 48, 56, 64, 72, 80, 88, 96, 112, 128, 144, 160, 176, 192, 224, 256, 288, 320, 352, 384, 448, 512, 576, 640, 704, 768, 896, 1024, 1152, 1280, 1408, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3584, 4096, 4608, 5120, 5632, 6144, 7168, 8192, 9216
(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=-4 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. The paradigm shift sequence for the R(-4,5) scheme is also the solution to the R(-2,4) scheme.
|
|
|
LINKS
|
|
|
|
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-6) for all n >= 12.
G.f.: x*(1+x)^2 * (1-x+x^2)^2 * (1+x+x^2)^2 / (1-2*x^6). - Colin Barker, Nov 18 2016
|
|
|
MATHEMATICA
|
Join[{1, 2, 3, 4, 5}, LinearRecurrence[{0, 0, 0, 0, 0, 2}, {6, 7, 8, 9, 10, 11}, 64]] (* Jean-François Alcover, Sep 25 2017 *)
|
|
|
PROG
|
(PARI) Vec(x*(1+x)^2 * (1-x+x^2)^2 * (1+x+x^2)^2 / (1-2*x^6) + O(x^100)) \\ Colin Barker, Nov 18 2016
|
|
|
CROSSREFS
|
Paradigm shift sequences with implementation step q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.
Paradigm shift sequences with negative bundling steps: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577.
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
