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A246097 Paradigm shift sequence for (4,3) production scheme with replacement. 9
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 24, 27, 30, 33, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 81, 90, 99, 108, 120, 132, 144, 160, 176, 192, 208, 224, 240, 260, 280, 300, 325, 360, 396, 432, 480, 528, 576, 640, 704, 768, 832, 896, 960, 1040, 1120, 1200, 1300, 1440, 1584, 1728, 1920, 2112, 2304, 2560, 2816, 3072, 3328, 3584, 3840, 4160, 4480, 4800, 5200, 5760, 6336, 6912, 7680, 8448, 9216, 10240, 11264, 12288, 13312, 14336, 15360, 16640, 17920, 19200, 20800, 23040 (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=3 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 = 4.

4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

LINKS

Table of n, a(n) for n=1..100.

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) ).

Recursive:  a(n) = 4*a(n-16) for all n >= 52.

CROSSREFS

Paradigm shift sequences with q=3: A029747, A029750, A246077, A246081, A246085, A246089, A246093, A246097, A246101.

Paradigm shift sequences with p=4: A193456, A246096, A246097, A246098, A246099.

Sequence in context: A078453 A052425 A272074 * A275776 A022465 A296762

Adjacent sequences:  A246094 A246095 A246096 * A246098 A246099 A246100

KEYWORD

nonn

AUTHOR

Jonathan T. Rowell, Aug 13 2014

STATUS

approved

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Last modified June 4 08:18 EDT 2020. Contains 334825 sequences. (Running on oeis4.)