|
|
A246103
|
|
Paradigm shift sequence for (5,5) production scheme with replacement.
|
|
13
|
|
|
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 117, 126, 135, 144, 153, 162, 171, 180, 192, 204, 216, 228, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 405, 432, 459, 486, 513, 540, 576, 612, 648, 684, 720, 768, 816, 864, 912, 960, 1024, 1088, 1152, 1216, 1280, 1344, 1408, 1472, 1539, 1620, 1728, 1836, 1944, 2052, 2160, 2304, 2448, 2592, 2736, 2880, 3072, 3264, 3456, 3648, 3840, 4096, 4352, 4608, 4864, 5120, 5376, 5632, 5888, 6156
(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=5 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 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
|
|
|
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-25) for all n >= 100.
|
|
CROSSREFS
|
Paradigm shift sequences with q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|