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A246080 Paradigm shift sequence for (0,2) production scheme with replacement. 9
1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 18, 21, 24, 30, 36, 45, 54, 63, 72, 90, 108, 135, 162, 189, 216, 270, 324, 405, 486, 567, 648, 810, 972, 1215, 1458, 1701, 1944, 2430, 2916, 3645, 4374, 5103, 5832, 7290, 8748, 10935, 13122, 15309, 17496, 21870, 26244 (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=0 steps), or implement the current bundled action (which requires q=2 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 = 3.

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

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3).

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) = 3*a(n-6) for all n >= 10.

G.f.: x*(1+x)^2 * (1+2*x^2+3*x^4+x^6) / (1-3*x^6). - Colin Barker, Nov 19 2016

PROG

(PARI) Vec(x*(1+x)^2 * (1+2*x^2+3*x^4+x^6) / (1-3*x^6) + O(x^100)) \\ Colin Barker, Nov 19 2016

CROSSREFS

Paradigm shift sequences with q=2: A029744, A029747, A246080, A246084, A246088, A246092, A246096, A246100.

Paradigm shift sequences with p=0: A000792, A246080, A246081, A246082, A246083.

Sequence in context: A279079 A029750 A266480 * A278619 A173925 A320319

Adjacent sequences:  A246077 A246078 A246079 * A246081 A246082 A246083

KEYWORD

nonn,easy

AUTHOR

Jonathan T. Rowell, Aug 13 2014

STATUS

approved

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Last modified July 9 13:27 EDT 2020. Contains 335543 sequences. (Running on oeis4.)