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A246083 Paradigm shift sequence for (0,5) 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, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 66, 72, 78, 84, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 198, 216, 234, 252, 270, 297, 324, 351, 378, 405, 432, 459, 486, 513, 540, 594, 648, 702, 756, 810, 891, 972, 1053, 1134, 1215, 1296, 1377, 1458 (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=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 = 3.

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

5. For large n, the sequence is recursively defined.

LINKS

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

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

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-15) for all n >= 25.

G.f.: x*(1 +x +x^2 +x^3 +x^4)^2 * (1 +2*x^5 +3*x^10 +x^15) / (1 -3*x^15). - Colin Barker, Nov 18 2016

PROG

(PARI) Vec(x*(1 +x +x^2 +x^3 +x^4)^2 * (1 +2*x^5 +3*x^10 +x^15) / (1 -3*x^15) + O(x^100)) \\ Colin Barker, Nov 18 2016

CROSSREFS

Paradigm shift sequences for q=5: A103969, A246074, A246075, A246076, A246079, A246083, A246087, A246091, A246095, A246099, A246103.

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

Sequence in context: A034837 A096105 A178157 * A246090 A175782 A272325

Adjacent sequences:  A246080 A246081 A246082 * A246084 A246085 A246086

KEYWORD

nonn,easy

AUTHOR

Jonathan T. Rowell, Aug 13 2014

STATUS

approved

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Last modified May 29 20:42 EDT 2020. Contains 334710 sequences. (Running on oeis4.)