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A246084 Paradigm shift sequence for (1,2) production scheme with replacement. 9
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 21, 24, 28, 32, 36, 45, 54, 63, 72, 84, 96, 112, 135, 162, 189, 216, 252, 288, 336, 405, 486, 567, 648, 756, 864, 1008, 1215, 1458, 1701, 1944, 2268, 2592, 3024, 3645, 4374, 5103, 5832, 6804, 7776, 9072, 10935, 13122, 15309, 17496, 20412, 23328, 27216, 32805, 39366, 45927 (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=1 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.

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 (0,0,0,0,0,0,3).

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-7) for all n >= 26.

G.f.: x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +5*x^7 +3*x^8 +x^9 +x^15 +2*x^16 +4*x^24) / (1 -3*x^7). - Colin Barker, Nov 19 2016

PROG

(PARI) Vec(x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +5*x^7 +3*x^8 +x^9 +x^15 +2*x^16 +4*x^24) / (1 -3*x^7) + 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=1: A178715, A246084, A246085, A246086, A246087.

Sequence in context: A102576 A101170 A236686 * A260768 A130224 A017903

Adjacent sequences:  A246081 A246082 A246083 * A246085 A246086 A246087

KEYWORD

nonn,easy

AUTHOR

Jonathan T. Rowell, Aug 13 2014

STATUS

approved

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)