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A246083
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Paradigm shift sequence for (0,5) production scheme with replacement.
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13
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Jonathan T. Rowell, Aug 13 2014
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STATUS
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approved
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