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A246088
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Paradigm shift sequence for (2,2) production scheme with replacement.
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8
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 54, 63, 72, 84, 96, 112, 128, 144, 162, 189, 216, 252, 288, 336, 384, 448, 512, 576, 648, 756, 864, 1008, 1152, 1344, 1536, 1792, 2048, 2304, 2592, 3024, 3456, 4032, 4608, 5376, 6144, 7168, 8192, 9216, 10368, 12096, 13824, 16128, 18432, 21504, 24576, 28672, 32768, 36864, 41472
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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=2 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 = 4.
4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,4).
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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) = 4*a(n-10) for all n >= 32.
G.f.: x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +7*x^10 +4*x^11 +3*x^12 +2*x^13 +x^14 +x^20 +6*x^21 +3*x^22 +2*x^29 +9*x^30) / ((1 -2*x^5) * (1 +2*x^5)). - Colin Barker, Nov 19 2016
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PROG
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(PARI) Vec(x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +6*x^5 +7*x^6 +8*x^7 +9*x^8 +10*x^9 +7*x^10 +4*x^11 +3*x^12 +2*x^13 +x^14 +x^20 +6*x^21 +3*x^22 +2*x^29 +9*x^30) / ((1 -2*x^5) * (1 +2*x^5)) + O(x^100)) \\ Colin Barker, Nov 19 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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