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A246078
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Paradigm shift sequence for (-1,4) production scheme with replacement.
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12
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 40, 44, 48, 54, 60, 66, 72, 81, 90, 99, 108, 120, 132, 144, 162, 180, 198, 216, 243, 270, 297, 324, 360, 396, 432, 486, 540, 594, 648, 729, 810, 891, 972, 1080, 1188, 1296, 1458, 1620, 1782, 1944, 2187, 2430, 2673, 2916, 3240, 3564, 3888, 4374
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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=-1 steps), or implement the current bundled action (which requires q=4 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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Index entries for linear recurrences with constant coefficients, signature (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-11) 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 +8*x^7 +9*x^8 +10*x^9 +11*x^10 +9*x^11 +7*x^12 +5*x^13 +4*x^14 +3*x^15 +2*x^16 +x^17 +x^23 +2*x^24) / (1 -3*x^11). - Colin Barker, Nov 18 2016
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MATHEMATICA
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CoefficientList[Series[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 + 11 x^10 + 9 x^11 + 7 x^12 + 5 x^13 + 4 x^14 + 3 x^15 + 2 x^16 + x^17 + x^23 + 2 x^24)/(1 - 3 x^11), {x, 0, 71}], x] (* Michael De Vlieger, Nov 18 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 +11*x^10 +9*x^11 +7*x^12 +5*x^13 +4*x^14 +3*x^15 +2*x^16 +x^17 +x^23 +2*x^24) / (1 -3*x^11) + O(x^100)) \\ Colin Barker, Nov 18 2016
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CROSSREFS
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Paradigm shift sequences with p<0: A103969, A246074, A246075, A246076, A246079, A029750, A246078, A029747, A246077, A029744, A029747, A131577.
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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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