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=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.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,3).
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-14) for all n >= 33.
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 +12*x^11 +13*x^12 +14*x^13 +12*x^14 +10*x^15 +8*x^16 +6*x^17 +5*x^18 +4*x^19 +3*x^20 +2*x^21 +x^22 +x^30 +2*x^31) / (1 -3*x^14). - Colin Barker, Nov 18 2016
MATHEMATICA
Join[Range[18], LinearRecurrence[PadLeft[{3}, 14], {20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 52, 56}, 55]] (* Jean-François Alcover, Sep 25 2017 *)
PROG
(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 +12*x^11 +13*x^12 +14*x^13 +12*x^14 +10*x^15 +8*x^16 +6*x^17 +5*x^18 +4*x^19 +3*x^20 +2*x^21 +x^22 +x^30 +2*x^31) / (1 -3*x^14) + O(x^100)) \\ Colin Barker, Nov 18 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jonathan T. Rowell, Aug 13 2014
STATUS
approved