

A193455


Paradigm shift sequence with procedure length p=3.


10



1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 16, 20, 25, 30, 36, 42, 49, 64, 80, 100, 125, 150, 180, 216, 256, 320, 400, 500, 625, 750, 900, 1080, 1296, 1600, 2000, 2500, 3125, 3750, 4500, 5400, 6480, 8000, 10000, 12500, 15625, 18750, 22500, 27000, 32400, 40000, 50000
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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 action, bundle all existing actions (which requires p=3 steps), or apply the current bundled action. The first use of a novel bundle erases (or makes obsolete) all prior actions. How many total actions (simple) can be applied in n time steps?"
1. This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as ParadigmShift Sequences with Procedural Lengths p=1 and 2, respectively.
2. The optimal number of pastes per copy, as measured by the geometric growth rate (p+z root of z), is z = 5. [Noninteger maximum between 4 and 5.]
3. The function a(n) = maximum value of the product of the terms k_i, where Sum (k_i) = n+ 3  3*i_max
4. All solutions will be of the form a(n) = m^b * (m+1)^d


LINKS



FORMULA

a(n) =
a(25) = 256 [C = 4 below]
a(1:24) = m^(CR) * (m+1)^R
where C = floor((n+6)/8) [min C=1],
R = n+3 mod C, m = floor((n+33*C)/C)
a(n>=26) = 4^b * 5^(C(b+d)) * 6^d
where C = floor((n+6)/8), R = n+6 mod 8,
b = max(0,3R), and d = max(0, R3)
Recursive: a(n) = 5*a(n8) for all n >= 34


EXAMPLE

For n=20, C = floor(26/8) = 3, R = (23 mod 3) = 2, m = floor (239/3) = floor(14/3)=4; therefore a(20) = 4^(32)*5^(2) = 4*5^2 = 100.
For n=25, the same general formula is used, but C=4 (instead of 3). R=28 mod 4 =0, m = floor(2812/4)=4; therefore a(25) = 4^4 = 256.
For n=35, C = floor(41/8)=5, R = 1, b = max(0,2)=2, d=max(0,2)=0; therefore a(35) = 4^2*5^(52)*6^0 = 2000.


PROG

(Python)
def a(n):
c=(n + 6)//8
if n<25:
if n<10: return n
r=(n + 3)%c
m=(n + 3  3*c)//c
return m**(c  r)*(m + 1)**r
elif n==25: return 256
else:
r=(n + 6)%8
b=max(0, 3  r)
d=max(0, r  3)
return 4**b*5**(c  (b + d))*6**d


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



