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 Paradigm-Shift 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. [Non-integer 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
Jonathan T. Rowell, Solution Sequences for the Keyboard Problem and its Generalizations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.10.7.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,5).
FORMULA
a(n) =
a(25) = 256 [C = 4 below]
a(1:24) = m^(C-R) * (m+1)^R
where C = floor((n+6)/8) [min C=1],
R = n+3 mod C, m = floor((n+3-3*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(0,3-R), and d = max(0, R-3)
Recursive: a(n) = 5*a(n-8) for all n >= 34
EXAMPLE
For n=20, C = floor(26/8) = 3, R = (23 mod 3) = 2, m = floor (23-9/3) = floor(14/3)=4; therefore a(20) = 4^(3-2)*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(28-12/4)=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^(5-2)*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
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 27 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jonathan T. Rowell, Jul 26 2011
STATUS
approved