The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A178715 a(n) = solution to the "Select All, Copy, Paste" problem: Given the ability to type a single letter, or to type individual "Select All", "Copy" or "Paste" command keystrokes, what is the maximal number of letters of text that can be obtained with n keystrokes? 36
1, 2, 3, 4, 5, 6, 9, 12, 16, 20, 27, 36, 48, 64, 81, 108, 144, 192, 256, 324, 432, 576, 768, 1024, 1296, 1728, 2304, 3072, 4096, 5184, 6912, 9216, 12288, 16384, 20736, 27648, 36864, 49152, 65536, 82944, 110592, 147456, 196608, 262144, 331776, 442368, 589824, 786432, 1048576, 1327104, 1769472, 2359296, 3145728, 4194304 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
It is assumed that we start with a single letter in the copy buffer.
Alternatively, a(n-1) = maximal value of Product (k_i-1) for any way of writing n = Sum k_i.
1. The description above assumes that the text is deselected after the Copy command is invoked.
2. This sequence is the solution to the equivalent problem formulated as {insert, "Select All+ Copy" macro (without deselection), Paste}.
3. This sequence is a "paradigm-shift" sequence with procedure length p =1 (in the sense of A193455).
4. The optimal number of pastes per copy, as measured by the geometric growth rate (p+z root of z), is z = 4. [noninteger maximum between 3 and 4]
5. The function a(n) = maximum value of the product of the terms k_i, where Sum (k_i) = n+1-i_max.
6. All solutions will be of the form a(n) = m^b * (m+1)^d.
LINKS
William Boyles, Table of n, a(n) for n = 1..8303 (terms 1..101 from Joerg Arndt) (all terms with at most 1000 digits)
Jonathan T. Rowell, Solution Sequences for the Keyboard Problem and its Generalizations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.10.7.
FORMULA
a(n) = 4*a(n-5) for n>=16.
a(n) =
a(5;10) = 5; 20 [C=1, 2 below, respectively]
a(n=1:14) = Q^(C-R)*(Q+1)^R
where C = floor(n/5)+1, R = n+1 mod C,
and Q = floor(n+1/C)-1
a(n>=15) = 3^(4-R)*4^(C-4+R)
where C = floor (n/5)+1, R = n mod 5.
G.f.: x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +2*x^5 +x^6 +3*x^10 +x^14) / (1 -4*x^5). - Colin Barker, Nov 19 2016
EXAMPLE
For n = 7 the a(7) = 9 solution is to type the seven keystrokes: paste, paste, paste, select-all, copy, paste paste which yields nine text characters.
Here is a table showing the pattern for n = 1 to 35. The first column is n and the second column is the number of characters that can be obtained with n keystrokes. The remainder of the line shows how to get the maximum, as follows. S = Select and C = Copy while a dot stands for Paste. The dots at the beginning of a line are equivalent to a single letter being typed, based on the assumption that at the start there is a single letter in the paste buffer.
01: 00001 .
02: 00002 ..
03: 00003 ...
04: 00004 ....
05: 00005 .....
06: 00006 ......
07: 00009 ...SC..
08: 00012 ....SC..
09: 00016 ....SC...
10: 00020 .....SC...
11: 00027 ...SC..SC..
12: 00036 ....SC..SC..
13: 00048 ....SC...SC..
14: 00064 ....SC...SC...
15: 00081 ...SC..SC..SC..
16: 00108 ....SC..SC..SC..
17: 00144 ....SC...SC..SC..
18: 00192 ....SC...SC...SC..
19: 00256 ....SC...SC...SC...
20: 00324 ....SC..SC..SC..SC..
21: 00432 ....SC...SC..SC..SC..
22: 00576 ....SC...SC...SC..SC..
23: 00768 ....SC...SC...SC...SC..
24: 01024 ....SC...SC...SC...SC...
25: 01296 ....SC...SC..SC..SC..SC..
26: 01728 ....SC...SC...SC..SC..SC..
27: 02304 ....SC...SC...SC...SC..SC..
28: 03072 ....SC...SC...SC...SC...SC..
29: 04096 ....SC...SC...SC...SC...SC...
30: 05184 ....SC...SC...SC..SC..SC..SC..
31: 06912 ....SC...SC...SC...SC..SC..SC..
32: 09216 ....SC...SC...SC...SC...SC..SC..
33: 12288 ....SC...SC...SC...SC...SC...SC..
34: 16384 ....SC...SC...SC...SC...SC...SC...
35: 20736 ....SC...SC...SC...SC..SC..SC..SC..
It appears that A000792 is the result if only one keystroke instead of two is required for the "Select All, Copy" operation. Here is the table. Here "C" means that all the previously typed characters are copied to the paste buffer.
01: 00001 .
02: 00002 ..
03: 00003 ...
04: 00004 ....
05: 00006 ...C.
06: 00009 ...C..
07: 00012 ....C..
08: 00018 ...C..C.
09: 00027 ...C..C..
10: 00036 ....C..C..
11: 00054 ...C..C..C.
12: 00081 ...C..C..C..
13: 00108 ....C..C..C..
14: 00162 ...C..C..C..C.
15: 00243 ...C..C..C..C..
16: 00324 ....C..C..C..C..
17: 00486 ...C..C..C..C..C.
18: 00729 ...C..C..C..C..C..
19: 00972 ....C..C..C..C..C..
20: 01458 ...C..C..C..C..C..C.
21: 02187 ...C..C..C..C..C..C..
22: 02916 ....C..C..C..C..C..C..
23: 04374 ...C..C..C..C..C..C..C.
24: 06561 ...C..C..C..C..C..C..C..
25: 08748 ....C..C..C..C..C..C..C..
26: 13122 ...C..C..C..C..C..C..C..C.
27: 19683 ...C..C..C..C..C..C..C..C..
28: 26244 ....C..C..C..C..C..C..C..C..
29: 39366 ...C..C..C..C..C..C..C..C..C.
30: 59049 ...C..C..C..C..C..C..C..C..C..
31: 78732 ....C..C..C..C..C..C..C..C..C..
MATHEMATICA
LinearRecurrence[{0, 0, 0, 0, 4}, {1, 2, 3, 4, 5, 6, 9, 12, 16, 20, 27, 36, 48, 64, 81}, 60] (* Harvey P. Dale, Apr 11 2017 *)
PROG
(PARI) Vec(x*(1 +2*x +3*x^2 +4*x^3 +5*x^4 +2*x^5 +x^6 +3*x^10 +x^14) / (1 -4*x^5) + O(x^100)) \\ Colin Barker, Nov 19 2016
(Python)
def a(n):
c=(n//5) + 1
if n<15:
if n==5: return 5
if n==10: return 20
r=(n + 1)%c
q=((n + 1)//c) - 1
return q**(c - r)*(q + 1)**r
else:
r=n%5
return 3**(4 - r)*4**(c - 4 + r)
print([a(n) for n in range(1, 102)]) # Indranil Ghosh, Jun 27 2017
CROSSREFS
See A193286 for another version. Cf. A000792.
A000792, A178715, A193286, A193455, A193456, and A193457 are paradigm shift sequences with procedure lengths p=0,1,...,5, respectively.
Cf. A367116 (squares summing to n).
Sequence in context: A307818 A358033 A057492 * A018123 A105859 A200445
KEYWORD
nonn,nice,easy
AUTHOR
Bill Blewett, Jan 11 2011
EXTENSIONS
Edited by N. J. A. Sloane, Jul 21 2011
Additional comment and formula from David Applegate, Jul 21 2011
Additional comments, formulas, and CrossRefs by Jonathan T. Rowell, Jul 30 2011
More terms from Joerg Arndt, Nov 15 2014
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 24 22:09 EDT 2024. Contains 372782 sequences. (Running on oeis4.)