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Puzzle: sequence without multiples of 3 - Albert Chan - 09-13-2019 07:05 PM
What is the simplest formula that can generate: 1,2, 4,5, 7,8, 10,11 ... ?
In other words, sequence never generate multiples of 3.
f(1) = 1, f(2) = 2, f(3) = 4, f(4) = 5, f(5) = 7, f(6) = 8 ...
What is f(10^6) ?
RE: Puzzle: sequence without multiples of 3 - Voldemar - 09-13-2019 08:57 PM
[attachment=7670]
RE: Puzzle: sequence without multiples of 3 - Voldemar - 09-13-2019 09:02 PM
f(10^6) = (3*333334-2)
k=333334
RE: Puzzle: sequence without multiples of 3 - Albert Chan - 09-13-2019 09:43 PM
(09-13-2019 09:02 PM)Voldemar Wrote: f(10^6) = (3*333334-2)
k=333334
No, f(k) = k-th numbers from the sequence 1,2, 4,5, 7,8, 10,11, 13,14, ...
Example, f(10) = 14
RE: Puzzle: sequence without multiples of 3 - Thomas Okken - 09-14-2019 12:30 AM
f(n) = floor((n-1)*1.5)+1
This wouldn't be much of a puzzle if the most obvious formula happened to be the solution, but I thought I'd get it out of the way. As a baseline, if you will.
UPDATE to add the answer to the second question: f(10^6) = 1,499,999
RE: Puzzle: sequence without multiples of 3 - John Keith - 09-14-2019 12:54 AM
A simple program to generate the sequence, which is A001651:
Code:
\<< \-> n
\<< 1 1 n
START DUP 1 + DUP 2 +
NEXT n 2 * 1 + \->LIST
\>>
\>>Returns 2n+1 terms. Not much use to compute f(10^6) though.
RE: Puzzle: sequence without multiples of 3 - Albert Chan - 09-14-2019 01:01 AM
Hi, Thomas Okken
You got it!
I saw the formula from a book review, The Irrationals, by Julian Havil
The formula itself is trivial, but the procedure to get it can be used for complicated sequences.
Example: for non-squares sequence
Code:
F=n² 1 4 9 16 25 36 49 64 81
n 1 2 3 4 5 6 7 8 9
f=n²-n 0 2 6 12 20 30 42 58 72
f* 1 1 2 2 2 2 3 3 3
n + f* 2 3 5 6 7 8 10 11 12This assumed f is non-decreasing function.
f* is max k such that f(k) < n, thus we have 2x1, 4x2, 6x3, 8x4 ...
→ f*(n) = floor(√(n) + 0.5)
→ non_squares(n) = n + f* = n + floor(√(n) + 0.5)
RE: Puzzle: sequence without multiples of 3 - Gerald H - 09-14-2019 05:37 AM
INT((3*N-1)/2) for HP 38G.
RE: Puzzle: sequence without multiples of 3 - Albert Chan - 09-14-2019 08:29 AM
Here is the HP-12C code that produce f(n), f(n) < 10^10
Code:
Enter Enter Enter
2 / INTG 2 × − Enter × ; n%2
2 − + 2 ÷ + ; f(n) = n + (n + n%2 − 2) / 2