A360845 - OEIS

A360845

Triangle read by rows: T(n,k) is the state of the k-th light after n steps of the light switch problem, 1 <= k <= A003418(n).

1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1

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OFFSET

COMMENTS

The light switch problem posits an infinite number of ordinally numbered lights which are initially off.

The 1st step turns all lights on.

The 2nd step turns every second one off leaving only odd lights illuminated.

The 3rd step reverses the state of every light having a number divisible by 3.

Every n-th step thereafter reverses the state of lights with numbers divisible by n.

The problem asks which lights are on as n is allowed to be arbitrarily large and the solution is all n where d(n) (A000005) is odd, i.e., the squares A000290. Alternatively, if 0 represents a light that is off and 1 a light that is on, the solution is represented by A010052 with offset 1.

This sequence considers intermediate solutions to arrive at A010052. After the n-th step, the lights will have a pattern which must repeats at most every LCM of {1..n} (sequence A003418). This sequence is an irregular triangle read by rows of the n-th repeating sequence.

LINKS

Table of n, a(n) for n=1..81.

This is illustrated from 2:10 in The Light Switch Problem Numberphile video

FORMULA

T(n,k) = (Sum_{d|k, d<=n} 1) mod 2.

T(n,k) = A138553(n,k) mod 2.

T(n,k) = A010052(k) for n >= k.

EXAMPLE

Triangle begins:

1,0;

1,0,0,0,1,1;

1,0,0,1,1,1,1,1,0,0,1,0;