A117967 -id:A117967 - cp's OEIS frontend

A297154 Balanced ternary representation of A083254(n), 2*phi(n)-n.

1, 0, 1, 0, 3, 7, 17, 0, 3, 7, 9, 8, 14, 7, 1, 0, 51, 21, 47, 8, 3, 7, 48, 19, 51, 7, 9, 8, 27, 67, 32, 0, 16, 7, 13, 24, 38, 7, 9, 19, 39, 63, 161, 8, 3, 7, 153, 68, 38, 20, 13, 8, 141, 63, 34, 19, 51, 7, 138, 56, 152, 7, 9, 0, 31, 55, 149, 8, 46, 71, 105, 57, 101, 7, 17, 8, 160, 60, 89, 68, 27, 7, 81, 72, 160, 7

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A083254, A117966, A117967, A117968.

Cf. also A297153, A295882.

(define (A297154 n) (let ((x (A083254 n))) (if (>= x 0) (A117967 x) (A117968 (- x)))))

If A083254(n) >= 0, then a(n) = A117967(A083254(n)), otherwise a(n) = A117968(-A083254(n)).

For all n >= 1, A117966(a(n)) = A083254(n).

A366793 Binary encoding of the ones in the balanced ternary representation of Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 3, 1, 0, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 4, 0, 2, 0, 1, 2, 0, 0, 3, 0, 1, 0, 2, 1, 0, 1, 2, 2, 0, 0, 3, 0, 2, 0, 1, 0, 3, 2, 0, 3, 0, 3, 4, 1, 0, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 4, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 3, 1, 2, 1, 0, 2, 1, 0, 4, 1, 0

Offset: 0

Antti Karttunen, Oct 24 2023

nonn

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

			A004718(254) = -7. In balanced ternary representation (see A117966) this is represented as -1*9 + 1*3 + -1*1. Taking the positive coefficients, and converting them to a binary string gives "10", which in base-2 (A007088) is equal to 2, therefore a(254) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A004718, A007088, A117966, A289813, A323909, A366794.

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ From the code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A323909(n) = { my(x = A004718(n)); if(x >= 0,A117967(x),A117968(-x)); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A366793(n) = A289813(A323909(n));

a(n) = A289813(A323909(n)).

A366794 Binary encoding of the twos (-1's) in the balanced ternary representation of Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 2, 3, 3, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 2, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 2, 4, 2, 0, 0, 2, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 2, 3, 3, 0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 1, 2, 0, 2, 0, 0, 0, 0, 2, 3, 3, 0, 0

Offset: 0

Antti Karttunen, Oct 24 2023

nonn

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

			A004718(254) = -7. In balanced ternary representation (see A117966) this is represented as -1*9 + 1*3 + -1*1. Taking the negative coefficients, and converting them to a binary string gives "101", which in base-2 (A007088) is equal to 5, therefore a(254) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A004718, A007088, A117966, A289814, A323909, A366793.

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ From the code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A323909(n) = { my(x = A004718(n)); if(x >= 0,A117967(x),A117968(-x)); };
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A366794(n) = A289814(A323909(n));

a(n) = A289814(A323909(n)).

cp's OEIS Frontend

A297154 Balanced ternary representation of A083254(n), 2*phi(n)-n.

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A366793 Binary encoding of the ones in the balanced ternary representation of Per Nørgård's "infinity sequence".

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A366794 Binary encoding of the twos (-1's) in the balanced ternary representation of Per Nørgård's "infinity sequence".

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Examples

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PARI

Formula