A283978 -id:A283978 - cp's OEIS frontend

A283988 a(n) = A002487(n-1) AND A002487(n), where AND is bitwise-and (A004198).

0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 4, 4, 3, 0, 0, 5, 2, 2, 5, 0, 0, 3, 4, 4, 1, 0, 4, 1, 0, 0, 3, 2, 2, 3, 8, 8, 5, 4, 4, 1, 0, 0, 1, 4, 4, 5, 8, 8, 3, 2, 2, 3, 0, 0, 1, 4, 0, 1, 6, 2, 1, 4, 8, 9, 4, 4, 11, 2, 2, 1, 0, 8, 1, 2, 10, 3, 0, 0, 5, 0, 0, 1, 0, 0, 3, 0, 0, 9, 2, 2, 9, 0, 0, 3, 0, 0, 1, 0, 0, 5, 0, 0, 3, 10, 2, 1, 8, 0, 1, 2, 2, 11, 4

Offset: 1

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Odd bisection of A283978.
Cf. A002487, A004198, A007306, A283986, A283987.
Cf. A283973 (positions of zeros), A283974 (nonzeros).

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[BitAnd[a[n - 1], a@ n], {n, 120}] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
for(n=1, 120, print1(bitand(A(n - 1), A(n)),", ")) \\ Indranil Ghosh, Mar 23 2017

from functools import reduce
def A283988(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0)))&sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n-1)[-1:2:-1],(1,0))) if n>1 else 0 # Chai Wah Wu, May 05 2023

(define (A283988 n) (A004198bi (A002487 (- n 1)) (A002487 n)))  ;; Where A004198bi implements bitwise-AND (A004198).

a(n) = A002487(n-1) AND A002487(n), where AND is bitwise-and (A004198).
a(n) = A283986(n) - A283987(n).
a(n) = A007306(n) - A283986(n) = (A007306(n) - A283987(n))/2.
a(n) = A283978((2*n)-1).

A283976 a(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 2, 3, 1, 3, 3, 3, 2, 3, 3, 3, 1, 5, 4, 7, 3, 7, 5, 7, 2, 7, 5, 7, 3, 7, 4, 5, 1, 5, 5, 5, 4, 7, 7, 7, 3, 11, 8, 13, 5, 7, 7, 7, 2, 7, 7, 7, 5, 13, 8, 11, 3, 7, 7, 7, 4, 5, 5, 5, 1, 7, 6, 7, 5, 13, 9, 13, 4, 15, 11, 15, 7, 15, 10, 11, 3, 11, 11, 11, 8, 13, 13, 13, 5, 13, 12, 15, 7, 15, 9, 11, 2, 11, 9, 15, 7, 15, 12, 13, 5, 13, 13, 13, 8

Offset: 0

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Bisections: A002487, A283986.

Cf. A002487, A003986, A283977, A283978.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[If[EvenQ@ n, a[n/2], BitOr[a[#], a[# + 1]] &[(n - 1)/2]], {n, 0, 108}] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
a(n) = if(n<2, n, if(n%2, bitor(A(n\2), A((n + 1)/2)), A(n\2)));
for(n=0, 101, print1(a(n),", ")) \\ Indranil Ghosh, Mar 23 2017

(define (A283976 n) (if (even? n) (A002487 n) (A003986bi (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))) ;; Where A003986bi implements bitwise-OR (A003986).

a(2n) = A002487(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).
a(n) = A283977(n) + A283978(n).
a(n) = A002487(n) - A283978(n).

A283977 a(2n) = A002487(n), a(2n+1) = A002487(n) XOR A002487(n+1), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 2, 3, 1, 2, 3, 1, 2, 1, 3, 2, 1, 5, 4, 7, 3, 6, 5, 7, 2, 7, 5, 6, 3, 7, 4, 5, 1, 4, 5, 1, 4, 3, 7, 4, 3, 11, 8, 13, 5, 2, 7, 5, 2, 5, 7, 2, 5, 13, 8, 11, 3, 4, 7, 3, 4, 1, 5, 4, 1, 7, 6, 3, 5, 12, 9, 13, 4, 15, 11, 12, 7, 13, 10, 9, 3, 8, 11, 3, 8, 5, 13, 8, 5, 9, 12, 11, 7, 14, 9, 11, 2, 11, 9, 14, 7, 11, 12, 9, 5, 8, 13, 5, 8, 3, 11, 8, 3

Offset: 0

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Bisections: A002487, A283987.

Cf. A003987, A283976, A283978.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[If[EvenQ@ n, a[n/2], BitXor[a[#], a[# + 1]] &[(n - 1)/2]], {n, 0, 112}] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
a(n) = if(n<2, n, if(n%2, bitxor(A(n\2), A((n + 1)/2)), A(n\2)));
for(n=0, 120, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 23 2017

(define (A283977 n) (if (even? n) (A002487 n) (A003987bi (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(2n) = A002487(2n) = A002487(n), a(2n+1) = A002487(n) XOR A002487(n+1), where XOR is bitwise-xor (A003987).
a(n) = A283976(n) - A283978(n).
a(n) = A002487(n) - 2*A283978(n).

cp's OEIS Frontend

A283988 a(n) = A002487(n-1) AND A002487(n), where AND is bitwise-and (A004198).

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A283976 a(2n) = A002487(n), a(2n+1) = A002487(n) OR A002487(n+1), where OR is bitwise-or (A003986).

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Formula

A283977 a(2n) = A002487(n), a(2n+1) = A002487(n) XOR A002487(n+1), where XOR is bitwise-xor (A003987).

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Mathematica

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Formula