A318703 -id:A318703 - cp's OEIS frontend

A318702 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let f(n) = Sum_{k=0..w} b_k * i^k * 2^floor(k/2) (where i denotes the imaginary unit); a(n) is the real part of f(n).

0, 1, 0, 1, -2, -1, -2, -1, 0, 1, 0, 1, -2, -1, -2, -1, 4, 5, 4, 5, 2, 3, 2, 3, 4, 5, 4, 5, 2, 3, 2, 3, 0, 1, 0, 1, -2, -1, -2, -1, 0, 1, 0, 1, -2, -1, -2, -1, 4, 5, 4, 5, 2, 3, 2, 3, 4, 5, 4, 5, 2, 3, 2, 3, -8, -7, -8, -7, -10, -9, -10, -9, -8, -7, -8, -7

Offset: 0

Rémy Sigrist, Sep 01 2018

See A318703 for the imaginary part of f.
See A318704 for the square of the modulus of f.
The function f defines a bijection from the nonnegative integers to the Gaussian integers.
This sequence has similarities with A316657.

Rémy Sigrist, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n
Rémy Sigrist, Colored scatterplot of (a(n), A318703(n)) for n = 0..2^19-1 (where the hue is function of n)

Cf. A053985, A059905, A316657, A318703, A318704.

Array[Re[Total@ MapIndexed[#1*I^(First@ #2 - 1)*2^Floor[(First@ #2 - 1)/2] &, Reverse@ IntegerDigits[#, 2]]] &, 76, 0] (* Michael De Vlieger, Sep 02 2018 *)

a(n) = my (b=Vecrev(binary(n))); real(sum(k=1, #b, b[k] * I^(k-1) * 2^floor((k-1)/2)))

a(n) = A053985(A059905(n)).

a(4 * k) = -2 * a(k) for any k >= 0.

A318704 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let f(n) = Sum_{k=0..w} b_k * i^k * 2^floor(k/2) (where i denotes the imaginary unit); a(n) is the square of the modulus of f(n).

Original entry on oeis.org

0, 1, 1, 2, 4, 1, 5, 2, 4, 5, 1, 2, 8, 5, 5, 2, 16, 25, 17, 26, 4, 9, 5, 10, 20, 29, 17, 26, 8, 13, 5, 10, 16, 17, 25, 26, 20, 17, 29, 26, 4, 5, 9, 10, 8, 5, 13, 10, 32, 41, 41, 50, 20, 25, 29, 34, 20, 29, 25, 34, 8, 13, 13, 18, 64, 49, 65, 50, 100, 81, 101

Offset: 0

Rémy Sigrist, Sep 01 2018

See A318702 for the real part of f and additional comments.

Rémy Sigrist, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A318702.

a(n) = my (b=Vecrev(binary(n))); norm(sum(k=1, #b, b[k] * I^(k-1) * 2^floor((k-1)/2)))

a(n) = A318702(n)^2 + A318703(n)^2.

a(4 * k) = 4 * a(k) for any k >= 0.

cp's OEIS Frontend

A318702 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let f(n) = Sum_{k=0..w} b_k * i^k * 2^floor(k/2) (where i denotes the imaginary unit); a(n) is the real part of f(n).

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