A318723 -id:A318723 - cp's OEIS frontend

A318722 Let f(0) = 0 and f(t4^k + u) = i^t ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the real part of g(n).

-1, -1, 0, -1, -2, -2, -2, -2, -1, 0, 1, 1, -2, -3, -3, -1, -1, -2, -3, -4, -4, -4, -4, -3, -3, -3, -2, -3, -4, -4, -4, -4, -3, -2, -1, -1, 1, 2, 2, 0, 0, 1, 2, 3, 3, 3, 3, 2, -4, -5, -5, -3, -3, -4, -5, -6, -6, -6, -6, -5, -2, -2, -3, -2, -1, -1, -1, -1, -2

Offset: 0

Rémy Sigrist, Sep 02 2018

See A318723 for the imaginary part of g.
See A318724 for the square of the modulus of g.
This sequence can be computed by considering the base 4 representation of A042968, hence the keyword base.
The function g runs uniquely through the set of Gaussian integers z such that Re(z) < 0 or Im(z) < 0.
The function g is related to the numbering of the cells in a Chair tiling (see representation of g(n) in Links section).
This sequence has similarities with A316657.

Rémy Sigrist, Table of n, a(n) for n = 0..12287
Rémy Sigrist, Colored scatterplot of (a(n), A318723(n)) for n = 0..3*4^9-1 (where the hue is function of n)
Rémy Sigrist, Colored scatterplot of (a(n), A318723(n)) for n = 0..3*4^9-1 (where the color is function of the sum of digits of A042968(n) in base 4)
Rémy Sigrist, Representation of g(n) for n = 0..3*4^2-1 in the complex plane, alongside the base 4 representation of A042968(n)
Tilings Encyclopedia, Chair

Cf. A042968, A048647, A318723, A318724.

a(n) = my (d=Vecrev(digits(1+n+n\3,4)), z=0); for (k=1, #d, if (d[k], z = I^d[k] * (-z + (1+I) * 2^(k-1)))); real((z-1-I)/2)

a(n) = A318723(n) iff the base 4 representation of A042968(n) contains only 0's and 2's.

If A048647(A042968(m)) = A042968(n), then a(m) = A318723(n) and A318723(m) = a(n).

A318724 Let f(0) = 0 and f(t4^k + u) = i^t ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the square of the modulus of g(n).

Original entry on oeis.org

1, 2, 1, 2, 5, 4, 5, 8, 5, 4, 5, 2, 8, 13, 10, 5, 10, 13, 18, 25, 20, 17, 16, 9, 13, 18, 13, 10, 17, 20, 25, 32, 25, 20, 17, 10, 10, 13, 8, 9, 16, 17, 20, 25, 18, 13, 10, 5, 32, 41, 34, 25, 34, 41, 50, 61, 52, 45, 40, 29, 29, 40, 45, 20, 17, 26, 37, 50, 53, 58

Offset: 0

Rémy Sigrist, Sep 02 2018

See A318722 for the real part of g and additional comments.

Rémy Sigrist, Table of n, a(n) for n = 0..12287

Cf. A042968, A048647, A318722, A318723.

a(n) = my (d=Vecrev(digits(1+n+n\3,4)), z=0); for (k=1, #d, if (d[k], z = I^d[k] * (-z + (1+I) * 2^(k-1)))); norm((z-1-I)/2)

a(n) = A318722(n)^2 + A318723(n)^2.

If A048647(A042968(m)) = A042968(n), then a(m) = a(n).

cp's OEIS Frontend

A318722 Let f(0) = 0 and f(t4^k + u) = i^t ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the real part of g(n).

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A318724 Let f(0) = 0 and f(t4^k + u) = i^t ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the square of the modulus of g(n).

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cp's OEIS Frontend

A318722 Let f(0) = 0 and f(t*4^k + u) = i^t * ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the real part of g(n).

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A318724 Let f(0) = 0 and f(t*4^k + u) = i^t * ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the square of the modulus of g(n).

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Programs

PARI

Formula

A318722 Let f(0) = 0 and f(t4^k + u) = i^t ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the real part of g(n).

A318724 Let f(0) = 0 and f(t4^k + u) = i^t ((1+i) * 2^k - f(u)) for any t in {1, 2, 3} and k >= 0 and u such that 0 <= u < 4^k (i denoting the imaginary unit); for any n >= 0, let g(n) = (f(A042968(n)) - 1 - i) / 2; a(n) is the square of the modulus of g(n).