A066321 -id:A066321 - cp's OEIS frontend

A256441 Binary representation of base-(i-1) expansion of -n: replace i-1 with 2 in base-(i-1) expansion of -n.

0, 29, 28, 17, 16, 205, 204, 193, 192, 221, 220, 209, 208, 7437, 7436, 7425, 7424, 7453, 7452, 7441, 7440, 7629, 7628, 7617, 7616, 7645, 7644, 7633, 7632, 7181, 7180, 7169, 7168, 7197, 7196, 7185, 7184, 7373, 7372, 7361, 7360, 7389, 7388, 7377, 7376, 4365

Offset: 0

Paul Tek, Mar 29 2015

Here i = sqrt(-1).
From Jianing Song, Jan 22 2023: (Start)
Also binary representation of base-(-1-i) expansion of -n.
Write out -n in base -4 (A212526), change each digit 0, 1, 2, 3 to 0000, 0001, 1100, 1101 respectively, then interpret as a binary number. (End)

			a(5) = 205 = 2^7 + 2^6 + 2^3 + 2^2 + 2^0 since (i-1)^7 + (i-1)^6 + (i-1)^3 + (i-1)^2 + (i-1)^0 = -5.

Paul Tek, Table of n, a(n) for n = 0..10000
Paul Tek, Perl program for this sequence
Wikipedia, Complex-base system

Cf. A066321.

a(n) = my(v = [-n,0], x=0, digit=0, a, b); while(v!=[0,0], a=v[1]; b=v[2]; v[1]=-2*(a\2)+b; v[2]=-(a\2); x+=(a%2)*2^digit; digit++); x \\ Jianing Song, Jan 22 2023; [a,b] represents the number a + b*(-1+i)

Perl
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See Links section.
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For n >= 1, a(4*n-0..3) = 16 * A066321(n) + 0, 1, 12, 13 respectively. - Jianing Song, Jan 22 2023

A066322 Variable terms in period-16 pattern of first differences of binary representation of base i-1 expansion of n.

Original entry on oeis.org

451, 3011, 451, 115651, 451, 3011, 451, -7229, 451, 3011, 451, -48189, 451, 3011, 451, -7229, 451, 3011, 451, 771011, 451, 3011, 451, -7229, 451, 3011, 451, -48189, 451, 3011, 451, -7229, 451, 3011, 451, 115651, 451, 3011, 451, -7229, 451, 3011, 451, -48189, 451, 3011, 451, -7229

Offset: 0

Marc LeBrun, Dec 14 2001

This sequence is very highly structured, but does it have a reasonable description?

Presumably, yes: see the comments to A066321. - Andrey Zabolotskiy, Feb 06 2017

			a(0) = A066321(0*16+4) - A066321(0*16+3) = 464 - 13 = 451

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 172, (Also exercise 16, p. 177, answer, p. 494)

Andrey Zabolotskiy, Table of n, a(n) for n = 0..65535

Cf. A066321, A282137.

A282137 Expansion of (24x^2-10x-1)/(16x^3-16x^2+x-1).

Original entry on oeis.org

1, 11, -29, -189, 451, 3011, -7229, -48189, 115651, 771011, -1850429, -12336189, 29606851, 197379011, -473709629, -3158064189, 7579354051, 50529027011, -121269664829, -808464432189, 1940314637251, 12935430915011, -31045034196029, -206966894640189

Offset: 0

Andrey Zabolotskiy, Feb 06 2017

Related to base i-1 representation of integers (Khmelnik encoding): presumably a(0) is the most common first difference of A066321 (occurs with density 1/2), a(1) is the second most common difference (density 1/4), a(2) has density 1/8, and so on; in particular, A066322 consists entirely of the terms a(n) with n>3.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-16,16).

Cf. A066321, A066322, A218723.

LinearRecurrence[{0,0,0,257,0,0,0,-256}, {1, 11, -29, -189, 451, 3011, -7229, -48189}, 24]
LinearRecurrence[{1, -16, 16}, {1, 11, -29}, 24]

Vec((1 - 2*x)*(1 + 12*x) / ((1 - x)*(1 + 16*x^2)) + O(x^30)) \\ Colin Barker, Feb 07 2017

print([[1, 11, -29, -189][n%4] + [450, 3000, -7200, -48000][n%4]*(256**(n//4)-1)//255 for n in range(24)])

a(k+8) - 257 * a(k+4) + 256 * a(k) = 0, for k >= 0. - Altug Alkan, Feb 07 2017
G.f.: (24*x^2-10*x-1)/(16*x^3-16*x^2+x-1).
From Colin Barker, Feb 07 2017: (Start)
a(n) = (-13 + (15+25*i)*(-4*i)^n + (15-25*i)*(4*i)^n) / 17 where i=sqrt(-1).
a(n) = a(n-1) - 16*a(n-2) + 16*a(n-3) for n>2.
(End)

A320283 Lexicographical ordering of pure imaginary integers in the base (-1+i) numeral system.

Original entry on oeis.org

0, 1, -2, -1, -4, -3, -6, -5, 8, 9, 6, 7, 4, 5, 2, 3, 16, 17, 14, 15, 12, 13, 10, 11, 24, 25, 22, 23, 20, 21, 18, 19, -32, -31, -34, -33, -36, -35, -38, -37, -24, -23, -26, -25, -28, -27, -30, -29, -16, -15, -18, -17, -20, -19, -22, -21, -8, -7, -10, -9, -12, -11, -14, -13, -64, -63, -66, -65, -68, -67, -70, -69

Offset: 0

Andreas K. Badea, Oct 09 2018

For ordering of pure real integers in same system see A073791.

All integers appear in this sequence.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..8191 (terms up to 255 from Andreas K. Badea)
Solomon I. Khmelnik, Specialized Digital Computer for Operations with Complex Numbers, Questions of Radio Electronics, 12 (1964), 60-82 [in Russian].
W. J. Penney, A "binary" system for complex numbers, NSA Technical Journal, Vol. X, No. 2 (1965), 13-15.
W. J. Penney, A "binary" system for complex numbers, JACM 12 (1965), 247-248.

Cf. A066321, A256441, A073791.

From Andrey Zabolotskiy, Jan 31 2019: (Start)
a(n) = A073791(2*n)/2.
a(n) = -a(4*n)/4.
a(n) = -4*a(floor(n/4)) + a(n mod 4). (End)

A355431 Numbers k whose binary expansion, when interpreted in base -1+i, gives a Gaussian prime.

Original entry on oeis.org

2, 5, 6, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 27, 31, 33, 37, 39, 41, 43, 49, 51, 53, 57, 58, 59, 63, 69, 71, 73, 77, 81, 83, 89, 97, 99, 101, 111, 113, 117, 119, 123, 127, 129, 131, 133, 137, 139, 141, 147, 159, 163, 169, 177, 183, 191, 193, 197, 201, 207

Offset: 1

John-Vincent Saddic, Jul 17 2022

Complex base -1+i is a bijection between integers k and Gaussian integers z(k) = A318438(k) + A318439(k)*i.
The present sequence is those k where z(k) is a Gaussian prime.
The Gaussian primes have an 8-way symmetry in the complex plane so that this sequence is also the Gaussian primes in the conjugate complex base -1-i.
The graphs on the complex plane (see links) show the Gaussian primes mapped and connected by lines in the order in which their indices appear in {a(n)}. The numbers in base -1+i tile the complex plane in the twin dragon fractal pattern, and the Gaussian primes are numerous such that the fractal is still discernible.
The only even terms are 2, 6, 14, and 58, since even terms correspond to Gaussian integers divisible by -1+i, and the base-(-1+i) expansions of -1+i, -1-i, 1+i, and 1-i are 10, 110, 1110, and 111010 respectively. - Jianing Song, Oct 02 2022

			123 is a term since z(123) = 2+7i is a Gaussian prime.
124 is not a term because z(124) = 2+4i is not a Gaussian prime.

John-Vincent Saddic, Graphs on the complex plane
John-Vincent Saddic, Julia program
John-Vincent Saddic, Python program

Cf. A066321 (real integers in base -1+i).

Julia
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Python
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A363955 When the base-2 representation of n is interpreted as a Gaussian integer x+yi in base (-1+i), both x and y are nonnegative.

Original entry on oeis.org

0, 1, 3, 8, 9, 10, 11, 12, 13, 14, 15, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 119, 120, 121, 122, 123, 124, 125, 126, 127, 256, 257, 258, 259

Offset: 1

Jeffrey Shallit, Jun 29 2023

nonn

			9 is in the sequence, since 9 in base 2 is 1001, which is 3+2i in base (-1+i).

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge, 2003, Section 14.5.

W. Penney, A "binary" system for complex numbers, J. ACM 12 (1965), 247-248.

Cf. A066321, A271472.

Select[Range[0, 260], Min[ReIm[FromDigits[IntegerDigits[#, 2], I - 1]]] >= 0 &] (* Amiram Eldar, Jun 29 2023 *)

There is an automaton of 21 states that accepts the base-2 representation of members of this sequence.

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A256441 Binary representation of base-(i-1) expansion of -n: replace i-1 with 2 in base-(i-1) expansion of -n.

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A066322 Variable terms in period-16 pattern of first differences of binary representation of base i-1 expansion of n.

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A282137 Expansion of (24x^2-10x-1)/(16x^3-16x^2+x-1).

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A320283 Lexicographical ordering of pure imaginary integers in the base (-1+i) numeral system.

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A355431 Numbers k whose binary expansion, when interpreted in base -1+i, gives a Gaussian prime.

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Keywords

Comments

Examples

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Programs

Julia

Python

A363955 When the base-2 representation of n is interpreted as a Gaussian integer x+yi in base (-1+i), both x and y are nonnegative.

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Keywords

Examples

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Mathematica

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