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User: Luis Rato

Luis Rato has authored 4 sequences.

A381490 Index of second half of decomposition of integers into pairs x(i)+y(j) based on A380008 and A380009, respectively.

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

Offset: 0

Luis Rato, Feb 25 2025

One coordinate of a recursive non-self-intersecting walk on the square lattice Z^2.

Every integer appears infinitely many times in the sequence.

			A380008( A381489(14)) + A380009(a(14)) = A380008(3) + A380009(2) = 6+8 = 14

Luis Rato, Plot of an NZ-order curve, containing the integers from 0 to 255.
Index entries for sequences related to coordinates of 2D curves

Cf. A381489

Cf. A380008 (indices of 0's), A380009.

n = A380008(A381489(n)) + A380009(a(n))

A381489 Index of first half of decomposition of integers into pairs x(i)+y(j) based on A380008 and A380009, respectively.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 4, 4, 5, 5, 6, 6, 7, 7, 0, 0, 1, 1, 2, 2, 3, 3, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 4, 4, 5, 5, 6, 6, 7, 7, 0, 0, 1, 1, 2, 2, 3, 3, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 4, 4, 5, 5, 6, 6, 7, 7

Offset: 0

Luis Rato, Feb 25 2025

One coordinate of a recursive non-self-intersecting walk on the square lattice Z^2.

Every integer appears infinitely many times in the sequence.

			A380008(a(14)) + A380009(A381490(14)) = A380008(3) + A380009(2) = 6+8 = 14.

Luis Rato, Plot of an NZ-order curve, containing the integers from 0 to 255.
Index entries for sequences related to coordinates of 2D curves

Cf. A380008, A380009, A381490.

n = A380008(a(n)) + A380009(A381490(n)).

A380008 Numbers t whose binary expansion Sum 2^e_i has exponents e_i which are odious numbers (A000069).

Original entry on oeis.org

0, 2, 4, 6, 16, 18, 20, 22, 128, 130, 132, 134, 144, 146, 148, 150, 256, 258, 260, 262, 272, 274, 276, 278, 384, 386, 388, 390, 400, 402, 404, 406, 2048, 2050, 2052, 2054, 2064, 2066, 2068, 2070, 2176, 2178, 2180, 2182, 2192, 2194, 2196, 2198, 2304, 2306, 2308, 2310, 2320, 2322, 2324, 2326, 2432, 2434, 2436, 2438, 2448, 2450, 2452, 2454

Offset: 0

Luis Rato, Jan 08 2025

These t in binary representation have 1s only in positions with 0s in the Thue-Morse sequence (A010059) with beginning of that sequence corresponding to least significant bit. a(n) can be derived from n by placing the bits of n into a(n) at those permitted positions.
a(n) can be represented in base 4 equal to binary representation of n with each digit multiplied by 1 or 2 according to the 1-2 Thue-Morse sequence A001285 starting in the least significant digit and transforming 1->2, and 2->1.
Any pair 2*p and 2*p+1 has one evil and the other odious number, so the bit at position p in n goes to either 2*p or 2*p+1 in a(n), according as which of those is odious.
Every integer k>=0 corresponds to a unique pair i,j with k = x(i) + y(j), with x(i)=a(i) and y(j)=A380009(j).
Sequences x(n) and y(n) have same growth rate and cross an infinite number of times.
Coordinate pairs (i,j), define a Morton space-filling curve, similar to Z-order curve.

			Considering the representation in base 4,
For n=11 = 1011_binary, a(11) -> 1021_base4 -> 2012_base4 = 134.
For n=12 = 1100_binary, a(12) -> 1200_base4 -> 2100_base4 = 144.
Considering all numbers are decomposed in binary, with exponents belonging to odious numbers: 1, 2, 4, 7,...
The sequence of terms together with their binary representation begins:
 n    a(n)      a(n)_bin
 0     0:         0 ~               0
 1     2:        10 ~             2^1
 2     4:       100 ~         2^2
 3     6:       110 ~         2^2+2^1
 4    16:     10000 ~     2^4
 5    18:     10010 ~     2^4   +2^1
 6    20:     10100 ~     2^4+2^2
 7    22:     10110 ~     2^4+2^2+2^1
 8   128:  10000000 ~ 2^7
 9   130:  10000010 ~ 2^7        +2^1
10   132:  10000100 ~ 2^7    +2^2
11   134:  10000110 ~ 2^7    +2^2+2^1
12   144:  10010000 ~ 2^7+2^4

Luis Rato, Plot of an NZ-order curve, containing the integers from 0 to 255.
Wikipedia, Morton code map, also known as Z-order curve.

Cf. A000069, A001285, A010059, A380009.

a(n) = { my (v = 0, e); while (n, n -= 2^e = exponent(n); v += 2^(2*e + if (hammingweight(e)%2, 0, 1));); return (v); } \\ Rémy Sigrist, Feb 02 2025

isok(t) = my(b=Vecrev(binary(t))); for (i=1, #b, if (b[i] && !(hammingweight(i-1)%2), return(0))); return(1); \\ Michel Marcus, Feb 10 2025

A380009 Numbers t whose binary expansion Sum 2^e_i has exponents e_i which are evil numbers (A001969).

Original entry on oeis.org

0, 1, 8, 9, 32, 33, 40, 41, 64, 65, 72, 73, 96, 97, 104, 105, 512, 513, 520, 521, 544, 545, 552, 553, 576, 577, 584, 585, 608, 609, 616, 617, 1024, 1025, 1032, 1033, 1056, 1057, 1064, 1065, 1088, 1089, 1096, 1097, 1120, 1121, 1128, 1129, 1536, 1537, 1544, 1545, 1568, 1569, 1576, 1577, 1600, 1601, 1608, 1609, 1632, 1633, 1640, 1641

Offset: 0

Luis Rato, Jan 09 2025

These t in binary representation have 1s only in positions with 0s in the Thue-Morse sequence (A010059) with beginning of that sequence corresponding to least significant bit. a(n) can be derived from n by placing the bits of n into a(n) at those permitted positions.
a(n) can be represented in base 4 equal to binary representation of n with each digit multiplied by 1 or 2 according to the 1-2 Thue-Morse sequence A001285 starting in the least significant digit.
Any pair 2*p and 2*p+1 has one evil and the other odious, so the bit at position p in n goes to either 2*p or 2*p+1 in a(n), according as which of those is evil.
Every integer k>=0 corresponds to a unique pair i,j with k = x(i) + y(j), with x(i)=a(i) and y(j)=A380008(j).
Sequences x(n) and y(n) have same growth rate and cross an infinite number of times.
Coordinate pairs (i,j), define a Morton space-filling curve, similar to Z-order curve.

			For n=11 = 1011_binary, a(11) = 1021_base4 = 41.
All numbers are also decomposed in binary, with exponents belonging to evil numbers: 0, 3, 5, 6, ...
The sequence of terms begins:
 n    a(n)      a(n)_bin
 0     0:         0 ~               0
 1     1:         1 ~             2^0
 2     8:      1000 ~         2^3
 3     9:      1001 ~         2^3+2^0
 4    32:    100000 ~     2^5
 5    33:    100001 ~     2^5    +2^0
 6    40:    101000 ~     2^5+2^3
 7    41:    101001 ~     2^5+2^3+2^0
 8    64:   1000000 ~ 2^6
 9    65:   1000001 ~ 2^6        +2^0
10    72:   1001000 ~ 2^6    +2^3
11    73:   1001001 ~ 2^6    +2^3+2^0

Luis Rato, Plot of an NZ-order curve, containing the integers from 0 to 255.
Wikipedia, Morton code map, also known as Z-order curve.

Cf. A000069, A001285, A001969, A010059, A380008.

isok(k) = my(b=binary(k), v=apply(x->#b-x, Vec(select(x->x, b, 1)))); #v == #select(x->(hammingweight(x)%2==0), v); \\ Michel Marcus, Jan 11 2025

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User: Luis Rato

A381490 Index of second half of decomposition of integers into pairs x(i)+y(j) based on A380008 and A380009, respectively.

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A381489 Index of first half of decomposition of integers into pairs x(i)+y(j) based on A380008 and A380009, respectively.

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A380008 Numbers t whose binary expansion Sum 2^e_i has exponents e_i which are odious numbers (A000069).

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A380009 Numbers t whose binary expansion Sum 2^e_i has exponents e_i which are evil numbers (A001969).

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PARI