Yosu Yurramendi - cp's OEIS frontend

A334999 Permutation of nonnegative integers that converts A162911 into A162912.

1, 3, 2, 5, 6, 7, 4, 9, 14, 11, 8, 13, 10, 15, 12, 17, 22, 19, 16, 21, 26, 23, 20, 25, 30, 27, 24, 29, 18, 31, 28, 33, 38, 35, 32, 37, 58, 39, 36, 41, 46, 43, 40, 45, 34, 47, 44, 49, 54, 51, 48, 53, 42, 55, 52, 57, 62, 59, 56, 61, 50, 63, 60

Offset: 1

Yosu Yurramendi, May 19 2020

nonn

For n > 0, A162911(a(n)) = A162912(n), as well as A162911(A054429(n)) = A162912(n). There is a self-inverse permutation, b, such that a(b(n)) = A054429(n), and a(n) = A054429(b(n)).

Yosu Yurramendi, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Cf. A334998 (inverse), A054429, A162911, A162912.

nmax <- 31
A334998 <- 1; A334999 <- 1
for(n in 1:31) {
  A334998[2*n  ] <- 2*A334999[n]+1
  A334998[2*n+1] <- 2*n
  A334999[2*n  ] <- 2*n+1
  A334999[2*n+1] <- 2*A334998[n]
}
A334998; A334999

A334998 Permutation of nonnegative integers that converts A162912 into A162911.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 6, 11, 8, 13, 10, 15, 12, 9, 14, 19, 16, 29, 18, 23, 20, 17, 22, 27, 24, 21, 26, 31, 28, 25, 30, 35, 32, 45, 34, 39, 36, 33, 38, 43, 40, 53, 42, 47, 44, 41, 46, 51, 48, 61, 50, 55, 52, 49, 54, 59, 56, 37, 58, 63, 60, 57, 62

Offset: 1

Yosu Yurramendi, May 19 2020

nonn

For n > 0, A162912(a(n)) = A162911(n), as well as A162912(A054429(n)) = A162911(n). There is a self-inverse permutation, b, such that a(b(n)) = A054429(n), and a(n) = A054429(b(n)).

Yosu Yurramendi, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Cf. A334999 (inverse), A054429, A162911, A162912.

#
  nmax <- 31
A334998 <- 1; A334999 <- 1
for(n in 1:31) {
  A334998[2*n  ] <- 2*A334999[n]+1
  A334998[2*n+1] <- 2*n
  A334999[2*n  ] <- 2*n+1
  A334999[2*n+1] <- 2*A334998[n]
}
A334998; A334999

A332769 Permutation of the positive integers: a(n) = A258996(A054429(n)) = A054429(A258996(n)).

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 6, 13, 12, 15, 14, 9, 8, 11, 10, 21, 20, 23, 22, 17, 16, 19, 18, 29, 28, 31, 30, 25, 24, 27, 26, 53, 52, 55, 54, 49, 48, 51, 50, 61, 60, 63, 62, 57, 56, 59, 58, 37, 36, 39, 38, 33, 32, 35, 34, 45, 44

Offset: 1

Yosu Yurramendi, Feb 23 2020

nonn

Sequence is self-inverse: a(a(n)) = n.
A002487(1+a(n)) = A162911(n) and A002487(a(n)) = A162912(n). So, a(n) generates the enumeration system of positive rationals based on Stern's sequence A002487 called 'drib'.
Given n, one can compute a(n) by taking into account the binary representation of n, and by flipping every second bit starting from the lowest until reaching the highest 1, which is not flipped.

			n = 23 =  10111_2
            x x
          10010_2 = 18 = a(n).
n = 33 = 100001_2
          x x x
         110100_2 = 52 = a(n).

Yosu Yurramendi, Table of n, a(n) for n = 1..8191

Similar R-programs: A258996, A284447.

a(n) = bitxor(n, 2<Kevin Ryde, Mar 30 2021

maxrow <- 6 # by choice
a <- 1
for(m in 0:maxrow) for(k in 0:(2^m-1)){
a[2^(m+1)+2*k  ] <- 2*a[2^(m+1)-1-k] + 1
a[2^(m+1)+2*k+1] <- 2*a[2^(m+1)-1-k]
}
a

# Given n, compute a(n) by taking into account the binary representation of n
maxblock <- 7 # by choice
a <- c(1, 3, 2)
for(n in 4:2^maxblock){
  ones <- which(as.integer(intToBits(n)) == 1)
  nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
  anbit <- nbit
  anbit[seq(1, length(anbit) - 1, 2)] <- 1 - anbit[seq(1, length(anbit) - 1, 2)]
  a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
a
# Yosu Yurramendi, Mar 30 2021

a(A054429(n)) = A054429(a(n)) = A258996(n),
a(A258996(n)) = A258996(a(n)) = A054429(n).
a(n) = A284447(A065190(n)) = A065190(A284447(n)),
a(A065190(n)) = A065190(a(n)) = A284447(n),
a(A284447(n)) = A284447(a(n)) = A065190(n).
a(A231551(n)) = A154437(n), a(A154437(n)) = A231551(n).
a(A153154(n)) = A284459(n), a(A284459(n)) = A153154(n).
a(1) = 1, a(2) = 3, a(3) = 2; for n > 3, a(2*n) = 2*a(A054429(n)) + 1, a(2*n+1) = 2*a(A054429(n)).
a(1) = 1; for m >= 0 and 0 <= k < 2^m, a(2^(m+1)+2*k) = 2*a(2^(m+1)-1-k) + 1, a(2^(m+1)+2*k+1) = 2*a(2^(m+1)-1-k).

A286921 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 10 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 10, 1, 55, 2575, 1, 550, 253000, 250525000, 1, 5050, 25007500, 250025500000, 2500000075000000, 1, 50500, 2500300000, 250002775000000, 25000000255000000000, 2500000000502500000000000, 1, 500500, 250000750000, 250000250500000000, 250000000000750000000000, 250000000000250500000000000000, 250000000000000000750000000000000000

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Computed using Burnsides orbit-counting lemma.

			Triangle begins:
==============================================================
n\m |   0   1      2          3              4
----|---------------------------------------------------------
0   |   1
1   |   1   10
2   |   1   55     2575
3   |   1   550    253000     250525000
4   |   1   5050   25007500   250025500000   2500000075000000
...

María Merino, Rows n=0..32 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433, A283434, A286893, A286895, A286919, A286920.

For even n and m: T(n,m) = (10^(m*n) + 3*10^(m*n/2))/4;
for even n and odd m: T(n,m) = (10^(m*n) + 10^((m*n+n)/2) + 2*10^(m*n/2))/4;
for odd n and even m: T(n,m) = (10^(m*n) + 10^((m*n+m)/2) + 2*10^(m*n/2))/4;
for odd n and m: T(n,m) = (10^(m*n) + 10^((m*n+n)/2) + 10^((m*n+m)/2) + 10^((m*n+1)/2))/4.

A286919 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 8 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 8, 1, 36, 1072, 1, 288, 66816, 33693696, 1, 2080, 4197376, 17184194560, 70368756760576, 1, 16640, 268517376, 8796399206400, 288230393868451840, 9444732983468915425280, 1, 131328, 17180065792, 4503616874348544, 1180591620768950910976, 309485009825866260538195968, 81129638414606695206587887255552

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Computed using Burnsides orbit-counting lemma.

			Triangle begins:
========================================================
n\m |   0   1      2        3             4
----|---------------------------------------------------
0   |   1
1   |   1   8
2   |   1   36     1072
3   |   1   288    66816    33693696
4   |   1   2080   4197376  17184194560   70368756760576
...

María Merino, Rows n=0..35 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433, A283434, A286893, A286895.

For even n and m: T(n,m) = (8^(m*n) + 3*8^(m*n/2))/4;
for even n and odd m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 2*8^(m*n/2))/4;
for odd n and even m: T(n,m) = (8^(m*n) + 8^((m*n+m)/2) + 2*8^(m*n/2))/4;
for odd n and m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 8^((m*n+m)/2) + 8^((m*n+1)/2))/4.

A286920 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 9 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 9, 1, 45, 1701, 1, 405, 134865, 97135605, 1, 3321, 10766601, 70618411521, 463255079498001, 1, 29889, 871858485, 51473762336565, 3039416437115008521, 179474497026544179696969, 1, 266085, 70607782701, 37523729625344145, 19941610769429949618201, 10597789568841677482963905405, 5632099886234793715531013441442501

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Computed using Burnsides orbit-counting lemma.

			Triangle begins:
==========================================================
n\m |   0   1     2         3              4
----|-----------------------------------------------------
0   |   1
1   |   1   9
2   |   1   45    1701
3   |   1   405   134865    97135605
4   |   1   3321  10766601  70618411521    463255079498001
...

María Merino, Rows n=0..33 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433, A283434, A286893, A286895, A286919.

For even n and m: T(n,m) = (9^(m*n) + 3*9^(m*n/2))/4;
for even n and odd m: T(n,m) = (9^(m*n) + 9^((m*n+n)/2) + 2*9^(m*n/2))/4;
for odd n and even m: T(n,m) = (9^(m*n) + 9^((m*n+m)/2) + 2*9^(m*n/2))/4;
for odd n and m: T(n,m) = (9^(m*n) + 9^((m*n+n)/2) + 9^((m*n+m)/2) + 9^((m*n+1)/2))/4.

A286893 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 6 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 6, 1, 21, 351, 1, 126, 12096, 2544696, 1, 666, 420876, 544638816, 705278736576, 1, 3996, 15132096, 117564302016, 914040184444416, 7107572245840091136, 1, 23436, 544230576, 25390538401536, 1184595336212990976, 55268479955808421134336, 2578606199622710056510488576

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Computed using Burnside's orbit-counting lemma.

			Triangle begins:
============================================================================
n\m |  0  1     2         3             4                5
----|-----------------------------------------------------------------------
0   |  1
1   |  1  6
2   |  1  21    351
3   |  1  126   12096     2544696
4   |  1  666   420876    544638816     705278736576
5   |  1  3996  15132096  117564302016  914040184444416  7107572245840091136
...

María Merino, Rows n=0..36 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433, A283434.

For even n and m: T(n,m) = (6^(m*n) + 3*6^(m*n/2))/4;
for even n and odd m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 2*6^(m*n/2))/4;
for odd n and even m: T(n,m) = (6^(m*n) + 6^((m*n+m)/2) + 2*6^(m*n/2))/4;
for odd n and m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 6^((m*n+m)/2) + 6^((m*n+1)/2))/4.

A286895 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 7 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 7, 1, 28, 637, 1, 196, 30184, 10151428, 1, 1225, 1443001, 3461821825, 8308236966001, 1, 8575, 70656628, 1186972525900, 19948070175962425, 335267157313994232775, 1, 58996, 3460410037, 407106879976216, 47895307855522569001, 5634835073082541702198396, 662932711464914589254954278237

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Computed using Burnside's orbit-counting lemma.

			Triangle begins:
============================================================================
n\m |  0  1    2        3             4                 5
----|-----------------------------------------------------------------------
0   |  1
1   |  1  7
2   |  1  28   637
3   |  1  196  30184    10151428
4   |  1  1225 1443001  3461821825    8308236966001
5   |  1  8575 70656628 1186972525900 19948070175962425 335267157313994232775
...

María Merino, Rows n=0..35 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433, A283434, A286893.

For even n and m: T(n,m) = (7^(m*n) + 3*7^(m*n/2))/4;
for even n and odd m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 2*7^(m*n/2))/4;
for odd n and even m: T(n,m) = (7^(m*n) + 7^((m*n+m)/2) + 2*7^(m*n/2))/4;
for odd n and m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 7^((m*n+m)/2) + 7^((m*n+1)/2))/4.

A283434 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 5 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 5, 1, 15, 175, 1, 75, 4125, 496875, 1, 325, 98125, 61140625, 38147265625, 1, 1625, 2446875, 7632421875, 23841923828125, 74505821533203125, 1, 7875, 61046875, 953736328125, 14901161376953125, 232830644622802734375, 3637978807094573974609375

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Computed using Burnside's orbit-counting lemma.

			Triangle begins:
============================================================================
n\m |   0   1      2         3            4                5
----|-----------------------------------------------------------------------
0   |   1
1   |   1   5
2   |   1   15     175
3   |   1   75     4125      496875
4   |   1   325    98125     61140625     38147265625
5   |   1   1625   2446875   7632421875   23841923828125   74505821533203125
...

María Merino, Rows n=0..38 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433.

For even n and m: T(n,m) = (5^(m*n) + 3*5^(m*n/2))/4;
for even n and odd m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 2*5^(m*n/2))/4;
for odd n and even m: T(n,m) = (5^(m*n) + 5^((m*n+m)/2) + 2*5^(m*n/2))/4;
for odd n and m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 5^((m*n+m)/2) + 5^((m*n+1)/2))/4.

A283433 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 4 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 4, 1, 10, 76, 1, 40, 1120, 67840, 1, 136, 16576, 4212736, 1073790976, 1, 544, 263680, 268779520, 274882625536, 281475530358784, 1, 2080, 4197376, 17184194560, 70368756760576, 288230393868451840, 1180591620768950910976

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Computed using Burnside's orbit-counting lemma.

			Triangle begins:
=======================================================================
n\m |  0   1       2        3           4               5
----|------------------------------------------------------------------
0   |  1
1   |  1    4
2   |  1    10     76
3   |  1    40     1120     67840
4   |  1    136    16576    4212736     1073790976
5   |  1    544    263680   268779520   274882625536    281475530358784
...

María Merino, Rows n=0..41 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432.

For even n and m: T(n,m) = (4^(m*n) + 3*4^(m*n/2))/4;
for even n and odd m: T(n,m) = (4^(m*n) + 4^((m*n+n)/2) + 2*4^(m*n/2))/4;
for odd n and even m: T(n,m) = (4^(m*n) + 4^((m*n+m)/2) + 2*4^(m*n/2))/4;
for odd n and m: T(n,m) = (4^(m*n) + 4^((m*n+n)/2) + 4^((m*n+m)/2) + 4^((m*n+1)/2))/4.

cp's OEIS Frontend

User: Yosu Yurramendi

A334999 Permutation of nonnegative integers that converts A162911 into A162912.

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A334998 Permutation of nonnegative integers that converts A162912 into A162911.

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A332769 Permutation of the positive integers: a(n) = A258996(A054429(n)) = A054429(A258996(n)).

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A286921 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 10 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A286919 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 8 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A286920 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 9 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A286893 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 6 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A286895 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 7 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A283434 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 5 colors and n>=m, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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