A345760 -id:A345760 - cp's OEIS frontend

A307163 Minimum number of intercalates in a diagonal Latin square of order n.

0, 0, 0, 12, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Eduard I. Vatutin, Mar 27 2019

An intercalate is a 2 X 2 subsquare of a Latin square.
Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A307164(n) <= A092237(n). - Eduard I. Vatutin, Sep 21 2020
Every intercalate is a partial loop and every partial loop is a loop, so 0 <= a(n) <= A307170(n) <= A307166(n). - Eduard I. Vatutin, Oct 19 2020
a(n)=0 for all orders n for which cyclic diagonal Latin squares exist (see A007310) due to all cyclic diagonal Latin squares don't have intercalates. - Eduard I. Vatutin, Aug 07 2023
a(n)=0 for all orders n for which diagonalized cyclic diagonal Latin squares exist (see A372922) due to all diagonalized cyclic diagonal Latin squares don't have intercalates. - Eduard I. Vatutin, Sep 24 2024
a(16) <= 2, a(17) = 0, a(18) <= 9, a(19) = 0, a(20) <= 1, a(21) <= 11, a(22) <= 9, a(23) = 0, a(24) <= 16, a(25) = 0, a(26) <= 29. - Eduard I. Vatutin, added Sep 10 2023, updated Mar 01 2025

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
E. I. Vatutin, About the minimum number of intercalates in a diagonal Latin squares of order 9 (in Russian)
E. I. Vatutin, On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops (in Russian)
Eduard I. Vatutin, About the heuristic approximation of the spectrum of number of intercalates in diagonal Latin squares of order 14 (in Russian)
Eduard I. Vatutin, About the minimum number of intercalates in diagonal Latin squares of order 15 (in Russian)
E. Vatutin, A. Belyshev, N. Nikitina, and M. Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, Communications in Computer and Information Science, Vol. 1304, Springer, 2020, pp. 127-146, DOI: 10.1007/978-3-030-66895-2_9.
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A007310, A307164, A092237, A307170, A307166, A345760, A372922.

a(9) added by Eduard I. Vatutin, Sep 21 2020
a(10)-a(13) added by Eduard I. Vatutin, Apr 01 2021
a(14)-a(15) added by Eduard I. Vatutin, Sep 24 2024

A307164 Maximum number of intercalates in a diagonal Latin square of order n.

Original entry on oeis.org

0, 0, 0, 12, 4, 9, 30, 112, 72

Offset: 1

Eduard I. Vatutin, Mar 27 2019

An intercalate is a 2 X 2 subsquare of a Latin square.
0 <= A307163(n) <= A307164(n) <= A092237(n). - Eduard I. Vatutin, Sep 21 2020
a(10) >= 109, a(11) >= 172, a(12) >= 324, a(13) >= 180, a(14) >= 391, a(15) >= 630, a(16) >= 960, a(17) >= 736, a(18) >= 547, a(19) >= 457, a(20) >= 1100, a(21) >= 785, a(22) >= 887, a(23) >= 899, a(24) >= 1680, a(25) >= 1700, a(26) >= 1299, a(27) >= 1372, a(28) >= 2892. - Eduard I. Vatutin, May 31 2021, updated Mar 02 2025
If, in theory, all unordered pairs of rows and columns form intercalate in their intersection, total number of intercalates will be (n*(n-1))^2, so a(n) <= (n*(n-1))^2, a(n) is asymptotically less than O(n^4). In practice a(n) << (n*(n-1))^2. - Eduard I. Vatutin, Mar 05 2025

			From _Eduard I. Vatutin_, May 31 2021: (Start)
One of the best known diagonal Latin squares of order n=5
  0 1 2 3 4
  4 2 0 1 3
  1 4 3 2 0
  3 0 1 4 2
  2 3 4 0 1
has 4 intercalates:
  . . 2 3 .   . . . . .   . . . . .   . . . . .
  . . . . .   . . 0 . 3   . . . . .   . . . . .
  . . 3 2 .   . . 3 . 0   1 . 3 . .   . 4 3 . .
  . . . . .   . . . . .   3 . 1 . .   . . . . .
  . . . . .   . . . . .   . . . . .   . 3 4 . .
so a(5)=4. (End)

Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, About the maximum number of intercalates in a diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the heuristic approximation of the spectrum of number of intercalates in diagonal Latin squares of order 14 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer, Cham (2020), 127-146.
Eduard I. Vatutin, Natalia N. Nikitina, and Maxim O. Manzuk, First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, Alexandr M. Albertyan, and Ilya I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021), Tula, 2021, pp. 7-17 (in Russian).
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
Index entries for sequences related to Latin squares and rectangles.

Cf. A092237, A307163, A345760.

a(9) added by Eduard I. Vatutin, Sep 21 2020

A344105 a(n) is the number of distinct numbers of transversals of order n diagonal Latin squares.

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 32, 73, 406

Offset: 1

Eduard I. Vatutin, Jun 22 2021

a(n) <= A287644(n) - A287645(n) + 1.
a(n) <= A287764(n).
Diagonal Latin squares are a special case of Latin squares, so a(n) <= A309344(n).
a(10) >= 459, a(11) >= 6437, a(12) >= 23707, a(13) >= 75891, a(14) >= 290681. - Eduard I. Vatutin, Oct 29 2021, updated Mar 01 2025
For all spectra of even orders all known values included in them are divisible by 2. For all spectra of orders n=6, n=10 and n=14 (and probably for all n=4k+2) all known values included in the corresponding spectra are divisible by 4. This leads to the following hypothesis: a(2k) <= (A287644(2k) - A287645(2k) + 2)/2 and a(4k+2) <= (A287644(4k+2) - A287645(4k+2) + 4)/4, where w(n) = A287644(n) - A287645(n) + 1 is a width of corresponding spectra and (w(n)+1)/2 is done to round the result of the division up. - Eduard I. Vatutin, Mar 21 2022

			For n=7 the number of transversals that a diagonal Latin square of order 7 may have is 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 41, 43, 45, 47, 55, or 133. Since there are 32 distinct values, a(7)=32.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, About the approximation of spectra of numerical characteristics of diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021), Tula, 2021, pp. 7-17. (in Russian)
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, and A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, and I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, and A. M. Albertyan, On the number of transversals in diagonal Latin squares of even orders (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
Index entries for sequences related to Latin squares and rectangles.

Cf. A287644, A287645, A287764, A309344, A345370, A345760, A345761.

a(8) added by Eduard I. Vatutin, Jul 14 2021

a(9) added by Eduard I. Vatutin, Nov 20 2022

A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 14, 47, 182

Offset: 1

Eduard I. Vatutin, Jun 16 2021

a(n) <= A287648(n) - A287647(n) + 1.
a(n) <= A287764(n).
Conjecture: a(12) = A287648(12) - A287647(12) + 1. - Natalia Makarova, Oct 26 2021
a(10) >= 736, a(11) >= 1344, a(12) >= 17693, a(13) >= 18241, a(14) >= 294053, a(15) >= 1958394, a(16) >= 13715. - Eduard I. Vatutin, Oct 29 2021, updated Mar 02 2025

			For n=7 the number of diagonal transversals that a diagonal Latin square of order 7 may have is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, or 27. Since there are 14 distinct values, a(7)=14.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, On the falsity of conjecture that spectra of diagonal transversals for diagonal Latin squares of order 12 is solid (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
E. I. Vatutin, Distributed diagonalization strategy for Latin squares, Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2023. pp. 309-311. (in Russian)
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287647, A287648, A309344, A344105, A345760, A345761, A349199.

a(8) added by Eduard I. Vatutin, Jul 15 2021

a(9) added by Eduard I. Vatutin, Oct 20 2022

A345761 a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 3, 31, 99

Offset: 1

Eduard I. Vatutin, Jun 26 2021

a(n) <= A287695(n) + 1.
a(n) <= A287764(n).
a(10) >= 10. It seems that a(10) = 10 due to long computational experiments within the Gerasim@Home volunteer distributed computing project did not reveal the existence of diagonal Latin squares of order 10 with the number of orthogonal diagonal Latin squares different from {0, 1, 2, 3, 4, 5, 6, 7, 8, 10}.
a(11) >= 112, a(12) >= 5079. - Eduard I. Vatutin, Nov 02 2021, updated Jan 23 2023

			For n=7 the number of orthogonal diagonal Latin squares that a diagonal Latin square of order 7 may have is 0, 1, or 3. Since there are 3 distinct values, a(7)=3.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, About the spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the lower bound for a spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 10 (in Russian).
Eduard I. Vatutin, About the lower bound for a spectrum of orthogonal diagonal Latin squares for one diagonal Latin squares of order 11 (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12).
Index entries for sequences related to Latin squares and rectangles.

Cf. A287695, A287764, A309344, A344105, A345370, A345760.

A354050 a(n) is the number of distinct numbers of intercalates that an orthogonal diagonal Latin square of order n can have.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 3, 26, 55

Offset: 1

Eduard I. Vatutin, May 16 2022

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate. Since all orthogonal diagonal Latin squares are diagonal Latin squares, a(n) <= A345760(n).

a(10) >= 74, a(11) >= 76, a(12) >= 190. - updated by Eduard I. Vatutin, Mar 01 2025

			For n=8 the number of intercalates that an orthogonal diagonal Latin square of order 8 may have is 2, 4, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 36, 40, 44, 48, 52, 56, 64, 80, or 112. Since there are 26 distinct values, a(8)=26.

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (4, 5, 7, 8, 9, 10, 11, 12).
Index entries for sequences related to Latin squares and rectangles.

Cf. A345760, A349199, A350585.

A368182 a(n) is the number of distinct numbers of intercalates in Latin squares of order n.

Original entry on oeis.org

1, 1, 1, 2, 2, 9, 23, 61

Offset: 1

Eduard I. Vatutin, Feb 15 2024

a(9)>=64, a(10)>=103, a(11)>=145, a(12)>=259, a(13)>=200, a(14)>=362, a(15)>=536, a(16)>=794, a(17)>=705, a(18)>=655, a(19)>=469, a(20)>=1362, a(21)>=985, a(22)>=1435, a(23)>=967, a(24)>=1754, a(25)>=1679, a(26)>=2040, a(28)>=2803. - Eduard I. Vatutin, added Aug 13 2024, updated Jun 25 2025

			For n=7, a Latin square of order 7 may have 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 26, 30, or 42 intercalates. There are 23 possibilities, so a(7)=23.

Eduard I. Vatutin, About the intercalates spectra in a Latin squares of orders 1-8 (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28).
Index entries for sequences related to Latin squares and rectangles.

Cf. A092237, A309344, A345760.

A382270 Maximum number of intercalates in a Brown's diagonal Latin square of order 2n.

Original entry on oeis.org

0, 12, 9, 112, 57

Offset: 1

Eduard I. Vatutin, Mar 20 2025

A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square (see A339641).
Brown's diagonal Latin squares are special case of plain symmetry diagonal Latin squares that do not exist for odd orders.
a(6)>=252, a(7)>=385, a(8)>=960, a(9)>=329, a(10)>=356, a(11)>=497, a(12)>=1008, a(13)>=497, a(14)>=524.

Eduard I. Vatutin, About the spectra of numerical characteristics of Brown's diagonal Latin squares (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A092237, A307163, A307164, A339641, A345760, A368182, A379665.

A382957 a(n) is the number of distinct numbers of intercalates in an extended self-orthogonal diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 3, 8, 52, 45

Offset: 1

Eduard I. Vatutin, Apr 10 2025

A self-orthogonal diagonal Latin square (SODLS, see A329685) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS, see A309210) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

			For n=7 the number of intercalates that an extended self-orthogonal diagonal Latin square of order 7 may have is 0, 10, or 18. Since there are 3 distinct values, a(7)=3.

Eduard I. Vatutin, Proving lists (1, 4, 5, 7, 8, 9, 10).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A309210, A309598, A309599, A329685, A345760, A382952.

cp's OEIS Frontend

A307163 Minimum number of intercalates in a diagonal Latin square of order n.

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A307164 Maximum number of intercalates in a diagonal Latin square of order n.

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A344105 a(n) is the number of distinct numbers of transversals of order n diagonal Latin squares.

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A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

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A345761 a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.

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A354050 a(n) is the number of distinct numbers of intercalates that an orthogonal diagonal Latin square of order n can have.

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A368182 a(n) is the number of distinct numbers of intercalates in Latin squares of order n.

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A382270 Maximum number of intercalates in a Brown's diagonal Latin square of order 2n.

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A382957 a(n) is the number of distinct numbers of intercalates in an extended self-orthogonal diagonal Latin squares of order n.

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