Andrey Zabolotskiy - cp's OEIS frontend

A383108 Number of crystallographic orbits in n dimensions, counting enantiomorphic pairs as distinct.

Original entry on oeis.org

1, 2, 30, 427

Offset: 0

Andrey Zabolotskiy, Apr 16 2025

Peter Engel, Geometric crystallography, D. Reidel Publishing Company, 1986. See Theorem 8.14 on p. 189.
Peter Engel, Geometric crystallography, in: P. M. Gruber and J. M. Wills, editors, Handbook of Convex Geometry, North-Holland, Amsterdam, 1993, Vol. B, pp. 989-1041. See Theorem 6.7 on p. 1027.

Online Dictionary of Crystallography, Crystallographic orbit.

Cf. A006227.

A382870 Minimum period of an optimum covering of the set of integers by translates of its subset with diameter no greater than n, maximized over such subsets.

Original entry on oeis.org

1, 2, 4, 5, 8, 8, 13, 13, 27, 27, 45, 53, 66, 109, 129, 147, 147, 170, 192, 250, 286, 317

Offset: 0

Andrey Zabolotskiy, Apr 07 2025

			For n = 3, the set S = {0, 1, 3} (diameter 3) covers the set of integers Z when translated by ...0, 1, 5, 6, 10, 11... with primitive period 5. Periodic coverings of Z by translates of S with smaller period are possible (e.g. by taking the entire Z as the set of translations) but they have greater density of overlaps and thus are not optimal. A different set can have a different period of an optimal covering, e.g. {0, 3} has the minimum period of 2 achieved by translations by ...0, 2, 4..., but a(n) maximizes over the subsets of diameter n, and the maximum is attained by S, so a(3) = 5.

Béla Bollobás, Svante Janson, and Oliver Riordan, On covering by translates of a set, Random Structures and Algorithms, 38 (2011), 33-67; arXiv:0910.3815 [math.CO], 2009-2010. See Table 1.

A381577 Unique sequence of 0's, 1's, and 2's such that for any terms x and y with x < y, the subsequence of x's and y's becomes the Thue-Morse sequence after substitution x -> 0, y -> 1.

Original entry on oeis.org

0, 1, 2, 2, 1, 0, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 0, 1, 2, 2, 1, 0, 2, 1, 0, 0, 1, 2, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 2, 1, 0, 0, 1, 2, 0, 1, 2, 2, 1, 0, 2, 1, 0, 0, 1, 2, 2, 1, 0

Offset: 0

Andrey Zabolotskiy, Feb 28 2025

Cf. the Thue-Morse sequence A010060 and its other ternary generalizations: A053838, A287150.

def A381577(n): return ((2,1,0) if (n//3).bit_count()&1 else (0,1,2))[n%3] # Chai Wah Wu, Feb 28 2025

In the Thue-Morse sequence, substitute 0 -> 012, 1 -> 210 once.

A377569 Number of simple graphs such that each connected component is nonseparable and the number of vertices minus the number of connected components equals n.

Original entry on oeis.org

1, 1, 2, 5, 16, 75, 560, 7772, 202546, 9955274, 911146844, 154541913254, 48588413940171, 28410569347709449, 31024350279787141361, 63532688288261802284578, 244915643061880269492533777, 1783405573307429828266152750816, 24605670701967180148649252153837623

Offset: 0

Andrey Zabolotskiy, Nov 01 2024

nonn

Euler transform of A002218, shifted by 1.

A376073 Number of solutions of the congruence y^2 + y == x^3 - x^2 (mod p) as p runs through the primes.

Original entry on oeis.org

4, 4, 4, 9, 10, 9, 19, 19, 24, 29, 24, 34, 49, 49, 39, 59, 54, 49, 74, 74, 69, 89, 89, 74, 104, 99, 119, 89, 99, 104, 119, 149, 144, 129, 159, 149, 164, 159, 179, 179, 194, 174, 174, 189, 199, 199, 199, 204, 209, 214, 209, 269, 249, 274, 259, 249, 259, 299, 279, 299

Offset: 1

Andrey Zabolotskiy, Sep 08 2024

nonn

Same as A272196, except for a(1).

Edward Frenkel, Love and math: the heart of hidden reality, Basic Books, 2013. See pages 86-89.

LMFDB, Elliptic curve with LMFDB label 11.a3.

Cf. A002070, A006571, A060457, A272196.

a(n) = prime(n) - A002070(n).

a(n) = A060457(prime(n)).

A375132 Number of nonaligned trivalent dissections of a rectangle into n rectangles.

Original entry on oeis.org

1, 1, 2, 6, 21, 101, 591, 4168, 32754, 282605

Offset: 1

Andrey Zabolotskiy, Jul 31 2024

J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.

Cf. A049021, A340984, A342141, A375129, A375130, A375131.

A375131 Number of trivalent dissections of a rectangle into n rectangles.

Original entry on oeis.org

1, 1, 2, 6, 22, 108, 668, 5026, 43005, 389803

Offset: 1

Andrey Zabolotskiy, Jul 31 2024

Among the dissections counted by A375129, this sequence counts only those without 4-way junctions. See A375129 for the details.

J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.

Cf. A049021, A340984, A342141, A375129, A375130, A375132.

A375130 Number of nonaligned dissections of a rectangle into n rectangles.

Original entry on oeis.org

1, 1, 2, 7, 23, 119, 735, 5527, 46204, 423724

Offset: 1

Andrey Zabolotskiy, Jul 31 2024

Among the dissections counted by A375129, this sequence counts only those without "alignments". See A375129 for the details.

J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.

Cf. A049021, A340984, A342141, A375129, A375131, A375132.

A375129 Number of combinatorially distinct ways to dissect a rectangle into n rectangles, taking into account the ordering of the lines that extend the sides.

Original entry on oeis.org

1, 1, 2, 7, 24, 126, 815, 6465, 58072, 578663

Offset: 1

Andrey Zabolotskiy, Jul 31 2024

Dissections related by rotations and reflections are considered equivalent (unlike in A342141).

			All dissections into n=4 pieces are shown in Peter Kagey's illustration, they are the same as the ones counted by A049021.
The following two dissections (labeled "Grating (3,3), 5 fronts, 0401, C_2" and "Grating (2,3), 5 fronts, 0401, K_4" in Bloch's catalog) into n=5 pieces
  (1) ┌─┬─┬─┐   (2) ┌─┬─┬─┐
      ├─┤ │ │       ├─┤ ├─┤
      │ │ ├─┤       └─┴─┴─┘
      └─┴─┴─┘
  are considered distinct by this sequence and by A375131, because the lines extending the inner horizontal sides go in the different order:
  (1) ┌─┬─┬─┐   (2) ┌─┬─┬─┐
      A─B │ │       A─B C─D
      │ │ C─D       └─┴─┴─┘
      └─┴─┴─┘
  in dissection (1), the line AB is above line CD, while in dissection (2) AB and CD is the same line. (One could also slide the side AB below CD, but this sequence would not distinguish that new dissection from (1) because it would be equivalent to the mirror image of (1).) However, A049021 views these two dissections as equivalent. A375130 and A375132 distinguish between these dissections but do not include dissection (2) at all because it has an "alignment": two internal sides AB and CD, even though they are not connected through a 4-way junction (or a sequence of sides with the same orientation, connected through 4-way junctions), still extend to coinciding lines.

J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.

C. J. Bloch, Catalogue of small rectangular plans, Environment and Planning B, 6 (1979), 155-190.
C. J. Bloch and R. Krishnamurti, The Counting of Rectangular Dissections, Environ. Plann. B, 5 (1978), 207-214.
Peter Kagey, Example of the a(4)=7 dissections into n=4 pieces.

Cf. A049021, A340984, A342141, A375130, A375131, A375132.

A371049 Low temperature series for spin-1/2 Ising partition function on body-centered cubic lattice.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 4, -4, 0, 28, -60, 44, 204, -750, 1084, 979, -8444, 18886, -7568, -82269, 280288, -348172, -576712, 3677331, -7445964, 569558, 41740944, -126624684

Offset: 1

Andrey Zabolotskiy, Mar 11 2024

The series is in the variable u = exp(-4J/kT).

The expansion of the logarithm of the g.f. of this sequence is given in Domb & Guttmann's Table 1 (with a reference to Sykes et al., 1965) and continued in Eq. (4.14) of Sykes et al., 1973.

Claude Itzykson and Jean-Michel Drouffe, Statistical field theory, vol. 2, Cambridge University Press, 1989. Eq. (120) is supposed to give the logarithm of the g.f., but its second half is erroneously switched with the second half of Eq. (121). These second halves are Eqs. (4.15) and (4.14) of Sykes et al., 1973.

C. Domb and A. J. Guttmann, Low-temperature series for the Ising model, J. Phys. C: Solid State Phys., 3 (1970), 1652-1660.
M. F. Sykes, J. W. Essam and D. S. Gaunt, Derivation of low-temperature expansions for the Ising model of a ferromagnet and an antiferromagnet, J. Math. Phys. 6 (1965), 283-298.
M. F. Sykes, D. S. Gaunt, J. W. Essam and C. J. Elliott, Derivation of low-temperature expansions for Ising model. VI. Three-dimensional lattices-temperature grouping, J. Phys. A: Math. Nucl. Gen., 6 (1973), 1507-1516.
Index entries for sequences related to b.c.c. lattice

Cf. A002891 (simple cubic), A002892 (f.c.c.); A003193 (magnetization), A002925 (ferromagnetic susceptibility), A007218 (antiferromagnetic susceptibility); A001406 (high temperature).

cp's OEIS Frontend

User: Andrey Zabolotskiy

A383108 Number of crystallographic orbits in n dimensions, counting enantiomorphic pairs as distinct.

Views

Author

Keywords

References

Links

Crossrefs

A382870 Minimum period of an optimum covering of the set of integers by translates of its subset with diameter no greater than n, maximized over such subsets.

Views

Author

Keywords

Examples

Links

A381577 Unique sequence of 0's, 1's, and 2's such that for any terms x and y with x < y, the subsequence of x's and y's becomes the Thue-Morse sequence after substitution x -> 0, y -> 1.

Views

Author

Keywords

Crossrefs

Programs

Python

Formula

A377569 Number of simple graphs such that each connected component is nonseparable and the number of vertices minus the number of connected components equals n.

Views

Author

Keywords

Crossrefs

A376073 Number of solutions of the congruence y^2 + y == x^3 - x^2 (mod p) as p runs through the primes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A375132 Number of nonaligned trivalent dissections of a rectangle into n rectangles.

Views

Author

Keywords

References

Crossrefs

A375131 Number of trivalent dissections of a rectangle into n rectangles.

Views

Author

Keywords

Comments

References

Crossrefs

A375130 Number of nonaligned dissections of a rectangle into n rectangles.

Views

Author

Keywords

Comments

References

Crossrefs

A375129 Number of combinatorially distinct ways to dissect a rectangle into n rectangles, taking into account the ordering of the lines that extend the sides.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A371049 Low temperature series for spin-1/2 Ising partition function on body-centered cubic lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs