A003792 -id:A003792 - cp's OEIS frontend

A019988 Number of ways of embedding a connected graph with n edges in the square lattice.

1, 2, 5, 16, 55, 222, 950, 4265, 19591, 91678, 434005, 2073783, 9979772, 48315186, 235088794, 1148891118, 5636168859, 27743309673

Offset: 1

Russ Cox

It is assumed that all edges have length one. - N. J. A. Sloane, Apr 17 2019
These are referred to as 'polysticks', 'polyedges' or 'polyforms'. - Jack W Grahl, Jul 24 2018
Number of connected subgraphs of the square lattice (or grid) containing n length-one line segments. Configurations differing only a rotation or reflection are not counted as different. The question may also be stated in terms of placing unit toothpicks in a connected arrangement on the square lattice. - N. J. A. Sloane, Apr 17 2019
The solution for n=5 features in the card game Digit. - Paweł Rafał Bieliński, Apr 17 2019

Brian R. Barwell, "Polysticks," Journal of Recreational Mathematics, 22 (1990), 165-175.

D. Goodger, An introduction to Polysticks
M. Keller, Counting polyforms
D. Knuth, Dancing Links, arXiv:cs/0011047 [cs.DS], 2000. (A discussion of backtracking algorithms which mentions some problems of polystick tiling.)
Ed Pegg, Jr., Illustrations of polyforms
N. J. A. Sloane, Illustration of a(1)-a(4)
Eric Weisstein's World of Mathematics, Polyedge
Wikicommons, Polysticks 5-sticks 6-sticks 7-sticks

If only translations (but not rotations) are factored, consider fixed polyedges (A096267).
If reflections are considered different, we obtain the one-sided polysticks, counted by (A151537). - Jack W Grahl, Jul 24 2018
Cf. A001997, A003792, A006372, A059103, A085632, A056841 (tree-like), A348095 (with cycles), A348096 (refined by symmetry), A181528.
See A336281 for another version.
6th row of A366766.

A348095(n) + A056841(n+1) = a(n). - R. J. Mathar, Sep 30 2021

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Feb 20 2002

a(18) from John Mason, Jun 01 2023

A003787 Order of universal Chevalley group A_n (3).

Original entry on oeis.org

1, 24, 5616, 12130560, 237783237120, 42064805779476480, 67034222101339041669120, 961721214905722855895197286400, 124190524600592082795473760093457612800, 144339416867688029764487130056208182629053235200

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Robert Steinberg, Lectures on Chevalley Groups, Dept. of Mathematics, Yale University, 1967, pp. 130-131.
Index entries for sequences related to groups.

Cf. A003788, A003789, A003790, A003791, A003792, A053290, A100220.

[&*[(3^n - 3^k): k in [0..n-1]]/2: n in [1..10]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}];
f[3, #] & /@ Range[0, 9] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053290(n)/2. - Ralf Stephan, Mar 30 2004
a(n) = A(3,n) where A(q,n) = q^(n*(n+1)/2) * Product_{k=2..n+1}(q^k-1). - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 3^(n*(n+2)), where c = (3/2) * A100220 = 0.840189116891... . - Amiram Eldar, Jul 07 2025

One more term from Sean A. Irvine, Sep 18 2015

A003789 Order of universal Chevalley group A_n (5).

Original entry on oeis.org

1, 120, 372000, 29016000000, 56653740000000000, 2766118855500000000000000, 3376566710423156250000000000000000, 103044374585338670859375000000000000000000000

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003790, A003791, A003792, A053292, A100222.

[&*[(5^n-5^k): k in [0..n-1]]/4: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[5, #] & /@ Range[0, 7] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053292(n)/4. - Ralf Stephan, Mar 30 2004
a(n) = A(5,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 5^(n*(n+2)), where c = (5/4) * A100222 = 0.950415994839... . - Amiram Eldar, Jul 07 2025

A003790 Order of universal Chevalley group A_n (7).

Original entry on oeis.org

1, 336, 5630688, 4635182361600, 187035198320488089600, 369826556020831611935738265600, 35832085525362833262818017603275320524800, 170115000551935077294273059250893063598899496222720000

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003789, A003791, A003792,A053293, A100220.

[&*[(7^n - 7^k): k in [0..n-1]]/6: n in [1..10]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[7, #] & /@ Range[0, 7] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053293(n)/6. - Ralf Stephan, Mar 30 2004
a(n) = A(7,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 7^(n*(n+2)), where c = (7/6) * A100220 = 0.840189116891... . - Amiram Eldar, Jul 07 2025

a(7) from Sean A. Irvine, Sep 18 2015

A052497 Number of nonsingular n X n matrices over GF(9).

Original entry on oeis.org

1, 8, 5760, 339655680, 1624314979123200, 629282246371356907929600, 19747506525777609095698646040576000, 50195501537943419769100848121708339934527488000

Offset: 0

Vladeta Jovovic, Mar 16 2000

G. C. Greubel, Table of n, a(n) for n = 0..30
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Cf. A027877, A053764.

Cf. A002884, A003792, A053290, A053291, A053292, A053293, A052496, A052498, A132037.

[1] cat [&*[(9^n - 9^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[(9^n - 9^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

{a(n) = prod(j=0,n-1, 9^n - 9^j)}; \\ G. C. Greubel, May 14 2019

[product(9^n - 9^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (9^n - 1)*(9^n - 9)*...*(9^n - 9^(n-1)).
a(n) = A053764(n)*A027877(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 9^(n^2), where c = A132037. - Amiram Eldar, Jul 06 2025

A003788 Order of universal Chevalley group A_n (4).

Original entry on oeis.org

1, 60, 60480, 987033600, 258492255436800, 1083930404878024704000, 72736898347485916060188672000, 78099458182389588115529148326215680000, 1341733356588640095264385107865053233298800640000

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Index entries for sequences related to groups.

Cf. A003787, A003789, A003790, A003791, A003792, A053291, A100221.

[&*[(4^n - 4^k): k in [0..n-1]]/3: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}];
f[4, #] & /@ Range[0, 8] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053291(n)/3. - Ralf Stephan, Mar 30 2004
a(n) = A(4,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 4^(n*(n+2)), where c = (4/3) * A100221 = 0.918050049493... . - Amiram Eldar, Jul 07 2025

One more term from Sean A. Irvine, Sep 18 2015

A003791 Order of universal Chevalley group A_n (8).

Original entry on oeis.org

1, 504, 16482816, 34558531338240, 4638226007491010887680, 39841906041871272087686291128320, 21903309038581548352789123727634573903790080

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003789, A003790, A003792, A052496, A132036.

[&*[(8^n - 8^k): k in [0..n-1]]/7: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[8, #] & /@ Range[0, 6] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A052496(n)/7. - Ralf Stephan, Mar 30 2004
a(n) = A(8,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 8^(n*(n+2)), where c = (8/7) * A132036 = 0.982178279315... . - Amiram Eldar, Jul 07 2025

cp's OEIS Frontend

A019988 Number of ways of embedding a connected graph with n edges in the square lattice.

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A003787 Order of universal Chevalley group A_n (3).

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A003789 Order of universal Chevalley group A_n (5).

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A003790 Order of universal Chevalley group A_n (7).

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A052497 Number of nonsingular n X n matrices over GF(9).

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A003788 Order of universal Chevalley group A_n (4).

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Magma

Mathematica

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A003791 Order of universal Chevalley group A_n (8).

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Magma

Mathematica