A212796 -id:A212796 - cp's OEIS frontend

A212798 Row 3 of array in A212796.

3, 294, 11664, 367500, 10609215, 292626432, 7839321861, 205683135000, 5312031978672, 135495143785470, 3421536337406913, 85686871818240000, 2130987634616000199, 52682956706683197258, 1295799745309605101520, 31730077997731715070000

Offset: 1

N. J. A. Sloane, May 27 2012

A linear divisibility sequence of order 10. - Peter Bala, May 04 2014

Seiichi Manyama, Table of n, a(n) for n = 1..200
Germain Kreweras, Complexite et circuits Euleriens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (58,-1131,8700,-29493,43734,-29493,8700,-1131,58,-1).

Cf. A212796, A054493.

seq(simplify(n/3*(-2 + ( (5 + sqrt(21))/2 )^n + ( (5 - sqrt(21))/2 )^n)^2), n = 1..14); # Peter Bala, May 04 2014

# Using graphillion
from graphillion import GraphSet
def make_CnXCk(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
        grids.append((i + (n - 1) * k, i))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
        grids.append((i + k - 1, i))
    return grids
def A212798(n):
    if n == 1: return 3
    if n == 2: return 294
    universe = make_CnXCk(n, 3)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A212798(n) for n in range(1, 30)])  # Seiichi Manyama, Nov 22 2020

From Peter Bala, May 04 2014: (Start)
a(n) = n/3*(-2 + ( (5 + sqrt(21))/2 )^n + ( (5 - sqrt(21))/2 )^n)^2 = 3*n*A054493(n-1)^2.
O.g.f.: 3*(x^8 + 40*x^7 - 665*x^6 - 866*x^5 + 5626*x^4 - 866*x^3 - 665*x^2 + 40*x + 1)/( (x - 1)^2*(x^2 - 5*x + 1)^2*(x^2 - 23*x + 1)^2 ). (End)

More terms from Peter Bala, May 04 2014

A212799 Row 4 of array in A212796.

Original entry on oeis.org

4, 2304, 367500, 42467328, 4381392020, 428652000000, 40643137651228, 3771854305099776, 344499209234302500, 31074298464967845120, 2774871814779003772844, 245741556726521856000000, 21611621448116558812137652, 1889376666754339457990201088, 164334311374716912516773437500

Offset: 1

N. J. A. Sloane, May 27 2012

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..24
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.

Cf. A003696, A212796, A212798.

Table[2^(6*n-4)*n*Product[Sin[j*Pi/4]^2 + Sin[k*Pi/n]^2, {j,1,3}, {k,1,n-1}], {n,1,20}]//Round (* Vaclav Kotesovec, Feb 26 2021 *)

# Using graphillion
from graphillion import GraphSet
def make_CnXCk(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
        grids.append((i + (n - 1) * k, i))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
        grids.append((i + k - 1, i))
    return grids
def A212799(n):
    if n == 1: return 4
    if n == 2: return 2304
    universe = make_CnXCk(4, n)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A212799(n) for n in range(1, 8)])  # Seiichi Manyama, Nov 22 2020

From Vaclav Kotesovec, Feb 26 2021: (Start)
a(n) ~ (21 + 12*sqrt(3) + 2*sqrt(2*(97 + 56*sqrt(3))))^n * n/4.
G.f.: 4*x*(1 + 310*x - 33278*x^2 + 785814*x^3 + 4923451*x^4 - 476492324*x^5 + 8394222196*x^6 - 74272031652*x^7 + 371582629705*x^8 - 981246223862*x^9 + 441533151262*x^10 + 6161037199338*x^11 - 23802532730757*x^12 + 46995963516168*x^13 - 58240430817576*x^14 + 46995963516168*x^15 - 23802532730757*x^16 + 6161037199338*x^17 + 441533151262*x^18 - 981246223862*x^19 + 371582629705*x^20 - 74272031652*x^21 + 8394222196*x^22 - 476492324*x^23 + 4923451*x^24 + 785814*x^25 - 33278*x^26 + 310*x^27 + x^28)/ ((1 - x)^2*(1 - 14*x + x^2)^2*(1 - 6*x + x^2)^2*(1 - 4*x + x^2)^2* (1 - 84*x + 230*x^2 - 84*x^3 + x^4)^2*(1 - 24*x + 50*x^2 - 24*x^3 + x^4)^2). (End)

a(10)-a(15) from Seiichi Manyama, Nov 22 2020

A212800 Number of spanning trees of the (n,n)-torus grid graph.

Original entry on oeis.org

1, 32, 11664, 42467328, 1562500000000, 587312954081280000, 2266101334892340404752384, 89927963805390785392395474173952, 36735015407753190053984060991247792275456, 154528563849617762057150663767149772800000000000000, 6695315138840257072470706538467584763944601124280722177130496

Offset: 1

N. J. A. Sloane, May 27 2012

nonn

Main diagonal of array in A212796.

Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Para. 4.
Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Torus Grid Graph

Cf. A212796, A340562.

Table[n^2 * Product[4*Sin[j*Pi/n]^2 + 4*Sin[k*Pi/n]^2, {k, 1, n-1}, {j, 1, n-1}], {n, 1, 12}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)

a(n) ~ Gamma(1/4)^4 * exp(4*G*n^2/Pi) / (16 * Pi^3), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

More terms from Eric W. Weisstein, May 10 2017

A358810 Number of spanning trees in C_5 X C_n.

Original entry on oeis.org

5, 16810, 10609215, 4381392020, 1562500000000, 522217835532030, 168437773747672835, 53095647535975155240, 16463182598208445194045, 5040439500800000000000000, 1527650417538030913166754055, 459160235715332056282793308860

Offset: 1

Seiichi Manyama, Dec 02 2022

nonn

Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Torus Grid Graph

Row 5 of A212796.

default(realprecision, 120);
T(n, k) = round(n*k*prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)));
a(n) = T(5, n);

A358811 Number of spanning trees in C_6 X C_n.

Original entry on oeis.org

6, 117600, 292626432, 428652000000, 522217835532030, 587312954081280000, 633426582213424399722, 665880333340217184000000, 687776414074843514847584256, 701129416495732552572667500000, 707405677027691828669857196745186

Offset: 1

Seiichi Manyama, Dec 02 2022

nonn

Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Torus Grid Graph

Row 6 of A212796.

default(realprecision, 120);
T(n, k) = round(n*k*prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)));
a(n) = T(6, n);

A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 49, 49, 1, 1, 288, 1296, 288, 1, 1, 1681, 30625, 30625, 1681, 1, 1, 9800, 707281, 2654208, 707281, 9800, 1, 1, 57121, 16257024, 219069601, 219069601, 16257024, 57121, 1, 1, 332928, 373301041, 17860500000, 62500000000, 17860500000, 373301041, 332928, 1

Offset: 1

Seiichi Manyama, Jan 11 2021

			Square array begins:
  1,    1,      1,         1,           1, ...
  1,    8,     49,       288,        1681, ...
  1,   49,   1296,     30625,      707281, ...
  1,  288,  30625,   2654208,   219069601, ...
  1, 1681, 707281, 219069601, 62500000000, ...

Rows and columns 1..2 give A000012, A001108.
Main diagonal gives A340562.
Cf. A212796, A340475, A340561.

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)))}

T(n,k) = T(k,n).

T(n,k) = A212796(n,k)/(n*k).

A212797 Number of spanning trees in C_2 X C_n.

Original entry on oeis.org

2, 32, 294, 2304, 16810, 117600, 799694, 5326848, 34928082, 226195360, 1450199542, 9220780800, 58221203066, 365440965344, 2282085037470, 14187697422336, 87860208024994, 542209573735200, 3335797263902918, 20465738163774720, 125247216613782858

Offset: 1

N. J. A. Sloane, May 27 2012

nonn

From Harry Richman and Alois P. Heinz, Jan 31 2023: (Start)
Row 2 of array in A212796.
a(n) is a divisibility sequence, i.e., if m divides n then a(m) divides a(n), since A001108 is one. (End)

Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.
Index entries for linear recurrences with constant coefficients, signature (14,-63,100,-63,14,-1).

Cf. A001108, A212796.

default(realprecision, 120);
{a(n) = round(n*2^(2*n-1)*prod(k=1, n-1, 1+sin(k*Pi/n)^2))} \\ Seiichi Manyama, Jan 13 2021

my(N=66, x='x+O('x^N)); Vec(2*x*deriv(x*(1+x)/((1-x)*(1-6*x+x^2)))) \\ Seiichi Manyama, Jan 13 2021

G.f.: 2*x*(1+2*x-14*x^2+2*x^3+x^4)/((x-1)^2*(1-6*x+x^2)^2) . - R. J. Mathar, Apr 16 2018
Conjecture: a(n) = 16*A001109(n+1) +3*A001109(n) -16*(A144133(n)-6*A144133(n-1)) -n = 3*A144133(n-1) -2*A144133(n-2) +3*A144133(n-3) -n. - R. J. Mathar, Apr 16 2018
From Seiichi Manyama, Jan 13 2021: (Start)
a(n) = 2 * n * A001108(n).
a(n) = 14*a(n-1) - 63*a(n-2) + 100*a(n-3) - 63*a(n-4) + 14*a(n-5) - a(n-6) for n > 6. (End)

More terms from Seiichi Manyama, Jan 13 2021

A358812 Number of spanning trees in C_7 X C_n.

Original entry on oeis.org

7, 799694, 7839321861, 40643137651228, 168437773747672835, 633426582213424399722, 2266101334892340404752384, 7871822605982542067643202616, 26818349084747196820449212376063, 90098172307754257628918141363625670, 299464785482715726798502702429093755197

Offset: 1

Seiichi Manyama, Dec 02 2022

nonn

Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Torus Grid Graph

Row 7 of A212796.

default(realprecision, 120);
T(n, k) = round(n*k*prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)));
a(n) = T(7, n);

A358813 Number of spanning trees in C_8 X C_n.

Original entry on oeis.org

8, 5326848, 205683135000, 3771854305099776, 53095647535975155240, 665880333340217184000000, 7871822605982542067643202616, 89927963805390785392395474173952, 1005049441217682470864686231147005000

Offset: 1

Seiichi Manyama, Dec 02 2022

nonn

Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Torus Grid Graph

Row 8 of A212796.

default(realprecision, 120);
T(n, k) = round(n*k*prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)));
a(n) = T(8, n);

A358814 Number of spanning trees in C_9 X C_n.

Original entry on oeis.org

9, 34928082, 5312031978672, 344499209234302500, 16463182598208445194045, 687776414074843514847584256, 26818349084747196820449212376063, 1005049441217682470864686231147005000, 36735015407753190053984060991247792275456

Offset: 1

Seiichi Manyama, Dec 02 2022

nonn

Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Torus Grid Graph

Row 9 of A212796.

default(realprecision, 120);
T(n, k) = round(n*k*prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)));
a(n) = T(9, n);

cp's OEIS Frontend

A212798 Row 3 of array in A212796.

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A212799 Row 4 of array in A212796.

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A212800 Number of spanning trees of the (n,n)-torus grid graph.

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A358810 Number of spanning trees in C_5 X C_n.

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A358811 Number of spanning trees in C_6 X C_n.

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A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4*sin(a*Pi/n)^2 + 4*sin(b*Pi/k)^2).

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A212797 Number of spanning trees in C_2 X C_n.

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PARI

PARI

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A358812 Number of spanning trees in C_7 X C_n.

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PARI

A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).