A231087 -id:A231087 - cp's OEIS frontend

A030221 Chebyshev even-indexed U-polynomials evaluated at sqrt(7)/2.

1, 6, 29, 139, 666, 3191, 15289, 73254, 350981, 1681651, 8057274, 38604719, 184966321, 886226886, 4246168109, 20344613659, 97476900186, 467039887271, 2237722536169, 10721572793574, 51370141431701, 246129134364931, 1179275530392954, 5650248517599839

Offset: 0

Wolfdieter Lang

a(n) = L(n,-5)*(-1)^n, where L is defined as in A108299; see also A004253 for L(n,+5). - Reinhard Zumkeller, Jun 01 2005
General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4; lim_{n->oo} a(n) = x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878. a(1)=5 gives A001834. a(1)=6 gives the present sequence. a(1)=7 gives A002315. a(1)=8 gives A033890. a(1)=9 gives A057080. a(1)=10 gives A057081. - Ctibor O. Zizka, Sep 02 2008
The primes in this sequence are 29, 139, 3191, 15289, 350981, 1681651, ... - Ctibor O. Zizka, Sep 02 2008
Inverse binomial transform of A030240. - Philippe Deléham, Nov 19 2009
For positive n, a(n) equals the permanent of the (2n)X(2n) matrix with sqrt(7)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
The aerated sequence (b(n))n>=1 = [1, 0, 6, 0, 29, 0, 139, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -3, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for a connection with Chebyshev polynomials. - Peter Bala, Mar 22 2015
From Wolfdieter Lang, Oct 26 2020: (Start)
((-1)^n)*a(n) = X(n) = ((-1)^n)*(S(n, 5) + S(n-1, 5)) and Y(n) = X(n-1) gives all integer solutions (modulo sign flip between X and Y) of X^2 + Y^2 + 5*X*Y = +7, for n = -oo..+oo, with Chebyshev S polynomials (see A049310), with S(-1, x) = 0, and S(-n, x) = - S(n-2, x), for n >= 2.
This binary indefinite quadratic form of discriminant 21, representing 7, has only this family of proper solutions (modulo sign flip), and no improper ones.
This comment is inspired by a paper by Robert K. Moniot (private communication). See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)

			G.f. = 1 + 6*x + 29*x^2 + 139*x^3 + 666*x^4 + 3191*x^5 + 15289*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681. (see page 678)
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 18.
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 13.
Tanya Khovanova, Recursive Sequences.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), rhs, m=6.
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,-1).

Cf. A004253, A004254, A100047, A054493 (partial sums), A049310, A003501 (first differences), A299109 (subsequence of primes).

I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015

A030221 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[1,6]);
    else
        5*procname(n-1)-procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Apr 30 2017

t[n_, k_?EvenQ] := I^k*Binomial[n-k/2, k/2]; t[n_, k_?OddQ] := -I^(k-1)*Binomial[n+(1-k)/2-1, (k-1)/2]; l[n_, x_] := Sum[t[n, k]*x^(n-k), {k, 0, n}]; a[n_] := (-1)^n*l[n, -5]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 05 2013, after Reinhard Zumkeller *)
a[ n_] := ChebyshevU[2 n, Sqrt[7]/2]; (* Michael Somos, Jan 22 2017 *)

{a(n) = simplify(polchebyshev(2*n, 2, quadgen(28)/2))}; /* Michael Somos, Jan 22 2017 */

[(lucas_number2(n,5,1)-lucas_number2(n-1,5,1))/3 for n in range(1,22)] # Zerinvary Lajos, Nov 10 2009

a(n) = 5*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.
a(n) = U(2*n, sqrt(7)/2).
G.f.: (1+x)/(x^2-5*x+1).
a(n) = A004254(n) + A004254(n+1).
a(n) ~ (1/2 + (1/6)*sqrt(21))*((1/2)*(5 + sqrt(21)))^n. - Joe Keane (jgk(AT)jgk.org), May 16 2002
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then a(n) = (-1)^n*q(n, -7). - Benoit Cloitre, Nov 10 2002
A054493(2*n) = a(n)^2 for all n in Z. - Michael Somos, Jan 22 2017
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jan 22 2017
0 = -7 + a(n)*(+a(n) - 5*a(n+1)) + a(n+1)*(+a(n+1)) for all n in Z. - Michael Somos, Jan 22 2017
a(n) = S(n, 5) + S(n-1, 5) = S(2*n, sqrt(7)) (see above in terms of U), for n >= 0 with S(-1, 5) = 0, where the coefficients of the Chebyshev S polynomials are given in A049310. - Wolfdieter Lang, Oct 26 2020
From Peter Bala, May 16 2025: (Start)
Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/7 (telescoping series: 7/(a(n) - 1/a(n)) = 1/A004254(n+1) + 1/A004254(n)).
Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(7/3) (telescoping product: Product_{k = 1..n} ((a(k) + 1)/(a(k) - 1))^2 = 7/3 * (1 - 8/A231087(n+1))). (End)

A231485 Number of perfect matchings in the graph C_5 X C_{2n}.

Original entry on oeis.org

722, 9922, 155682, 2540032, 41934482, 694861522, 11527389122, 191304901282, 3175220160032, 52703408458882, 874800747092322, 14520494659638322, 241020661471736882, 4000620276282860032, 66404949893677073282, 1102233473331064193122, 18295603728585969257522

Offset: 2

Sergey Perepechko, Nov 09 2013

Colin Barker, Table of n, a(n) for n = 2..819
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
Index entries for linear recurrences with constant coefficients, signature (29,-261,1029,-2001,2001,-1029,261,-29,1).

Cf. A220864, A231087.

CoefficientList[Series[2*(361 - 5508*x + 28193*x^2 - 64021*x^3 + 70770*x^4 - 38841*x^5 + 10278*x^6 - 1173*x^7 + 41*x^8)/((1 - x)*(1 - 9*x + 21*x^2 - 9*x^3 + x^4)*(1 - 19*x + 41*x^2 - 19*x^3 + x^4)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jul 17 2025 *)

Vec(2*x^2*(361-5508*x+28193*x^2-64021*x^3+70770*x^4-38841*x^5+10278*x^6-1173*x^7+41*x^8)/((1-x)*(1-9*x+21*x^2-9*x^3+x^4)*(1-19*x+41*x^2-19*x^3+x^4)) + O(x^100)) \\ Colin Barker, Dec 13 2014

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 5, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/5)^2)))); \\ Seiichi Manyama, Feb 14 2021

G.f.: 2*x^2*(361-5508*x+28193*x^2-64021*x^3+70770*x^4-38841*x^5+10278*x^6-1173*x^7+41*x^8)/((1-x)*(1-9*x+21*x^2-9*x^3+x^4)*(1-19*x+41*x^2-19*x^3+x^4)).
From Seiichi Manyama, Feb 14 2021: (Start)
a(n) = sqrt( Product_{j=1..n} Product_{k=1..5} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/5)^2) ).
a(n) = 28*a(n-1) - 233*a(n-2) + 796*a(n-3) - 1205*a(n-4) + 796*a(n-5) - 233*a(n-6) + 28*a(n-7) - a(n-8) + 200. (End)

A232804 Number of perfect matchings in the graph C_6 x C_n.

Original entry on oeis.org

224, 3108, 9922, 90176, 401998, 3113860, 16091936, 114557000, 643041038, 4357599552, 25689719122, 169094614280, 1026275640544, 6640849944580, 40998347400722, 262671237617216, 1637828186763038, 10433179552323108, 65428999765032736, 415409841636546440, 2613799160004664798, 16563343174199239744

Offset: 3

Sergey Perepechko, Nov 30 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 3..500
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.

Row n=3 of A341741.

Cf. A220864, A231087, A231485.

G.f: 2*x^3*(112+882*x-8955*x^2-22184*x^3+151298*x^4+192108*x^5-1004174*x^6-773678*x^7+3077791*x^8+1598624*x^9-4646368*x^10-1738444*x^11+3589216*x^12+ 1010882*x^13-1408253*x^14-318388*x^15+271982*x^16+52648*x^17-23250*x^18-4062*x^19+601*x^20+100*x^21)/((1-x)*(1+x)*(1+5*x+x^2)*(1-5*x+x^2)*(1-2*x-x^2)* (1+2*x-x^2)*(1+x-x^2)*(1-x-x^2)*(1-5*x^2+x^4)*(1-6*x-3*x^2+6*x^3+x^4)).

A341533 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k} (4sin((2a-1)Pi/(2n))^2 + 4sin((2b-1)*Pi/k)^2) ).

Original entry on oeis.org

2, 8, 2, 14, 36, 2, 36, 50, 200, 2, 82, 256, 224, 1156, 2, 200, 722, 2916, 1058, 6728, 2, 478, 2916, 9922, 38416, 5054, 39204, 2, 1156, 10082, 80000, 155682, 527076, 24200, 228488, 2, 2786, 38416, 401998, 2775556, 2540032, 7311616, 115934, 1331716, 2

Offset: 1

Seiichi Manyama, Feb 13 2021

			Square array begins:
  2,     8,    14,      36,       82,        200, ...
  2,    36,    50,     256,      722,       2916, ...
  2,   200,   224,    2916,     9922,      80000, ...
  2,  1156,  1058,   38416,   155682,    2775556, ...
  2,  6728,  5054,  527076,  2540032,  105125000, ...
  2, 39204, 24200, 7311616, 41934482, 4115479104, ...

Columns 1..7 give A007395, A341543, A231087, A341544, A231485, A341545, A230033.
Main diagonal gives A341535.
Cf. A340475.

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

A230033 Number of perfect matchings in the graph C_7 X C_{2n}.

Original entry on oeis.org

10082, 401998, 19681538, 1034315998, 55820091938, 3044533460992, 166779871224962, 9152970837103102, 502711247500143362, 27619744381029252622, 1517688682641434229698, 83401213534557960429502, 4583249488240161816039552, 251871805990373105011941118, 13841645914590329223808310018, 760670944425011837491619633038

Offset: 2

Sergey Perepechko, Dec 20 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 2..500
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.

Column k=7 of A341533.

Cf. A231087, A220864, A231485, A232804, A281583, A308761.

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 7, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/7)^2)))); \\ Seiichi Manyama, Feb 14 2021

G.f.: 2*x^2*(5041 -499700*x +20440353*x^2 -466963360*x^3 +6751799885*x^4 -66182756655*x^5 +459438362278*x^6 -2327864968019*x^7 +8797357131438*x^8 -25192378831195*x^9 +55291405473782*x^10 -93750343061691*x^11 +123440474579985*x^12 -126568817064424*x^13 +101127542456783*x^14 -62874205910076*x^15 +30308779015615*x^16 -11259345843608*x^17 +3194422598067*x^18 -683503915153*x^19 +108424368962*x^20 -12458825709*x^21 +1004282914*x^22 -54198917*x^23 +1818498*x^24 -33157*x^25 +239*x^26)/((1 -x)*(1 -13*x +57*x^2 -97*x^3 +57*x^4 -13*x^5 +x^6)*(1 -71*x +952*x^2 -3976*x^3 +6384*x^4 -3976*x^5 +952*x^6 -71*x^7 +x^8)*(1 -54*x +1039*x^2 -9096*x^3 +39037*x^4 -90378*x^5 +118951*x^6 -90378*x^7 +39037*x^8 -9096*x^9 +1039*x^10 -54*x^11 +x^12)).

a(n) = sqrt( Product_{j=1..n} Product_{k=1..7} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/7)^2) ). - Seiichi Manyama, Feb 14 2021

A253678 Number of perfect matchings in the graph C_8 X C_n.

Original entry on oeis.org

1058, 39952, 155682, 3113860, 19681538, 311853312, 2415542018, 33898728836, 294554220578, 3827188349968, 35866638601250, 442299574618756, 4365923647238658, 51942700201804032, 531410627302657538, 6169093269471927940, 64681086501382749218, 738453913359765339152, 7872683691901209561122, 88873260229652630182276

Offset: 3

Sergey Perepechko, Jan 09 2015

nonn

S. N. Perepechko, Combinatorial properties of dimer problem on tori (in Russian). Mathematical physics and its applications, The fourth int. conf. Samara, 2014, 280-281.

Seiichi Manyama, Table of n, a(n) for n = 3..500
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Perfect Matching
Eric Weisstein's World of Mathematics, Torus Grid Graph

Row n=4 of A341741.

Cf. A231087, A220864, A231485, A232804, A230033.

a(n) = 14*a(n-1) + 145*a(n-2) - 2492*a(n-3) - 5832*a(n-4) + 164332*a(n-5) + 6360*a(n-6) - 5592188*a(n-7) + 5575094*a(n-8) + 111829704*a(n-9) - 176471286*a(n-10) - 1404071060*a(n-11) + 2757391176*a(n-12) + 11493707876*a(n-13) - 26094214040*a(n-14) - 62666476628*a(n-15) + 161092194209*a(n-16) + 229194775110*a(n-17) - 673504262865*a(n-18) - 556186915928*a(n-19) + 1946775340976*a(n-20) + 855365272888*a(n-21) - 3933950269712*a(n-22) - 705783359960*a(n-23) + 5586898052980*a(n-24) - 5586898052980*a(n-26) + 705783359960*a(n-27) + 3933950269712*a(n-28) - 855365272888*a(n-29) - 1946775340976*a(n-30) + 556186915928*a(n-31) + 673504262865*a(n-32) - 229194775110*a(n-33) - 161092194209*a(n-34) + 62666476628*a(n-35) + 26094214040*a(n-36) - 11493707876*a(n-37) - 2757391176*a(n-38) + 1404071060*a(n-39) + 176471286*a(n-40) - 111829704*a(n-41) - 5575094*a(n-42) + 5592188*a(n-43) - 6360*a(n-44) - 164332*a(n-45) + 5832*a(n-46) + 2492*a(n-47) - 145*a(n-48) - 14*a(n-49) + a(n-50).

G.f.: 2*x^3*(529 + 12570*x - 278528*x^2 - 1111096*x^3 + 29622124*x^4 + 15949216*x^5 - 1354335880*x^6 + 1073870160*x^7 + 33231636934*x^8 - 49093408612*x^9 - 484852497568*x^10 + 922702092728*x^11 + 4448623050276*x^12 - 9889298009728*x^13 - 26519860399096*x^14 + 66909591407824*x^15 + 104242913448099*x^16 - 300153880511538*x^17 - 268804327853184*x^18 + 917127529551440*x^19 + 437177534552376*x^20 - 1937370697752896*x^21 - 386856893695952*x^22 + 2851262465341600*x^23 + 31463729114724*x^24 - 2933939639544920*x^25 + 353114911609152*x^26 + 2113468417316080*x^27 - 452714140134072*x^28 - 1064902306141568*x^29 + 302352881352848*x^30 + 373692292484128*x^31 - 126783009087417*x^32 - 90391126093930*x^33 + 35100066280832*x^34 + 14772327002472*x^35 - 6497628908516*x^36 - 1572040067936*x^37 + 799287715544*x^38 + 101192826896*x^39 - 63992712074*x^40 - 3215530756*x^41 + 3212411488*x^42 - 3162664*x^43 - 94666796*x^44 + 3355392*x^45 + 1438440*x^46 - 83696*x^47 - 8091*x^48 + 578*x^49)/((1-x)*(1+x)*(1+4*x+x^2)*(1-4*x+x^2)*(1-2*x-x^2)*(1+2*x-x^2)*(1+8*x+16*x^2+8*x^3+x^4)* (1-14*x+34*x^2-14*x^3+x^4)*(1-8*x+16*x^2-8*x^3+x^4)*(1-4*x^2+x^4)*(1+4*x-4*x^2-4*x^3+x^4)*(1+8*x-10*x^2-8*x^3+x^4)*(1-4*x-4*x^2+4*x^3+x^4)*(1-8*x-10*x^2+8*x^3+x^4)*(1-14*x^2+34*x^4-14*x^6+x^8)).

A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

Original entry on oeis.org

2, 8, 2, 14, 36, 2, 36, 50, 200, 2, 82, 272, 224, 1156, 2, 200, 722, 3108, 1058, 6728, 2, 478, 3108, 9922, 39952, 5054, 39204, 2, 1156, 10082, 90176, 155682, 537636, 24200, 228488, 2, 2786, 39952, 401998, 3113860, 2540032, 7379216, 115934, 1331716, 2

Offset: 1

Seiichi Manyama, Feb 18 2021

Dimer tilings of 2n x k toroidal grid.

			Square array begins:
  2,     8,    14,      36,       82,        200, ...
  2,    36,    50,     272,      722,       3108, ...
  2,   200,   224,    3108,     9922,      90176, ...
  2,  1156,  1058,   39952,   155682,    3113860, ...
  2,  6728,  5054,  537636,  2540032,  114557000, ...
  2, 39204, 24200, 7379216, 41934482, 4357599552, ...

Seiichi Manyama, Antidiagonals n = 1..50, flattened
P. W. Kasteleyn, The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225. See Eq. (25).
Index entries for sequences related to dominoes

Columns 1..12 give A007395, A162484(2*n), A231087, A220864(2*n), A231485, A232804(2*n), A230033, A253678(2*n), A281583, A281679(2*n), A308761, A309018(2*n).
Rows 1..6 give A162484, A220864, A232804, A253678, A281679, A309018.
T(n,2*n) gives A335586.
Cf. A341533, A341738, A341739.
Cf. A099390, A103997, A103999.

T(n,k) = A341533(n,k)/2 + A341738(n,k) + 2 * ((k+1) mod 2) * A341739(n,ceiling(k/2)).
T(n, 2k) = T(k, 2n).
If k is odd, T(n,k) = A341533(n,k) = 2*A341738(n,k).

New name from Andrey Zabolotskiy, Dec 26 2021

A281583 Number of perfect matchings in the graph C_9 X C_{2n}.

Original entry on oeis.org

140450, 16091936, 2415542018, 400448833106, 69206906601800, 12190695635108354, 2167175327735637122, 387018647188487143424, 69272289588070930561250, 12413316310203106546620386, 2225719417041514241075539592, 399192630631160441128470998546

Offset: 2

Sergey Perepechko, Jan 25 2017

nonn

For odd values of m the order of recurrence relation for the number of perfect matchings in C_{m} X C_{2n} graph does not exceed 3^floor(m/2).

Seiichi Manyama, Table of n, a(n) for n = 2..443
S. N. Perepechko, The number of perfect matchings on C_m X C_n graphs, (in Russian), Information Processes, 2016, V.16, No.4, pp.333-361.
Sergey Perepechko, Generating function, in Maple notation.

Column k=9 of A341533.

Cf. A253678, A230033, A232804, A231485, A220864, A231087.

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 9, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/9)^2)))); \\ Seiichi Manyama, Feb 14 2021

a(n) = sqrt( Product_{j=1..n} Product_{k=1..9} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/9)^2) ). - Seiichi Manyama, Feb 14 2021

A281679 Number of perfect matchings in the graph C_10 X C_n.

Original entry on oeis.org

5054, 537636, 2540032, 114557000, 1034315998, 33898728836, 400448833106, 11203604497408, 152849502772958, 3876306765700644, 58099728840105682, 1375359477482867528, 22057225099289357824, 496348449090698237956, 8370856315868909044082, 181385918483215101487880

Offset: 3

Sergey Perepechko, Jan 26 2017

nonn

For even values of m the order of recurrence relation for the number of perfect matchings in C_m X C_n graph does not exceed (3^delta(m/2) + 2*(3/5)^(1 - delta(m/2)))*5^floor(m/4) + 1. Here delta(k) equals 1 for odd values of k and 0 otherwise. If m=10 the above estimate gives 126 for the order of recurrence relation while the exact value equals 118.

Seiichi Manyama, Table of n, a(n) for n = 3..500
S. N. Perepechko, The number of perfect matchings on C_m X C_n graphs, (in Russian), Information Processes, 2016, V. 16, No. 4, pp. 333-361.
Sergey Perepechko, Generating function, in Maple notation.
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Perfect Matching
Eric Weisstein's World of Mathematics, Torus Grid Graph

Row n=5 of A341741.

Cf. A220864, A230033, A231087, A231485, A232804, A253678, A281583.

A341493 a(n) = ( Product_{j=1..n} Product_{k=1..n+1} (4sin((2j-1)Pi/n)^2 + 4sin((2k-1)Pi/(n+1))^2) )^(1/4).

Original entry on oeis.org

1, 2, 14, 50, 722, 9922, 401998, 19681538, 2415542018, 400448833106, 152849502772958, 83804387156528018, 100644292294423977842, 180483873668860889130642, 686161117968330536875295134, 4001215836806010384390623471618

Offset: 0

Seiichi Manyama, Feb 13 2021

nonn

Number of perfect matchings in the graph C_n X C_{n+1} for n > 0.

S. N. Perepechko, The number of perfect matchings on C_m X C_n graphs, (in Russian), Information Processes, 2016, V. 16, No. 4, pp. 333-361.
Eric Weisstein's World of Mathematics, Perfect Matching
Eric Weisstein's World of Mathematics, Torus Grid Graph

Cf. A162484, A220864, A230033, A231087, A231485, A232804, A253678, A281583, A281679, A308761, A309018, A335586.

Table[Product[4*Sin[(2*j - 1)*Pi/n]^2 + 4*Sin[(2*k - 1)*Pi/(n+1)]^2, {k, 1, n+1}, {j, 1, n}]^(1/4), {n, 0, 15}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)

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

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

cp's OEIS Frontend

A030221 Chebyshev even-indexed U-polynomials evaluated at sqrt(7)/2.

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A231485 Number of perfect matchings in the graph C_5 X C_{2n}.

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A232804 Number of perfect matchings in the graph C_6 x C_n.

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A341533 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k} (4*sin((2*a-1)*Pi/(2*n))^2 + 4*sin((2*b-1)*Pi/k)^2) ).

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A230033 Number of perfect matchings in the graph C_7 X C_{2n}.

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A253678 Number of perfect matchings in the graph C_8 X C_n.

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A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

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A281583 Number of perfect matchings in the graph C_9 X C_{2n}.

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A281679 Number of perfect matchings in the graph C_10 X C_n.

A341533 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k} (4sin((2a-1)Pi/(2n))^2 + 4sin((2b-1)*Pi/k)^2) ).

A341493 a(n) = ( Product_{j=1..n} Product_{k=1..n+1} (4sin((2j-1)Pi/n)^2 + 4sin((2k-1)Pi/(n+1))^2) )^(1/4).