David J. Seal - cp's OEIS frontend

A331905 Number of spanning trees in the multigraph cube of an n-cycle.

1, 4, 12, 128, 605, 3072, 16807, 82944, 412164, 2035220, 9864899, 47579136, 227902597, 1084320412, 5134860060, 24207040512, 113664879137, 531895993344, 2481300851179, 11543181696640, 53565699079956, 248005494380204, 1145875775104967, 5284358088818688

Offset: 1

David J. Seal, Jan 31 2020

nonn

The multigraph cube of an n-cycle has n nodes V1, V2, ... Vn, with one edge Vi to Vj for each pair (i,j) such that j = i+1, i+2 or i+3 modulo n. It is a multigraph when n <= 6 because this produces instances of multiple edges between the same two vertices, and it also produces loops if n <= 3.
Baron et al. (1985) describes the corresponding sequence A169630 for the multigraph square of a cycle.
I conjecture that a(n) = gcd(n,2) * n * (A005822(n))^2. [This is correct - see the Formula section. - N. J. A. Sloane, Feb 06 2020]
Terms a(7) to a(18) calculated by Brendan McKay, and terms a(1) to a(6) by David J. Seal, in both cases using Kirchhoff's matrix tree theorem.

			The multigraph cube of a 4-cycle has four vertices, with two edges between each pair of distinct vertices - i.e., it is a doubled-edge cover of the complete graph on 4 vertices. The complete graph on 4 vertices has 4^2 = 16 spanning trees, and each of those spanning trees corresponds to 8 spanning trees of the multigraph tree because there are independent choices of 2 multigraph edges to be made for each of the three edges in the graph's spanning tree. So a(4) = 16 * 8 = 128.

Alois P. Heinz, Table of n, a(n) for n = 1..1546
G. Baron et al., The number of spanning trees in the square of a cycle, Fib. Quart., 23 (1985), 258-264.
Yoshiaki Doi et al., A note on spanning trees
Min Li, Zhibing Chen, Xiaoqing Ruan, and Xuerong Yong, The formulas for the number of spanning trees in circulant graphs, Disc. Math. 338 (11) (2015) 1883-1906, Lemma 1.

Cf. A005822, A169630 (corresponding sequence for the multigraph square of an n-cycle).

a:= n-> ((<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|4|1|4>>^iquo(n, 2, 'd').
       <[<0, 1, 4, 16>, <1, 2, 11, 49>][d+1]>)[1, 1])^2*n*(2-irem(n, 2)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 06 2020

The following formulas were provided by Tsuyoshi Miezaki on Feb 05 2020 (see Doi et al. link). Let z1=(-3+sqrt(-7))/4, z2=(-3-sqrt(-7))/4; T(n,z) = cos(n*arccos(z)). Then a(n) = (2*n/7)*(T(n,z1)-1)*(T(n,z2)-1). Furthermore a(n) = 2*n*A005822(n)^2 if n is even, or n*A005822(n)^2 if n is odd. - N. J. A. Sloane, Feb 06 2020

More terms from Alois P. Heinz, Feb 06 2020

A292703 Values of n such that prime(n) does not divide any 10-digit pandigital number (i.e. any value in A050278).

Original entry on oeis.org

10545, 11602, 16237, 17984, 19978, 21788, 28046, 28666, 28669, 28693, 28928, 29629, 30698, 30896, 32869, 33438, 34699, 35373, 37198, 37300, 37639, 39273, 39477, 39755, 39756, 41859, 42003, 42219, 42490, 42538, 42619, 42624

Offset: 1

David J. Seal, Sep 21 2017

This is the complement of the finite list of n such that prime(n) divides one or more 10-digit pandigital numbers. That finite list has been obtained by computer; it contains 1102173 numbers, with the first number that is not in the list being 10545 and the last that is in the list being 55537259.
A292471 is the corresponding list of primes.
These are the values of n for which A180489(n) has more than 10 digits, and also the values of n for which A274328(n) = 0.

			a(1) = 10545 because prime(10545) = 111119 does not divide any of the 10-digit pandigital numbers 1023456789, 1023456798, ..., 9876543210, and all smaller primes do divide at least one of them.

Cf. A050278, A061604, A180489, A274328, A292471.

A292471 Primes that do not divide any 10-digit pandigital number (i.e. any value in A050278).

Original entry on oeis.org

111119, 123457, 178889, 199999, 224467, 246913, 325477, 333337, 333367, 333667, 336667, 345679, 359147, 361909, 387403, 394549, 411113, 419753, 443221, 444449, 449161, 470551, 473219, 476647, 476659, 504323, 506173, 509053, 512683, 513269, 514289, 514357

Offset: 1

David J. Seal, Sep 21 2017

This is the complement in A000040 of the finite list of primes that divide one or more 10-digit pandigital numbers. That finite list has been obtained by computer; it contains 1102173 primes, with the first prime that is not in the list being prime(10545) = 111119 and the last that is in the list being prime(55537259) = 1097393447.

			a(1) = 111119 because 111119 is prime and does not divide any of the 10-digit pandigital numbers 1023456789, 1023456798, ..., 9876543210, and all smaller primes do divide at least one of them.

Cf. A050278, A061604, A180489, A274328, A292703.

A292281 Number of magic labelings of the prism graph I X C_6 having magic sum n.

Original entry on oeis.org

1, 20, 167, 867, 3322, 10309, 27410, 64770, 139479, 278674, 523457, 933725, 1594008, 2620411, 4168756, 6444020, 9711165, 14307456, 20656363, 29283143, 40832198, 56086305, 75987814, 101661910, 134442035, 175897566, 227863845, 292474657, 372197252, 469870007, 588742824

Offset: 0

David J. Seal, Sep 13 2017

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A019298, A061927, A244497, A244879, A244873, A289992.

f[n_] := SeriesCoefficient[(1 + 11 x + 24 x^2 + 11 x^3 + x^4)/(1 - x)^7, {x, 0, n}]; Table[f[n] + 2 Sum[f[i], {i, 0, n - 1}], {n, 0, 24}] (* Michael De Vlieger, Sep 15 2017 *)

a(n) = A244879(n) + 2*Sum_{i=0..n-1} A244879(i).
From Colin Barker, Sep 13 2017: (Start)
G.f.: (1 + x)*(1 + 11*x + 24*x^2 + 11*x^3 + x^4) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>7.
(End)
[Proof of the g.f. follows from the g.f. of A244879 with the resummation demonstrated in A289992: g.f. = A244879(x)*(1+2*x/(1-x)). - R. J. Mathar, Mar 09 2025]

A289992 Number of magic labelings of the prism graph I X C_8 having magic sum n.

Original entry on oeis.org

1, 49, 746, 6122, 34067, 144963, 506772, 1524628, 4074949, 9898229, 22220990, 46695870, 92769495, 175610631, 318756136, 557659432, 944355593, 1553488697, 2489980818, 3898657938, 5976186139, 8985711691, 13274641084, 19296041660, 27634190285

Offset: 0

David J. Seal, Sep 13 2017

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A019298, A061927, A244497, A292281, A244873, A244880.

a(n) = A244880(n) + 2*Sum_{i=0..n-1} A244880(i).
From Colin Barker, Sep 13 2017: (Start)
G.f.: (1 + x)*(1 + 6*x + x^2)*(1 + 32*x + 70*x^2 + 32*x^3 + x^4) / (1 - x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>9. (End)
[Proof of the g.f. follows from the convolution formula and insertion of the g.f. A244880(x): Sum_{n>=0} a(n)x^n = Sum_{n>=0} A244880(n)*x^n +2*Sum_{n>=0} Sum_{i=0..n-1} A244880(i)*x^n = A244880(x) +2*Sum_{i>=0} Sum_{n>=i+1} A244880(i)*x^n = A244880(x) +2*Sum_{i>=0} A244880(i)*x^(i+1) Sum_{n>=0} x^n = A244880(x)+2*A244880(x)*x/(1+x) = A244880(x)*(1+2*x/(1-x)). R. J. Mathar, Mar 09 2025]

A288847 Numbers whose trajectories under the map x -> A230625(x) never reach a prime.

Original entry on oeis.org

217, 255, 446, 558, 717, 735, 775, 945, 958, 1007, 1062, 1115, 1269, 1344, 1503, 1984, 2215, 2358, 3003, 3751, 3858, 4131, 4471, 5144, 6174, 6627, 6915, 6923, 7033, 7073, 7139, 7434, 7530, 7778, 8125, 8142, 8239, 8335, 8575, 8967, 9186, 9303, 10040, 10179, 10856, 11907, 12081, 12248

Offset: 1

David J. Seal, Jun 18 2017

Sequence suggested by N. J. A. Sloane on the SeqFan mailing list. These are also the numbers n for which A230626(n) = -1 and A230627(n) = A287875(n) = 0. All currently-listed terms (those <= 3858) enter one of the two loops 1007 <-> 1269 and 1503 <-> 3751.

Further values added (Jun 19 2017) based on Sean A. Irvine's extension of the b-file for A230626.

			217, 255, 945, 1007 and 1269 are in the sequence because under the map x -> A230625(x):
217 = 7*31 = binary 111*11111 -> binary 11111111 = 255
255 = 3*5*17 = binary 11*101*10001 -> binary 1110110001 = 945
945 = 3^3*5*7 = binary 11^11*101*111 -> binary 1111101111 = 1007
1007 = 19*53 = binary 10011*110101 -> binary 10011110101 = 1269
1269 = 3^3*47 = binary 11^11*101111 -> binary 1111101111 = 1007

Cf. A230625, A230626, A230627, A287874, A287875, A287878.

a(43)-a(48) from Chai Wah Wu, Jul 13 2017

cp's OEIS Frontend

User: David J. Seal

A331905 Number of spanning trees in the multigraph cube of an n-cycle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A292703 Values of n such that prime(n) does not divide any 10-digit pandigital number (i.e. any value in A050278).

Views

Author

Keywords

Comments

Examples

Crossrefs

A292471 Primes that do not divide any 10-digit pandigital number (i.e. any value in A050278).

Views

Author

Keywords

Comments

Examples

Crossrefs

A292281 Number of magic labelings of the prism graph I X C_6 having magic sum n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289992 Number of magic labelings of the prism graph I X C_8 having magic sum n.

Views

Author

Keywords

Links

Crossrefs

Formula

A288847 Numbers whose trajectories under the map x -> A230625(x) never reach a prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions