A244497 -id:A244497 - cp's OEIS frontend

A293310 Number of magic labelings of the graph LOOP X C_9 (see comments) having magic sum n, n >= 0.

1, 76, 1460, 13604, 81555, 363606, 1310974, 4029310, 10936124, 26868719, 60843972, 128724276, 257103166, 488789593, 890341484, 1562177132, 2651877099, 4371379686, 7018869628, 11006262508, 16893296453, 25429357976, 37604290362

Offset: 0

L. Edson Jeffery, Oct 06 2017

nonn

The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.

The generating function for this sequence was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews et al.

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A293311, A293312.

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293309 (magic labelings of LOOP X C_k, for k=1..8,10).

CoefficientList[Series[(1 + 67*z + 811*z^2 + 3049*z^3 + 4609*z^4 + 3049*z^5 + 811*z^6 + 67*z^7 + z^8)/((1 + z)*(1 - z)^10), {z, 0, 22}], z]

my(x='x+O('x^99));Vec((1+67*x+811*x^2+3049*x^3+4609*x^4+3049*x^5+811*x^6+67*x^7+x^8)/((1+x)*(1-x)^10)) \\ Altug Alkan, Oct 11 2017

G.f.: (1 + 67*z + 811*z^2 + 3049*z^3 + 4609*z^4 + 3049*z^5 + 811*z^6 + 67*z^7 + z^8)/((1 + z)*(1 - z)^10).

A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 2, 4, 1, 4, 6, 2, 5, 1, 7, 11, 10, 3, 6, 1, 11, 26, 23, 15, 3, 7, 1, 18, 57, 70, 42, 21, 4, 8, 1, 29, 129, 197, 155, 69, 28, 4, 9, 1, 47, 289, 571, 533, 301, 106, 36, 5, 10, 1, 76, 650, 1640, 1884, 1223, 532, 154, 45, 5, 11

Offset: 1

L. Edson Jeffery, Oct 10 2017

Conjecture: For all n >= 1, for all k >= 2, A(n, k) = A293311(k, n); i.e., A(n, k) = number of magic labelings of the graph LOOP X C_k with magic sum n - 1.

			Array begins:
.   1 1  1   1    1     1      1       1       1        1         1
.   2 1  3   4    7    11     18      29      47       76       123
.   3 2  6  11   26    57    129     289     650     1460      3281
.   4 2 10  23   70   197    571    1640    4726    13604     39175
.   5 3 15  42  155   533   1884    6604   23219    81555    286555
.   6 3 21  69  301  1223   5103   21122   87677   363606   1508401
.   7 4 28 106  532  2494  11998   57271  274132  1310974   6271378
.   8 4 36 154  876  4654  25362  137155  743724  4029310  21836366
.   9 5 45 215 1365  8105  49347  298184 1806597 10936124  66220705
.  10 5 55 290 2035 13355  89848  599954 4016683 26868719 179784715
.  11 6 66 381 2926 21031 154935 1132942 8306078 60843972 445824731
.  ...

Cf. A293311.
Cf. A000012, A000032, A274975, A188128, A189237 (rows 1..5).
Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (columns k = 0,2..10 (conjectured)).

s[0, x_] := 1; s[1, x_] := x; s[k_, x_] := x*s[k - 1, x] - s[k - 2, x]; c[n_, j_] := 2 (-1)^(j - 1) Cos[j*Pi/(2 n + 1)]; a[n_, k_] := Round[Sum[s[n - 1, c[n, j]]^(k), {j, n}]];
(* Array: *)
Grid[Table[a[n, k], {n, 11}, {k, 0, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[a[n, k - n], {k, 11}, {n, k}]]

Let S(0, x) = 1, S(1, x) = x, S(k, x) = x*S(k - 1, x) - S(k - 2, x) (the S-polynomials of Wolfdieter Lang) and c(n, j) = 2*(-1)^(j - 1)*cos(j*Pi/(2*n + 1)). Then A(n, k) = Sum_{j=1..n} S(n - 1, c(n, j))^(k), n >= 1, k >= 0.

A380963 The number of perimeter-magic pentagons of order 3 with magic sum n.

Original entry on oeis.org

1, 9, 33, 75, 233, 374, 742, 1294, 2042, 3029, 4931, 6535, 9507, 13214, 17577, 22762, 31335, 38341, 49660, 62791, 77689, 94239, 119151, 139727, 170553, 204832, 242122, 282811, 340914, 388834, 456668, 530819, 609982, 694982, 810204, 906951, 1038672

Offset: 14

Derek Holton and Alex Holton, Feb 09 2025

nonn

The requirements are that there are 3 integers at each side of the pentagon (2 of them shared by adjacent sides), which sum up to n. All 10 integers on the 5 sides must be distinct. Pentagons obtained by reflections or rotations are considered to be the same.

If the 10 integers do not need to be distinct and if solutions by rotations around the five-fold symmetry axis and flips are considered distinct, there are A244497(n-3) perimeter-magic pentagons. - R. J. Mathar, Mar 10 2025

			for n = 14, a(14) = 1           5
                              6    7
                            3         2
                             10      8
                              1  9  4

R. J. Mathar, Generating perimeter-magic polygons, C++ (2025)

Cf. A380962 (perimeter-magic squares), A380853 (perimeter-magic triangles), A380964 (perimeter-magic hexagons).

cp's OEIS Frontend

A293310 Number of magic labelings of the graph LOOP X C_9 (see comments) having magic sum n, n >= 0.

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A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

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A380963 The number of perimeter-magic pentagons of order 3 with magic sum n.

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