A285009 -id:A285009 - cp's OEIS frontend

A342384 Irregular triangle T read by rows: T(n, k) is the number of n-th order magic triangles with magic constant equal to A285009(n) + k, with 0 < k <= 3*n - 5.

0, 1, 1, 1, 1, 2, 0, 4, 6, 4, 0, 2, 18, 38, 71, 108, 115, 115, 108, 71, 38, 18, 155, 351, 695, 1067, 1475, 1815, 2007, 1815, 1475, 1067, 695, 351, 155, 1891, 4768, 9872, 15370, 22527, 30096, 35731, 37957, 37957, 35731, 30096, 22527, 15370, 9872, 4768, 1891

Offset: 2

Stefano Spezia, Mar 10 2021

			The triangle begins:
    0;
    1,   1,   1,    1;
    2,   0,   4,    6,    4,    0,    2;
   18,  38,  71,  108,  115,  115,  108,   71,   38,   18;
  155, 351, 695, 1067, 1475, 1815, 2007, 1815, 1475, 1067, 695, 351, 155;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..118 (rows 2..10)
Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20.

Cf. A016777 (row length), A179805, A285009, A341740, A342467 (row sums).

\\ See A342467 for program code.
{ for(n=2, 6, print(A342384row(n))) } \\ Andrew Howroyd, Feb 05 2022

Terms a(14) and beyond from Andrew Howroyd, Feb 05 2022

A342467 a(n) is the number of n-th order magic triangles.

Original entry on oeis.org

0, 4, 18, 700, 13123, 316424, 7317145, 176476738, 4279366371

Offset: 2

Stefano Spezia, Mar 13 2021

The Trotter reference gives the value 1356 = 76 * 18 for a(5), which is incorrect since 76 is the number of corner groupings and 18 is the maximum number of solutions in any grouping. - Andrew Howroyd, Feb 05 2022

Andrew Howroyd, PARI Program, Feb 2022.
Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20.

Row sums of A342384.

Cf. A016777, A179805, A285009, A341740, A351223.

PARI
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\\ See Links.
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a(n) < A351223(n). - Stefano Spezia, Feb 05 2022

a(5) corrected by Andrew Howroyd and Stefano Spezia, Feb 05 2022

a(6)-a(10) from Andrew Howroyd, Feb 05 2022

A342757 Array read by ascending antidiagonals: T(k, n) is the minimum value of the magic constant in a perimeter-magic k-gon of order n.

Original entry on oeis.org

9, 12, 17, 14, 22, 28, 17, 27, 37, 42, 19, 32, 45, 55, 59, 22, 37, 54, 68, 78, 79, 24, 42, 62, 81, 96, 104, 102, 27, 47, 71, 94, 115, 129, 135, 128, 29, 52, 79, 107, 133, 154, 167, 169, 157, 32, 57, 88, 120, 152, 179, 200, 210, 208, 189, 34, 62, 96, 133, 170, 204, 232, 251, 258, 250, 224

Offset: 3

Stefano Spezia, Mar 21 2021

			The array begins:
k\n|  3   4   5   6    7 ...
---+------------------------
3  |  9  17  28  42   59 ...
4  | 12  22  37  55   78 ...
5  | 14  27  45  68   96 ...
6  | 17  32  54  81  115 ...
7  | 19  37  62  94  133 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10 and 12).

Cf. A016873 (n = 4), A285009 (k = 3), A342719, A342758 (maximum).

T[k_,n_]:= ((1-Mod[k,2])Mod[n,2]+k*(n^2-2*n+2)+n)/2; Table[T[k+3-n,n],{k,3,13},{n,3,k}]//Flatten

G.f.: (x^2*(-3*y^3 + 2*y - 1) - x*(2*y^3 + y^2 - 2*y + 1) + (y - 1)*y)/((x - 1)^2*(x + 1)*(y - 1)^3*(y + 1)).
T(k, n) = (n^2/2 - n + 1)*k + n/2 if n is even or both n and k are odd.
T(k, n) = (n^2/2 - n + 1)*k + (n + 1)/2 if n is odd and k is even.
T(k, n) = ((1 - (k mod 2))*(n mod 2) + k*(n^2 - 2*n + 2) + n)/2.

A341740 a(n) is the maximum value of the magic constant in a normal magic triangle of order n.

Original entry on oeis.org

12, 23, 37, 54, 74, 97, 123, 152, 184, 219, 257, 298, 342, 389, 439, 492, 548, 607, 669, 734, 802, 873, 947, 1024, 1104, 1187, 1273, 1362, 1454, 1549, 1647, 1748, 1852, 1959, 2069, 2182, 2298, 2417, 2539, 2664, 2792, 2923, 3057, 3194, 3334, 3477, 3623, 3772, 3924

Offset: 3

Stefano Spezia, Feb 18 2021

Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005449, A016777, A095794, A130808, A140090, A179805, A285009.

LinearRecurrence[{3,-3,1},{12,23,37},49]

O.g.f.: x^3*(12 - 13*x + 4*x^2)/(1 - x)^3.
E.g.f.: 3 + x - 2*x^2 - exp(x)*(6 - 4*x - 3*x^2)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
a(n) = (3*n^2 + n - 6)/2 for n > 2.
a(n) = A285009(n) + A016777(n-2) - 1 for n > 3.
a(n) = A095794(n) - 2 = A140090(n-1) - 1. - Hugo Pfoertner, Feb 18 2021

cp's OEIS Frontend

A342384 Irregular triangle T read by rows: T(n, k) is the number of n-th order magic triangles with magic constant equal to A285009(n) + k, with 0 < k <= 3*n - 5.

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A342467 a(n) is the number of n-th order magic triangles.

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A342757 Array read by ascending antidiagonals: T(k, n) is the minimum value of the magic constant in a perimeter-magic k-gon of order n.

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A341740 a(n) is the maximum value of the magic constant in a normal magic triangle of order n.

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Mathematica

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