A179805 -id:A179805 - cp's OEIS frontend

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

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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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

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

A179901 Triangle read by rows, antidiagonals of an array generated from (1, r, r, r, ...) convolved with (1, 0, r, r, r, ...).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 6, 8, 4, 1, 5, 8, 15, 12, 5, 1, 6, 10, 24, 24, 16, 6, 1, 7, 12, 35, 40, 33, 20, 7, 1, 8, 14, 48, 60, 56, 42, 24, 8, 1, 9, 16, 63, 84, 85, 70, 51, 28, 9, 1, 10, 18, 80, 112, 120, 110, 88, 60, 32, 10, 1, 11, 20, 99, 144, 161, 156, 135, 104, 69, 36, 11

Offset: 1

Gary W. Adamson, Jul 31 2010

Row sums = A179902: (1, 2, 5, 11, 23, 46, 87, 155, ...).

			First few rows of the array:
.
1,.1,..2,...3,...4,...5,...6,...7,....8,...
1,.2,..4,...8,..12,..16...20,..24,...28,... = A019442
1,.3,..6,..15,..24,..33,..42,..51,...60,... = A179805
1,.4,..8,..24,..40,..56,..70,..88,..104,...
.
Example: row 4 = (1, 4, 8, 24, ...) = (1, 4, 4, 4, ...) * (1, 0, 4, 4, 4, ...) = (1, r, 2*r, (2*r + r^2), ...).
.
First few rows of the triangle:
.
1,
1, 1;
1, 2, 2;
1, 3, 4, 3;
1, 4, 6, 8, 4;
1, 5, 8, 15, 12, 5;
1, 6, 10, 24, 24, 16, 6;
1, 7, 12, 35, 40, 33, 20, 7;
1, 8, 14, 48, 60, 56, 42, 24, 8;
1, 9, 16, 63, 84, 85, 70, 51, 28, 9;
1, 10, 18, 80, 112, 120, 110, 88, 60, 32, 10;
1, 11, 20, 99, 144, 161, 156, 135, 104, 69, 36, 11;
1, 12, 22, 120, 180, 208, 210, 192, 160, 120, 78, 40, 12;
1, 13, 24, 143, 220, 261, 272, 259, 228, 185, 136, 87, 44, 13;
...

Cf. A019442, A179805

Triangle read by rows, antidiagonals of an array generated from (1, r, r, r, ...) convolved with (1, 0, r, r, r, ...), such that the r-th row of the array = (1, r, 2*r, ...) then for n > 3, a(n) = r^2 + a(n-1).

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.

Original entry on oeis.org

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

A343053 Table read by ascending antidiagonals: T(k, n) is the maximum vertex sum in a perimeter-magic k-gon of order n.

Original entry on oeis.org

15, 24, 24, 40, 42, 33, 54, 65, 56, 42, 77, 93, 90, 74, 51, 96, 126, 126, 115, 88, 60, 126, 164, 175, 165, 140, 106, 69, 150, 207, 224, 224, 198, 165, 120, 78, 187, 255, 288, 292, 273, 237, 190, 138, 87, 216, 308, 350, 369, 352, 322, 270, 215, 152, 96, 260, 366, 429, 455, 450, 420, 371, 309, 240, 170, 105

Offset: 3

Stefano Spezia, Apr 03 2021

			The table begins:
k\n|   3    4    5    6    7 ...
---+------------------------
3  |  15   24   33   42   51 ...
4  |  24   42   56   74   88 ...
5  |  40   65   90  115  140 ...
6  |  54   93  126  165  198 ...
7  |  77  126  175  224  273 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 6 and 8).

Cf. A005475 (n = 4), A022267 (n = 6), A059270, A179805 (k = 3), A343052 (minimum).

T[k_,n_]:=k(1+k(2n-3)-Mod[n,2](1-Mod[k,2]))/2; Table[T[k+3-n,n],{k,3,14},{n,3,k}]//Flatten

T(k, n) = k*(1 + k*(2n - 3) - (n mod 2)*(1 - (k mod 2)))/2.

T(n, n) = A059270(n-1).

cp's OEIS Frontend

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

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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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A179901 Triangle read by rows, antidiagonals of an array generated from (1, r, r, r, ...) convolved with (1, 0, r, r, r, ...).

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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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Author

Keywords

Examples

Links

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Programs

PARI

Extensions

A343053 Table read by ascending antidiagonals: T(k, n) is the maximum vertex sum in a perimeter-magic k-gon of order n.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula