cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A342819 Table read by ascending antidiagonals: T(k, n) is the number of distinct values of the magic constant in a perimeter-magic k-gon of order n.

Original entry on oeis.org

4, 4, 7, 6, 9, 10, 6, 11, 12, 13, 8, 13, 16, 17, 16, 8, 15, 18, 21, 20, 19, 10, 17, 22, 25, 26, 25, 22, 10, 19, 24, 29, 30, 31, 28, 25, 12, 21, 28, 33, 36, 37, 36, 33, 28, 12, 23, 30, 37, 40, 43, 42, 41, 36, 31, 14, 25, 34, 41, 46, 49, 50, 49, 46, 41, 34, 14, 27, 36, 45, 50, 55, 56, 57, 54, 51, 44, 37
Offset: 3

Views

Author

Stefano Spezia, Mar 22 2021

Keywords

Examples

			The table begins:
k\n|  3   4   5   6   7 ...
---+-------------------
3  |  4   7  10  13  16 ...
4  |  4   9  12  17  20 ...
5  |  6  11  16  21  26 ...
6  |  6  13  18  25  30 ...
7  |  8  15  22  29  36 ...
...

Links

  • Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10-13).

Crossrefs

Cf. A005408 (n = 4), A016813 (n = 6), A016921 (n = 8), A017077 (n = 10), A146951 (n = 7), A238290 (n = 9), A342757, A342758.

Programs

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

Formula

O.g.f.: (1 - y + 2*x*(y^2 + y - 1) + x^2*(4*y^2 + y - 3))/((1 - x)^2*(1 + x)*(1 - y)^2*(1 + y)).
E.g.f.: (1 + x*(y - 2))*cosh(x + y) + cosh(y)*sinh(x) + x*(y - 2)*sinh(x + y).
T(k, n) = k*(n - 2) + ((k mod 2) - 1)*(n mod 2) + 1.
T(k, n) = A342758(k, n) - A342757(k, n) + 1.

A356790 Table read by antidiagonals: T(n,k) (n >= 1, k >= 1) is the number of regions formed by straight line segments when connecting the k-1 points along the top side of a rectangle to each of the k-1 points along the bottom side that divide these sides into k equal parts, along with straight lines that directly connect the n-1 points along the left side to the diametrically opposite point on the right side that divide these sides into n equal parts.

Original entry on oeis.org

1, 2, 2, 6, 4, 3, 18, 10, 6, 4, 48, 24, 16, 8, 5, 106, 56, 34, 20, 10, 6, 216, 116, 70, 44, 26, 12, 7, 382, 228, 134, 84, 58, 30, 14, 8, 650, 396, 250, 152, 112, 60, 36, 16, 9, 1030, 666, 422, 272, 190, 112, 78, 40, 18, 10, 1564, 1048, 696, 448, 320, 196, 150, 84, 46, 20, 11
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Sep 04 2022

Keywords

Examples

			The table begins:
1,  2,  6,  18,  48,  106, 216, 382, 650,  1030, 1564, 2258, 3210, 4386, 5926, ...
2,  4,  10, 24,  56,  116, 228, 396, 666,  1048, 1584, 2280, 3234, 4412, 5954, ...
3,  6,  16, 34,  70,  134, 250, 422, 696,  1082, 1622, 2322, 3280, 4462, 6008, ...
4,  8,  20, 44,  84,  152, 272, 448, 726,  1116, 1660, 2364, 3326, 4512, 6062, ...
5,  10, 26, 58,  112, 190, 320, 506, 794,  1194, 1748, 2462, 3434, 4630, 6190, ...
6,  12, 30, 60,  112, 196, 326, 512, 800,  1200, 1754, 2468, 3440, 4636, 6196, ...
7,  14, 36, 78,  150, 258, 418, 626, 936,  1358, 1934, 2670, 3664, 4882, 6464, ...
8,  16, 40, 84,  152, 256, 414, 632, 942,  1364, 1940, 2676, 3670, 4888, 6470, ...
9,  18, 46, 94,  172, 290, 468, 710, 1050, 1490, 2084, 2838, 3850, 5086, 6686, ...
10, 20, 50, 104, 188, 304, 480, 720, 1060, 1516, 2112, 2868, 3882, 5120, 6722, ...
11, 22, 56, 118, 218, 366, 586, 878, 1280, 1794, 2454, 3258, 4320, 5606, 7256, ...
12, 24, 60, 120, 208, 336, 518, 764, 1114, 1580, 2204, 2992, 4020, 5272, 6888, ...
.
.
See the attached table for further terms.

Links

Crossrefs

Cf. A146951, A355798, A306302, A290131, A331452, A355902.

Formula

T(1,k) = A306302(k-2) + 2, k >= 2.
T(2,k) = 2*A355902(k-2) + 4 = A306302(k-2) + 2*k, k >= 2.
T(n,1) = n.
T(n,2) = 2n.
T(n,3) = A146951(n).

A131501 Xm/CV where Xm is a point of maximum error using an approximation method for x^(1/2) which I have found and CV is the population coefficient of variation from my list of error values.

Original entry on oeis.org

6, 10, 16, 20, 26, 30, 36, 40, 46, 50
Offset: 1

Views

Author

Anthony J. Browne (tony2theipi(AT)yahoo.com), Aug 13 2007

Keywords

Comments

I am no expert at sequences, but my work is forcing me to be. I need only an equation to represent this sequence and I believe I will have completed my goal, as well as found a new approximation technique for square roots. It views them in a whole new way and should prove interesting to more formal mathematicians. This work has taken me 2.5 years and I would appreciate any help in its finalization.
a(n) = A146951(n) for 1 <= n <= 10, but more terms would be needed to justify such a hypothesis. - Georg Fischer, Nov 03 2018

Crossrefs

Cf. A146951.

Formula

The terms shown satisfy a(n) = 10n-4 if n is odd, a(n) = 10n-10 if n is even. - N. J. A. Sloane, Aug 15 2007
a(n) = 10*n - a(n-1) - 4, a(1)=6. - Vincenzo Librandi, Nov 23 2010
Showing 1-3 of 3 results.

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