A343053 -id:A343053 - cp's OEIS frontend

A377521 Antidiagonal sums of A343053.

0, 0, 0, 15, 48, 115, 217, 385, 611, 945, 1366, 1947, 2650, 3575, 4663, 6045, 7637, 9605, 11836, 14535, 17556, 21147, 25125, 29785, 34903, 40825, 47282, 54675, 62686, 71775, 81571, 92597, 104425, 117645, 131768, 147455, 164152, 182595, 202161, 223665, 246411, 271297, 297550

Offset: 0

Stefano Spezia, Jan 03 2025

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A343053.

LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,0,0,15,48,115,217,385,611,945},43]

a(n) = (2 - n)*(12*(1 + (-1)^n) - 105*n + 3*(-1)^n*n - 32*n^2 - 4*n^3)/48 for n > 1.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 9.
G.f.: x^3*(15 + 18*x - 11*x^2 - 19*x^3 + 9*x^4 + 7*x^5 - 3*x^6)/((1 - x)^5*(1 + x)^3).
E.g.f.: (1 - 3*x + 23*x^2/8 + x^3 + x^4/12)*cosh(x) - x*(81 - 72*x - 24*x^2 - 2*x^3)*sinh(x)/24 - 1 + 3*x.

A343052 Table read by ascending antidiagonals: T(k, n) is the minimum vertex sum in a perimeter-magic k-gon of order n.

Original entry on oeis.org

6, 12, 6, 15, 10, 6, 24, 15, 12, 6, 28, 21, 15, 10, 6, 40, 28, 24, 15, 12, 6, 45, 36, 28, 21, 15, 10, 6, 60, 45, 40, 28, 24, 15, 12, 6, 66, 55, 45, 36, 28, 21, 15, 10, 6, 84, 66, 60, 45, 40, 28, 24, 15, 12, 6, 91, 78, 66, 55, 45, 36, 28, 21, 15, 10, 6, 112, 91, 84, 66, 60, 45, 40, 28, 24, 15, 12, 6

Offset: 3

Stefano Spezia, Apr 03 2021

			The table begins:
k\n|   3   4   5   6   7 ...
---+--------------------
3  |   6   6   6   6   6 ...
4  |  12  10  12  10  12 ...
5  |  15  15  15  15  15 ...
6  |  24  21  24  21  24 ...
7  |  28  28  28  28  28 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 5 and 7).

Cf. A000217 (n = 4), A010722 (k = 3), A010854 (k = 5), A010867 (k = 7), A265225, A343053 (maximum).

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

O.g.f.: x*(1 + x^2 + y + x*(2 + 3*y))/((1 - x)^3*(1 + x)^2*(1 - y^2)).
E.g.f.: x*((5 + 2*x)*cosh(x + y) - cosh(x - y) + 2*(2 + x)*sinh(x + y))/4.
T(k, n) = k*(1 + k + (n mod 2)*(1 - (k mod 2)))/2.
T(k, 3) = A265225(k-1) (conjectured).

cp's OEIS Frontend

A377521 Antidiagonal sums of A343053.

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