A064842 -id:A064842 - cp's OEIS frontend

A064843 a(n) = A064842(n)/2.

0, 1, 3, 9, 18, 33, 53, 81, 116, 161, 215, 281, 358, 449, 553, 673, 808, 961, 1131, 1321, 1530, 1761, 2013, 2289, 2588, 2913, 3263, 3641, 4046, 4481, 4945, 5441, 5968, 6529, 7123, 7753, 8418, 9121, 9861, 10641, 11460, 12321, 13223, 14169

Offset: 1

N. J. A. Sloane, Oct 25 2001

Partial sums of A084265. - N. J. A. Sloane, Jul 20 2008

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A064842, A084265.

G.f.: -x^2*(-1+x^3-2*x^2) / ( (1+x)*(x-1)^4 ). - R. J. Mathar, Nov 26 2012

A110610 Maximal value of sum(p(i)p(i+1),i=1..n), where p(n+1)=p(1), as p ranges over all permutations of {1,2,...,n}.

Original entry on oeis.org

1, 4, 11, 25, 48, 82, 129, 191, 270, 368, 487, 629, 796, 990, 1213, 1467, 1754, 2076, 2435, 2833, 3272, 3754, 4281, 4855, 5478, 6152, 6879, 7661, 8500, 9398, 10357, 11379, 12466, 13620, 14843, 16137, 17504, 18946, 20465, 22063, 23742, 25504

Offset: 1

Emeric Deutsch, Jul 30 2005

			a(4)=25 because the values of the sum for the permutations of {1,2,3,4} are 21 (8 times), 24 (8 times) and 25 (8 times).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, The Fifty-Seventh William Lowell Putnam Competition, Amer. Math. Monthly, 104, 1997, 744-754, Problem B-3.
Vasile Mihai and Michael Woltermann, Problem 10725: The Smoothest and Roughest Permutations, Amer. Math. Monthly, 108 (March 2001), pp. 272-273.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A016825, A110611.

Cf. also A064842, A087035, A185173.

a:=proc(n) if n=1 then 1 else (2*n^3+3*n^2-11*n+18)/6 fi end: seq(a(n),n=1..50);

Rest@ CoefficientList[Series[x (1 + x) (1 - x + 2 x^2 - x^3)/(1 - x)^4, {x, 0, 42}], x] (* Michael De Vlieger, Jan 29 2022 *)

a(1)=1; a(n)=(2n^3+3n^2-11n+18)/6 for n>=2.

G.f.: x*(1+x)*(1-x+2*x^2-x^3)/(1-x)^4. [Colin Barker, Jul 24 2012]

A110611 Minimal value of sum(p(i)p(i+1),i=1..n), where p(n+1)=p(1), as p ranges over all permutations of {1,2,...,n}.

Original entry on oeis.org

1, 4, 11, 21, 37, 58, 87, 123, 169, 224, 291, 369, 461, 566, 687, 823, 977, 1148, 1339, 1549, 1781, 2034, 2311, 2611, 2937, 3288, 3667, 4073, 4509, 4974, 5471, 5999, 6561, 7156, 7787, 8453, 9157, 9898, 10679, 11499, 12361, 13264, 14211, 15201, 16237

Offset: 1

Emeric Deutsch, Jul 30 2005

			a(4)=21 because the values of the sum for the permutations of {1,2,3,4} are 21 (8 times), 24 (8 times) and 25 (8 times).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, The Fifty-Seventh William Lowell Putnam Competition, Amer. Math. Monthly, 104, 1997, 744-754, Problem B-3.
Vasile Mihai and Michael Woltermann, Problem 10725: The Smoothest and Roughest Permutations, Amer. Math. Monthly, 108 (March 2001), pp. 272-273.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A064842, A110610.

I:=[1, 4, 11, 21, 37]; [n le 5 select I[n] else 3*Self(n-1)-2*Self(n-2)-2*Self(n-3)+3*Self(n-4)-Self(n-5): n in [1..50]]; // Vincenzo Librandi, May 11 2012

a:=proc(n) if n mod 2 = 0 then (n^3+3*n^2+5*n-6)/6 else (n^3+3*n^2+5*n-3)/6 fi end: seq(a(n),n=1..52);

CoefficientList[Series[(1+x+x^2-2*x^3+x^4)/((1-x)^4*(1+x)),{x,0,50}],x] (* Vincenzo Librandi, May 11 2012 *)

a(n) = (n^3+3*n^2+5*n-6)/6 if n is even; a(n)=(n^3+3*n^2+5*n-3)/6 if n is odd.
G.f.: x*(1+x+x^2-2*x^3+x^4)/((1-x)^4*(1+x)). [Colin Barker, May 10 2012]
a(n) = (2*n^3+6*n^2+10*n-9-3*(-1)^n)/12. - Luce ETIENNE, Jul 26 2014

A358212 a(n) is the maximal possible sum of squares of the side lengths of an n^2-gon supported on a subset 1 <= x,y <= n of an integer lattice.

Original entry on oeis.org

4, 10, 36, 98, 232

Offset: 2

Giedrius Alkauskas, Nov 04 2022

Examples show that a(7) >= 462, a(8) >= 842, a(9) >= 1424, a(10) >= 2242.

Asymptotics: liminf a(n)/n^4 >= 8/27, limsup a(n)/n^4 <= 2/3.

Oliver Mantas Ališauskas, Grid connector, Web application for this problem.
Oliver Mantas Ališauskas, Giedrius Alkauskas, and Valdas Dičiūnas, Full Grid Lattice Polygons with Maximal Sum of Squares of Edge-Lengths, arXiv:2311.03011 [math.CO], 2023-2024.
S. Chow, A. Gafni, and P. Gafni, Connecting the dots: maximal polygons on a square grid, Math. Mag. 94 (2021), no. 2, 118-124.
G. L. Cohen and E. Tonkes, Dartboard arrangements, Elect. J. Combin., 8(2) (2001), #R4.

Cf. A209077, A110611, A064842, A226595, A226596.

a(5) from Giedrius Alkauskas, Oct 09 2023

a(6) from Giedrius Alkauskas, Nov 30 2023

cp's OEIS Frontend

A064843 a(n) = A064842(n)/2.

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A110610 Maximal value of sum(p(i)p(i+1),i=1..n), where p(n+1)=p(1), as p ranges over all permutations of {1,2,...,n}.

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A110611 Minimal value of sum(p(i)p(i+1),i=1..n), where p(n+1)=p(1), as p ranges over all permutations of {1,2,...,n}.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A358212 a(n) is the maximal possible sum of squares of the side lengths of an n^2-gon supported on a subset 1 <= x,y <= n of an integer lattice.

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Author

Keywords

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