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.

User: Bill Correll, Jr.

Bill Correll, Jr.'s wiki page.

Bill Correll, Jr. has authored 3 sequences.

A243643 Number of non-degenerate parallelograms in an n X n permutation array.

Original entry on oeis.org

0, 0, 0, 6, 62, 362, 1266, 3692, 8904, 19376, 38144, 70574, 122478, 203722, 324554, 500868, 749128, 1094204, 1560240, 2183562, 2998598, 4055502, 5402450, 7107528, 9233488, 11870016, 15100040, 19037586, 23787678, 29495558, 36290898, 44354248, 53846392, 64985376
Offset: 1

Views

Author

Christopher N. Swanson, Randy Ho, Bill Correll, Jr., Jun 08 2014

Keywords

Comments

A non-degenerate parallelogram has positive area.

Links

A234471 Number of sets of 4 distinct collinear points in an n X n permutation array in which leftmost pair has same spacing as rightmost pair.

Original entry on oeis.org

0, 0, 0, 2, 10, 30, 86, 180, 344, 624, 1056, 1626, 2522, 3646, 5118, 7164, 9784, 12868, 16848, 21438, 27202, 34298, 42550, 51800, 63184, 76256, 91032, 108086, 127850, 149442, 174902, 202552, 233608, 268512, 306664, 349394, 398226, 450510, 507038, 569204
Offset: 1

Views

Author

Christopher N. Swanson, Randy Ho, and Bill Correll, Jr., Jun 08 2014

Keywords

Comments

Configuration of points forms a degenerate parallelogram.

Links

Crossrefs

Cf. A243643.

A212892 a(n) = n^4/8 if n is even, a(n) = (n^2-1)^2/8 if n is odd.

Original entry on oeis.org

0, 0, 2, 8, 32, 72, 162, 288, 512, 800, 1250, 1800, 2592, 3528, 4802, 6272, 8192, 10368, 13122, 16200, 20000, 24200, 29282, 34848, 41472, 48672, 57122, 66248, 76832, 88200, 101250, 115200, 131072, 147968, 167042, 187272, 209952, 233928, 260642, 288800, 320000, 352800, 388962, 426888
Offset: 0

Views

Author

Clark Kimberling, May 30 2012, and Bill Correll, Jr., Jun 07 2014

Keywords

Comments

The sequence a(0)=0, a(1)=2, a(2)=8, a(3)=32, ... arises as the number of quadruples (w,x,y,z) with all terms in {0,...,n} and w-x, x-y, and y-z all odd. For a guide to related sequences, see A211795.
The sequence a(3)=2, a(4)=8, a(5)=32, ... is the number of L3 configurations in an n X n permutation array. An L3 configuration is defined to be a set of 3 equally-spaced, collinear points in a permutation array. L3 configurations were first enumerated by Davies in his study of the density of Costas arrays. They constitute a violation of the definition of a Costas array, so Costas arrays cannot have any. Davies's work went unpublished until it appeared in the survey paper by Drakakis. - Bill Correll, Jr., Jun 07 2014
Every term is even.

Links

Crossrefs

Cf. A211795.
Cf. A030179.

Programs

  • Maple
    f:=n->if n mod 2 = 0 then n^4/8 else (n^2-1)^2/8; fi; [seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 06 2015
  • Mathematica
    (* If the sequence begins a(0)=0, a(1)=2, a(2)=8, a(3)=32, ... *)
t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[Mod[w - x, 2] == Mod[x - y, 2] == Mod[y - z, 2] == 1, s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 40]]   (* A212892 *)
m/2 (* integers *)

Formula

If the sequence begins a(0)=0, a(1)=2, a(2)=8, a(3)=32, ...: 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). G.f.: f(x)/g(x), where f(x) = x+2*x^2+6*x^3+2*x^4+x^5 and g(x) = ((1-x)^5)*(1+x)^3.
Sum_{n>=2} 1/a(n) = Pi^4/180 + Pi^2/6 - 3/2. - Amiram Eldar, Sep 08 2022
a(n) = 2*floor(n^2/4)^2 = 2*A030179(n). - Ridouane Oudra, Sep 12 2023

Extensions

Formed by merging two entries that arose in different contexts. Thanks to Alois P. Heinz, Mar 04 2015 for noticing that the sequences were essentially identical. - N. J. A. Sloane, Mar 06 2015

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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