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: Al Zimmermann

Al Zimmermann's wiki page.

Al Zimmermann has authored 4 sequences.

A355015 The least integer with cost n, using the cost function used in sequence A354914.

Original entry on oeis.org

1, 2, 3, 7, 23, 719, 1169951
Offset: 0

Author

Stan Wagon, Joseph DeVincentis, and Al Zimmermann, Jun 15 2022

Keywords

Comments

The values up to 1169951 were computed by Joseph DeVincentis, Stan Wagon, and Al Zimmermann. The values appear to rise roughly quadratically, so the next one might be near 10^12 and impossible to find. It is known that the sequence is infinite: that is, the cost function is not bounded.

Examples

			Example: a(4) = 23 because 23 can be reached by the path 1, 2, 3, 4, 5, 20, 23, which has 4 addition steps, and one can check that each smaller number has cost at most 3.
		

Crossrefs

Cf. A354914.

Extensions

a(6) corrected by Stan Wagon, Feb 15 2023

A115866 a(n) = g(n,n,n) where g(a, b, c) is defined as follows: if a = 0 or b = 0 or c = 0 then return 1 otherwise return g(a, b, c-1) + g(a, b-1, c) + g(a-1, b, c) + g(a, b-1, c-1) + g(a-1, b, c-1) + g(a-1, b-1, c) + g(a-1, b-1, c-1).

Original entry on oeis.org

1, 7, 157, 5419, 220561, 9763807, 454635973, 21894817147, 1080094827649, 54250971690007, 2763339510402637, 142338478909290187, 7399210542653679985, 387578046480606144079, 20433042381373273363477, 1083193405190852829195259, 57697563083258107660231681
Offset: 0

Author

Al Zimmermann, Apr 02 2006

Keywords

Comments

A generalization of the recurrence in A001850. The original description of this sequence was the same as that of A126086. The correct explanation for these terms was provided by Nick Hobson, Mar 03 2007.

Crossrefs

Column k=3 of A263159.

Programs

  • Maple
    g():= seq(convert(n, base, 2)[1..3], n=9..15):
    b:= proc(l) option remember;
          `if`(l[1]=0, 1, add(b(sort(l-h)), h=g()))
        end:
    a:= n-> b([n$3]):
    seq(a(n), n=0..25);  # Alois P. Heinz, Oct 14 2015
  • Mathematica
    g[] = Table[Reverse[IntegerDigits[n, 2]][[;; 3]], {n, 2^3 + 1, 2^4 - 1}];
    b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[]}]];
    a[n_] := b[Table[n, {3}]];
    a /@ Range[0, 25] (* Jean-François Alcover, Apr 25 2020, after Alois P. Heinz *)

Formula

D-finite with recurrence: 2*(n-1)^2*(2*n-1)*(243*n^5 - 3159*n^4 + 16254*n^3 - 41325*n^2 + 51838*n - 25620)*a(n) = (53703*n^8 - 887922*n^7 + 6273882*n^6 - 24692601*n^5 + 59070956*n^4 - 87717383*n^3 + 78694087*n^2 - 38816698*n + 8003688)*a(n-1) + (94527*n^8 - 1549611*n^7 + 10848681*n^6 - 42278007*n^5 + 100087538*n^4 - 147021644*n^3 + 130465402*n^2 - 63678226*n + 13003980)*a(n-2) - (31833*n^8 - 541890*n^7 + 3945213*n^6 - 16007835*n^5 + 39486422*n^4 - 60435299*n^3 + 55812796*n^2 - 28273516*n + 5965068)*a(n-3) + (n-3)*(3159*n^7 - 48114*n^6 + 301212*n^5 - 1002003*n^4 + 1908157*n^3 - 2073535*n^2 + 1184960*n - 272792)*a(n-4) - 2*(n-4)^2*(n-3)*(243*n^5 - 1944*n^4 + 6048*n^3 - 9087*n^2 + 6529*n - 1769)*a(n-5). - Vaclav Kotesovec, Nov 27 2016
a(n) ~ (12*2^(2/3)+15*2^(1/3)+19)^n / (2^(4/3)*3^(1/2)*Pi*n). - Vaclav Kotesovec, Nov 27 2016

Extensions

Edited by N. J. A. Sloane following email from Nick Hobson, Mar 03 2007
More terms from Alois P. Heinz, Sep 30 2015

A068940 Maximum number of chess queens that can be placed on a 3-dimensional chessboard of order n so that no two queens attack each other.

Original entry on oeis.org

1, 1, 4, 7, 13, 21, 32, 48, 67, 91, 121, 133, 169
Offset: 1

Author

Al Zimmermann, Mar 08 2002

Keywords

Crossrefs

A068941 Maximum number of chess queens that can be placed on a 4-dimensional chessboard of order n so that no two attack each other.

Original entry on oeis.org

1, 1, 6, 16, 38, 80, 145
Offset: 1

Author

Al Zimmermann, Mar 08 2002

Keywords

Crossrefs

Extensions

a(5) from Alexis Langlois-Rémillard, Christoph Müßig and Érika Roldán added by Christoph Muessig, Nov 11 2022
a(6)-a(7) from Alexis Langlois-Rémillard, Christoph Müßig and Érika Roldán added by Christoph Muessig, Apr 02 2023