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.

Showing 1-3 of 3 results.

A057058 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; each k is an R(i(k),j(k)) and a(n)=i(A057027(n)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 2, 3, 1, 5, 2, 4, 3, 1, 6, 2, 5, 3, 4, 1, 7, 2, 6, 3, 5, 4, 1, 8, 2, 7, 3, 6, 4, 5, 1, 9, 2, 8, 3, 7, 4, 6, 5, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 1, 11, 2, 10, 3, 9, 4, 8, 5, 7, 6, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 1, 13, 2, 12, 3, 11, 4
Offset: 1

Views

Author

Clark Kimberling, Jul 30 2000

Keywords

Comments

Since A057027 is a permutation of the natural numbers, every natural number occurs infinitely many times in this sequence.
Consider the triangle TN := 1; 1, -2; 1, -3, 2; 1, -4, 2, -3; ... Antidiagonal sums give A129819(n+2). TN arises in studying the equation (E) dy/dx=Q(n,x,y)/P(n,x,y) involving saddle-points quantities, P and Q are bidimensional polynomials n=2,3,4.. . (E) leads also for instance to the one-dimension polynomials in A129326, A129587, A130679. - Paul Curtz, Aug 16 2008
First inverse function (numbers of rows) for pairing function A194982. - Boris Putievskiy, Jan 10 2013

Links

Crossrefs

Cf. A057059, A194982.

Formula

From Boris Putievskiy, Jan 10 2013: (Start)
a(n) = -((A002260(n)+1)/2)*((-1)^A002260(n)-1)/2+(A004736(n)+A002260(n)/2)*((-1)^A002260(n)+1)/2.
a(n) = -((i+1)/2)*((-1)^i-1)/2+(j+i/2)*((-1)^i+1)/2, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2). (End)

A158442 Triangle T(n,k) = [x^k] n!*(n+1+x^n)*Sum_{i=0..n-1} x^i/(i+1).

Original entry on oeis.org

2, 1, 6, 3, 2, 1, 24, 12, 8, 6, 3, 2, 120, 60, 40, 30, 24, 12, 8, 6, 720, 360, 240, 180, 144, 120, 60, 40, 30, 24, 5040, 2520, 1680, 1260, 1008, 840, 720, 360, 240, 180, 144, 120, 40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040, 2520, 1680, 1260, 1008, 840, 720, 362880
Offset: 1

Views

Author

Paul Curtz, Mar 19 2009

Keywords

Comments

The coefficient in front of x^k of the polynomial n!*(n+1+x^n)*Sum_{i=0..n-1} x^i/(i+1), columns k=0..2n-1.

Examples

			The triangle starts
    2,  1;
    6,  3,  2,  1;
   24, 12,  8,  6,  3,  2;
  120, 60, 40, 30, 24, 12,  8,  6;

Crossrefs

Cf. A130679 (table Q), A158442.

Programs

  • Maple
    P := proc(n,k) (n+1+x^n)*add( x^i/(i+1),i=0..n-1) ; coeftayl(expand(%),x=0,k) ; end:
T := proc(n,k) n!*P(n,k) ; end:
for n from 1 to 10 do for k from 0 to 2*n-1 do printf("%d,",T(n,k)) ; od: od: # R. J. Mathar, Apr 09 2009

Formula

Row sums: (n+2)*A000254(n).

Extensions

Edited by R. J. Mathar, Apr 09 2009

A158440 Triangle T(n,k) read by rows: row n contains n times n+1 followed by n 1's.

Original entry on oeis.org

2, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 1, 6, 6, 6, 6, 6, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, 8, 8, 8, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 1, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11
Offset: 1

Views

Author

Paul Curtz, Mar 19 2009

Keywords

Comments

These are essentially the unreduced numerators of the coefficients of P(2n-1,x) of A130679, with denominators represented by A122197.

Examples

			The triangle has 2n columns in row n. It starts:
  2, 1;
  3, 3, 1, 1;
  4, 4, 4, 1, 1, 1;
  5, 5, 5, 5, 1, 1, 1, 1;

Crossrefs

Cf. A003057.

Programs

  • Mathematica
    Flatten[Table[Join[PadRight[{},n,n+1],PadRight[{},n,1]],{n,12}]] (* Harvey P. Dale, Feb 18 2013 *)

Extensions

Edited by R. J. Mathar, Apr 09 2009
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.