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.

A021069 Decimal expansion of 1/65.

Original entry on oeis.org

0, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Without the leading 0 also the decimal expansion of 2/13.

Examples

			0.0153846153846...  - _Natan Arie Consigli_, Sep 18 2016

Links

Crossrefs

Cf. A010734, A020806, A021017, A021095, A128834, A130806, A144472.

Programs

Formula

Equals 2 - 24/13. See Táfula link. - Michel Marcus, May 31 2024
G.f.: x*(1 + 4*x - 2*x^2 + 6*x^3)/((1 - x)*(1 + x)*(1 - x + x^2)). - Stefano Spezia, Apr 30 2025

A193002 Triangle T(n,k)=0 (k odd), T(0,0)=-3, T(n,0)=1 (n > 0) and T(n,k) = T(n-1,k) - T(n-2,k-2).

Original entry on oeis.org

-3, 1, 0, 1, 0, 3, 1, 0, 2, 0, 1, 0, 1, 0, -3, 1, 0, 0, 0, -5, 0, 1, 0, -1, 0, -6, 0, 3, 1, 0, -2, 0, -6, 0, 8, 0, 1, 0, -3, 0, -5, 0, 14, 0, -3, 1, 0, -4, 0, -3, 0, 20, 0, -11, 0, 1, 0, -5, 0, 0, 0, 25, 0, -25, 0, 3, 1, 0, -6
Offset: 0

Views

Author

Paul Curtz, Jul 14 2011

Keywords

Comments

Consider an array with recurrence BB(m,n) = BB(m,n-1) + BB(m-1,n), m >= 0:
3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,
3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7,
3, 5, 6, 6, 5, 3, 0, -4, -9, -15, -22,
3, 8, 14, 20, 25, 28, 28, 24, 15, 0, -22,
3, 11, 25, 45, 70, 98, 126, 150, 165, 165, 143,
3, 14, 39, 84, 154, 252, 378, 528, 693, 858, 1001,
with BB(m,n) = (3m-n)*binomial(n+m-1,n)/m if m > 0. So the BB are polynomials of degree m in n:
BB(1,n) = -(n-3)/1,
BB(2,n) = -(n-6)*(n+1)/2, (see A055999)
BB(3,n) = -(n-9)*(n+1)*(n+2)/6,
BB(4,n) = -(n-12)*(n+1)*(n+2)*(n+3)/24,
BB(5,n) = -(n-15)*(n+1)*(n+2)*(n+3)*(n+4)/120.
Columns in the array are A010701, A016789, A095794, A005564, A059302.
T(n,k) is a zero-padded, column-shifted, sign-modified transpose of this array.

Examples

			Triangle begins
  -3;
   1,   0;
   1,   0,   3;
   1,   0,   2,   0;
   1,   0,   1,   0,  -3;
   1,   0,   0,   0,  -5,   0;
   1,   0,  -1,   0,  -6,   0,   3;
   1,   0,  -2,   0,  -6,   0,   8,   0;
   1,   0,  -3,   0,  -5,   0,  14,   0,  -3;
   1,   0,  -4,   0,  -3,   0,  20,   0, -11,   0;

Crossrefs

Cf. A174559.

Programs

  • Maple
    BB := proc(m,n) if m=0 then if n= 0 then 3 ; else -1; end if; else (3*m-n)*binomial(n+m-1,n)/m ; end if; end proc:
A193002 := proc(n,k) if type(k,'odd') then 0; else (-1)^(1+k/2)*BB(k/2,n-k) ; end if; end proc:
seq(seq(A193002(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 30 2011

Formula

Sum_{k=0..n} T(n,k) = A130806(n+5). (row sums)
Sum_{k=0..n} (-1)^(k/2)*T(n,k) = -A000032(n-2). (alternating row sums)
T(n,k) = (-1)^(1+k/2)*BB(k/2,n-k). - R. J. Mathar, Aug 30 2011
T(n,2k) = (-1)^(1+k)*(5-n/k)*binomial(n-k-1,k-1), k > 0. - R. J. Mathar, Aug 30 2011

A121512 a(n) = a(n-1) + a(n-3) - a(n-4) for n>4, with a(1)=1, a(2)=4, a(3)=10, a(4)=4.

Original entry on oeis.org

1, 4, 10, 4, 7, 13, 7, 10, 16, 10, 13, 19, 13, 16, 22, 16, 19, 25, 19, 22, 28, 22, 25, 31, 25, 28, 34, 28, 31, 37, 31, 34, 40, 34, 37, 43, 37, 40, 46, 40, 43, 49, 43, 46, 52, 46, 49, 55, 49, 52, 58, 52, 55, 61, 55, 58, 64, 58, 61, 67, 61, 64, 70, 64, 67, 73
Offset: 1

Views

Author

Roger L. Bagula, Sep 07 2006

Keywords

Comments

This sequence gives a linearly increasing triangle shaped form on plotting.

Links

Crossrefs

Cf. A005448, A130806.

Programs

  • Mathematica
    M = {{0, 1, 0}, {0, 0, 1}, {1, 0, 0}} v[1] = {1, 4, 10} v[n_] := v[n] = M.v[n - 1] + {0, 0, 3} a = Table[v[n][[1]], {n, 1, 50}]

Formula

G.f.: -x*(-1-3*x-6*x^2+7*x^3)/((1+x+x^2)*(x-1)^2). [Oct 14 2009]
a(n) = n+3 + (-1)^n * A130806(n+1). [Oct 14 2009]
a(n) = (24*n + 75 - 18*cos(2*(n+1)*Pi/3) + 9*cos(2*Pi*sin(2*n*Pi/3)/sqrt(3)) + 50*sqrt(3)*sin(2*(n+1)*Pi/3))/24. - Wesley Ivan Hurt, Oct 05 2017

Extensions

Definition replaced by recurrence - The Assoc. Editors of the OEIS, Oct 14 2009
Showing 1-3 of 3 results.

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