A133081 -id:A133081 - cp's OEIS frontend

A133090 A133081 * [1,2,3,...].

1, 1, 5, 3, 9, 5, 13, 7, 17, 9, 21, 11, 25, 13, 29, 15, 33, 17, 37, 19, 41, 21, 45, 23, 49, 25, 53, 27, 57, 29, 61, 31, 65, 33, 69, 35, 73, 37, 77, 39, 81, 41, 85, 43, 89, 45, 93, 47, 97, 49, 101, 51, 105, 53, 109, 55, 113, 57, 117, 59, 121, 61, 125, 63, 129, 65, 133, 67, 137, 69, 141, 71

Offset: 1

Gary W. Adamson, Sep 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A133081.

seq((3/2)*n - (n/2)*(-1)^n - 1, n=1..60); # Ridouane Oudra, Sep 08 2023

Table[Riffle[{4*n + 1}, {2*n + 1} ], {n, 0, 50}] // Flatten (* G. C. Greubel, Oct 21 2017 *)

def A133090(n): return (n<<1)-1 if n&1 else n-1 # Chai Wah Wu, Apr 04 2024

2k+1 interleaved with 4k+1.
a(n) = (3/2)*n - (n/2)*(-1)^n - 1. - Ridouane Oudra, Sep 08 2023
From Stefano Spezia, Dec 04 2023: (Start)
O.g.f.: x*(1 + x + 3*x^2 + x^3)/((1 - x)^2*(1 + x)^2).
E.g.f.: 1 + (2*x - 1)*cosh(x) + (x - 1)*sinh(x). (End)

Terms a(17) onward added by G. C. Greubel, Oct 21 2017

A133080 Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Sep 08 2007

A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, ...).
Inverse of A133080: subdiagonal changes to (-1, 0, -1, 0, -1, ...); main diagonal unchanged.
A133080^(-1) * [1,2,3,...] = A093178: (1, 1, 3, 1, 5, 1, 7, 1, 9, ...).
In A133081, diagonal terms are switched with subdiagonal terms.

			First few rows of the triangle are:
  1;
  1, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 0, 0, 1;
  0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000034 (row sums), A114753, A093178, A133081.

A133080 := proc(n,k)
    if n = k then
        1;
    elif  k=n-1 and type(n,even) then
        1;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Jun 20 2015

T[n_, k_] := If[k == n, 1, If[k == n - 1, (1 + (-1)^n)/2 , 0]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

T(n, k) = if (k==n, 1, if (k == (n-1), 1 - (n % 2), 0)); \\ Michel Marcus, Feb 13 2014

firstrows(n) = {my(res = vector(binomial(n + 1, 2)), t=0); for(i=1, n, t+=i; res[t] = 1; if(i%2==0, res[t-1]=1)) ;res} \\ David A. Corneth, Oct 21 2017

Infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (1,0,1,0,1,...) in the subdiagonal.
Odd rows, (n-1) zeros followed by "1". Even rows, (n-2) zeros followed by "1, 1".
T(n,n)=1. T(n,k)=0 if 1 <= k < n-1. T(n,n-1)=1 if n even. T(n,n-1)=0 if n odd. - R. J. Mathar, Feb 14 2015

cp's OEIS Frontend

A133090 A133081 * [1,2,3,...].

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