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.

A117184 Riordan array ((1+x)c(x^2)/sqrt(1-4x^2),xc(x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 4, 1, 1, 10, 4, 5, 1, 1, 10, 15, 5, 6, 1, 1, 35, 15, 21, 6, 7, 1, 1, 35, 56, 21, 28, 7, 8, 1, 1, 126, 56, 84, 28, 36, 8, 9, 1, 1, 126, 210, 84, 120, 36, 45, 9, 10, 1, 1, 462, 210, 330, 120, 165, 45, 55, 10, 11, 1, 1
Offset: 0

Views

Author

Paul Barry, Mar 01 2006

Keywords

Comments

Row sums are A117186. Diagonal sums are A117187. Inverse is A117185.

Examples

			Triangle begins
1,
1, 1,
3, 1, 1,
3, 4, 1, 1,
10, 4, 5, 1, 1,
10, 15, 5, 6, 1, 1,
35, 15, 21, 6, 7, 1, 1,
35, 56, 21, 28, 7, 8, 1, 1

Programs

  • Mathematica
    c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 + #) c[#^2]/Sqrt[1 - 4 #^2]&, # c[#^2]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Formula

Number triangle T(n,k)=C(n+1,(n+k)/2+1)(1+(-1)^(n-k))/2+C(n,(n+k)/2+1/2)(1-(-1)^(n-k))/2; Column k has e.g.f. Bessel_I(k,2x)+Bessel_I(k+1,2x)+Bessel_I(k+2,2x).

A117188 Expansion of (1-x^2)/(1+x^2+x^4).

Original entry on oeis.org

1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0
Offset: 0

Views

Author

Paul Barry, Mar 01 2006

Keywords

Comments

Periodic: repeat [1, 0, -2, 0, 1, 0].
Minton mentions that the subsequence a(2^i), i >= 1, oscillates between -2 and 1 (and does not converge 2-adically). - N. J. A. Sloane, Jul 09 2014

Links

Crossrefs

Row sums of A117185.

Programs

Formula

G.f.: (1 - 2*x^2 + x^4)/(1-x^6).
a(n) = (1 + (-1)^n)/(-2 + 4^(floor((n-1)/3) - 2*floor((n-1)/6))). - Tani Akinari, Aug 02 2013
a(n) = -a(n-2) - a(n-4) for n >= 4. - N. J. A. Sloane, Jul 09 2014
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = cos(n*Pi/2) * (cos(n*Pi/6) + sqrt(3)*sin(n*Pi/6)). (End)
E.g.f.: cos(sqrt(3)*x/2)*cosh(x/2) - sqrt(3)*sin(sqrt(3)*x/2)*sinh(x/2). - Ilya Gutkovskiy, Jun 27 2016
a(n) = cos((n+1)*Pi/3) - cos(2*(n+1)*Pi/3). - Ridouane Oudra, Dec 14 2021

A117584 Generalized Pellian triangle.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 7, 12, 1, 5, 9, 17, 29, 1, 6, 11, 22, 41, 70, 1, 7, 13, 27, 53, 99, 169, 1, 8, 15, 32, 65, 128, 239, 408, 1, 9, 17, 37, 77, 157, 309, 577, 985, 1, 10, 19, 42, 89, 186, 379, 746, 1393, 2378
Offset: 1

Views

Author

Gary W. Adamson, Mar 29 2006

Keywords

Examples

			First few rows of the triangle are:
  1;
  1, 2;
  1, 3,  5;
  1, 4,  7, 12;
  1, 5,  9, 17, 29;
  1, 6, 11, 22, 41, 70;
  1, 7, 13, 27, 53, 99, 169;
  ...
The triangle rows are antidiagonals of the generalized Pellian array:
  1, 2,  5, 12, 29, ...
  1, 3,  7, 17, 41, ...
  1, 4,  9, 22, 53, ...
  1, 5, 11, 27, 65, ...
  ...
For example, in the row (1, 5, 11, 27, 65, ...), 65 = 2*27 + 11.

Links

Crossrefs

Diagonals include A000129, A001333, A048654, A048655, A048693.
Cf. A117185.

Programs

  • Magma
    P:= func< n | Round( ((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2*Sqrt(2)) ) >;
T:= func< n,k | P(k) + (n-1)*P(k-1) >;
[T(n-k+1, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 05 2021
  • Mathematica
    T[n_, k_]:= Fibonacci[k, 2] + (n-1)*Fibonacci[k-1, 2];
Table[T[n-k+1, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 05 2021 *)
  • Sage
    def T(n,k): return lucas_number1(k,2,-1) + (n-1)*lucas_number1(k-1,2,-1)
flatten([[T(n-k+1, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 05 2021

Formula

Antidiagonals of the generalized Pellian array. First row of the array = A000129: (1, 2, 5, 12, ...). n-th row of the array starts (1, n+1, ...); as a Pellian sequence.
From G. C. Greubel, Jul 05 2021: (Start)
T(n, k) = P(k) + (n-1)*P(k-1), where P(n) = A000129(n) (square array).
Sum_{k=1..n} T(n-k+1, k) = A117185(n). (End)
Showing 1-3 of 3 results.

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

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