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.

Previous Showing 11-14 of 14 results.

A028285 Elements to right of central elements in 4-Pascal triangle A028275 that are not 1.

Original entry on oeis.org

5, 6, 16, 7, 23, 8, 55, 31, 9, 86, 40, 10, 196, 126, 50, 11, 322, 176, 61, 12, 714, 498, 237, 73, 13, 1212, 735, 310, 86, 14, 2640, 1947, 1045, 396, 100, 15, 4587, 2992, 1441, 496, 115, 16, 9867, 7579, 4433, 1937, 611, 131, 17, 17446, 12012, 6370, 2548, 742
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers

A028287 Odd elements (greater than 1) to right of central elements in 4-Pascal triangle A028275.

Original entry on oeis.org

5, 7, 23, 55, 31, 9, 11, 61, 237, 73, 13, 735, 1947, 1045, 15, 4587, 1441, 115, 9867, 7579, 4433, 1937, 611, 131, 17, 19, 185, 1241, 205, 21, 6477, 28101, 7923, 23, 105621, 9595, 271, 353685, 141645, 45619, 11515, 2191, 295, 25, 1076559, 3023603
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Extensions

More terms from James Sellers
Title improved by Sean A. Irvine, Dec 17 2019

A028262 Elements in 3-Pascal triangle (by row).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 8, 5, 1, 1, 6, 13, 13, 6, 1, 1, 7, 19, 26, 19, 7, 1, 1, 8, 26, 45, 45, 26, 8, 1, 1, 9, 34, 71, 90, 71, 34, 9, 1, 1, 10, 43, 105, 161, 161, 105, 43, 10, 1, 1, 11, 53, 148, 266, 322, 266, 148, 53, 11, 1, 1, 12, 64, 201, 414, 588, 588, 414, 201, 64, 12, 1
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Triangle begins:
  1;
  1 1;
  1 3 1;
  1 4 4 1;
  1 5 8 5 1;
  ...

Links

Crossrefs

Cf. A007318, A028275, A072405.

Programs

  • Haskell
    a028262 n k = a028262_tabl !! n !! k
a028262_row n = a028262_tabl !! n
a028262_tabl = [1] : [1,1] : iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1,3,1]
-- Reinhard Zumkeller, Aug 02 2012
  • Magma
    T:= func< n,k | n lt 2 select 1 else Binomial(n, k) + Binomial(n-2, k-1) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 28 2021
  • Mathematica
    T[n_, k_]:= If[n==1, 1, Binomial[n, k] + Binomial[n-2, k-1]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jan 28 2015 *)
  • Sage
    def T(n,k): return 1 if n<2 else binomial(n,k) + binomial(n-2,k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021

Formula

After the 3rd row, use Pascal's rule.
From Ralf Stephan, Jan 31 2005: (Start)
T(n, k) = C(n, k) + C(n-2, k-1).
G.f.: (1 + x^2*y)/(1 - x*(1+y)). (End)
T(n+2,k+1) = A007318(n,k) - A007318(n+2,k+1); 0 < k < n. - Reinhard Zumkeller, Aug 02 2012
Sum_{k=0..n} T(n,k) = (n+1)*[n<2] + 5*2^(n-2)*[n>=2]. - G. C. Greubel, Apr 28 2021

Extensions

More terms from James Sellers

A028313 Elements in the 5-Pascal triangle (by row).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 6, 6, 1, 1, 7, 12, 7, 1, 1, 8, 19, 19, 8, 1, 1, 9, 27, 38, 27, 9, 1, 1, 10, 36, 65, 65, 36, 10, 1, 1, 11, 46, 101, 130, 101, 46, 11, 1, 1, 12, 57, 147, 231, 231, 147, 57, 12, 1, 1, 13, 69, 204, 378, 462, 378, 204, 69, 13, 1, 1, 14, 82, 273, 582, 840, 840, 582, 273, 82, 14, 1
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  5,  1;
  1,  6,  6,   1;
  1,  7, 12,   7,   1;
  1,  8, 19,  19,   8,   1;
  1,  9, 27,  38,  27,   9,   1;
  1, 10, 36,  65,  65,  36,  10,  1;
  1, 11, 46, 101, 130, 101,  46, 11,  1;
  1, 12, 57, 147, 231, 231, 147, 57, 12,  1;

Links

Crossrefs

Cf. A000007, A005009, A010892, A022112, A028275, A028314, A028315.
Cf. A028316, A028317, A028318, A028319, A028320, A028321, A028322.
Cf. A028323, A028324, A028325, A029653, A051472, A051936, A051937.
Cf. A178915.

Programs

  • Magma
    [n le 1 select 1 else Binomial(n,k) +3*Binomial(n-2,k-1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 05 2024
  • Mathematica
    Table[If[n<2, 1, Binomial[n,k] +3*Binomial[n-2,k-1]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2024 *)
  • SageMath
    def A028313(n,k): return 1 if n<2 else binomial(n,k) + 3*binomial(n-2,k-1)
flatten([[A028313(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 05 2024

Formula

From Ralf Stephan, Jan 31 2005: (Start)
T(n, k) = C(n, k) + 3*C(n-2, k-1), with T(0, k) = T(1, k) = 1.
G.f.: (1 + 3*x^2*y)/(1 - x*(1+y)). (End)
From G. C. Greubel, Jan 05 2024: (Start)
T(n, n-k) = T(n, k).
T(n, n-1) = n + 3*(1 - [n=1]) = A178915(n+3), n >= 1.
T(n, n-2) = A051936(n+2), n >= 2.
T(n, n-3) = A051937(n+1), n >= 3.
T(2*n, n) = A028322(n).
Sum_{k=0..n} T(n, k) = A005009(n-2) - (3/4)*[n=0] - (3/2)*[n=1].
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n) - 3*[n=2].
Sum_{k=0..floor(n/2)} T(n-k, k) = A022112(n-2) + 3*([n=0] - [n=1]).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = 4*A010892(n) - 3*([n=0] + [n=1]). (End)

Extensions

More terms from Sam Alexander (pink2001x(AT)hotmail.com)
Previous Showing 11-14 of 14 results.

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