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 11-15 of 15 results.

A028320 Distinct even elements in the 5-Pascal triangle A028313.

Original entry on oeis.org

6, 12, 8, 38, 10, 36, 46, 130, 204, 378, 462, 14, 82, 582, 840, 96, 1422, 1680, 16, 1210, 3102, 562, 6204, 18, 144, 5148, 8866, 162, 2912, 14014, 23166, 20, 1176, 4740, 14028, 31668, 55848, 78078, 87230, 22, 222, 5916, 18768, 45696, 87516, 133926
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Links

Crossrefs

Cf. A028313, A028314, A028315, A028316, A028317, A028318, A028319.
Cf. A028321, A028322, A028323, A028324, A028325, A051472.

Programs

  • Mathematica
    DeleteDuplicates[Table[If[n<2, 1, Binomial[n,k] +3*Binomial[n-2,k-1]], {n,0,30}, {k,0,n}]//Flatten]//Select[EvenQ] (* G. C. Greubel, Jul 13 2024 *)
  • SageMath
    def A028323(n, k): return 1 if n<2 else binomial(n, k) + 3*binomial(n-2, k-1)
b=flatten([[A028323(n, k) for k in range(n+1)] for n in range(31)])
def a(seq): # order preserving
    nd = [] # no duplicates
    [nd.append(i) for i in seq if not nd.count(i) and i%2==0]
    return nd
a(b) # A028320 # G. C. Greubel, Jul 13 2024

Extensions

More terms from James Sellers, Dec 08 1999

A028321 Even elements to the right of the central elements of the 5-Pascal triangle A028313.

Original entry on oeis.org

6, 8, 36, 10, 46, 12, 378, 204, 840, 582, 82, 14, 1422, 96, 3102, 1210, 16, 562, 8866, 5148, 144, 18, 14014, 2912, 162, 20, 78078, 55848, 31668, 14028, 4740, 1176, 165308, 133926, 87516, 45696, 18768, 5916, 222, 22, 299234, 221442, 133212
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Links

Crossrefs

Cf. A028313, A028314, A028315, A028316, A028317, A028318, A028319.
Cf. A028320, A028322, A028323, A028324, A028325, A051472.

Programs

  • Magma
    T:= func< n, k | n le 1 select 1 else Binomial(n, k) + 3*Binomial(n-2, k-1) >; // T = A028323
b:=[T(n, k): k in [1+Floor(n/2)..n], n in [0..100]];
[b[n]: n in [1..150] | (b[n] mod 2) eq 0]; // G. C. Greubel, Jul 02 2024
  • Mathematica
    b:= Table[If[n<2, 1, Binomial[n,k] +3*Binomial[n-2,k-1]], {n,0,30}, {k, Floor[n/2]+1, n}]//Flatten;
Select[b, EvenQ] (* G. C. Greubel, Jul 02 2024 *)
  • SageMath
    def A028323(n, k): return binomial(n, k) + 3*binomial(n-2, k-1) - 3*int(n==0)
b=flatten([[A028323(n, k) for k in range(1+(n//2),n+1)] for n in range(101)])
[b[n] for n in (1..150) if b[n]%2==0] # G. C. Greubel, Jul 02 2024

Extensions

More terms from James Sellers

A028275 Elements in 4-Pascal triangle (by row).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 10, 6, 1, 1, 7, 16, 16, 7, 1, 1, 8, 23, 32, 23, 8, 1, 1, 9, 31, 55, 55, 31, 9, 1, 1, 10, 40, 86, 110, 86, 40, 10, 1, 1, 11, 50, 126, 196, 196, 126, 50, 11, 1, 1, 12, 61, 176, 322, 392, 322, 176, 61, 12, 1, 1, 13, 73, 237, 498, 714, 714, 498, 237
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			1; 1 1; 1 4 1; 1 5 5 1; 1 6 10 6 1; ...

Crossrefs

Cf. A028262, A028313.

Formula

Apart from first 3 rows, use the Pascal rule.
T(n, k) = C(n, k) + 2C(n-2, k-1). G.f.: (1+2x^2y) / [1-x(1+y)]. - Ralf Stephan, Jan 31 2005

Extensions

More terms from Ben Baugher (s1191623(AT)cedarville.edu)

A051472 a(n) = A028317(n)/2.

Original entry on oeis.org

3, 3, 6, 4, 4, 19, 5, 18, 18, 5, 23, 65, 23, 6, 6, 102, 189, 231, 189, 102, 7, 41, 291, 420, 420, 291, 41, 7, 48, 711, 840, 711, 48, 8, 605, 1551, 1551, 605, 8, 281, 3102, 281, 9, 72, 2574, 4433, 4433, 2574, 72, 9, 81, 1456, 7007, 11583, 7007, 1456, 81, 10, 10, 588
Offset: 0

Views

Author

James Sellers

Keywords

Examples

			Even elements of (1/2)*A028317 as an irregular triangle:
   3,  3;
   6;
   4,  4;
  19;
   5, 18, 18, 5;
  23, 65, 23;
   6,  6;
  ...

Links

Crossrefs

Cf. A028313, A028317.

Programs

  • Magma
    A028313:= func< n, k | n le 1 select 1 else Binomial(n, k) +3*Binomial(n-2, k-1) >;
a:=[A028313(n, k): k in [0..n], n in [0..100]];
[a[n]/2: n in [1..200] | (a[n] mod 2) eq 0]; // G. C. Greubel, Jan 06 2024
  • Mathematica
    A028313[n_, k_]:= If[n<2, 1, Binomial[n,k] +3*Binomial[n-2,k-1]];
f= Table[A028313[n,k], {n,0,100}, {k,0,n}]//Flatten;
b[n_]:= DeleteCases[{f[[n+1]]}, _?OddQ]/2;
Table[b[n], {n,0,200}]//Flatten (* G. C. Greubel, Jan 06 2024 *)
  • SageMath
    def A028313(n, k): return 1 if n<2 else binomial(n, k) + 3*binomial(n-2, k-1)
a=flatten([[A028313(n, k) for k in range(n+1)] for n in range(101)])
[a[n]/2 for n in (0..200) if a[n]%2==0] # G. C. Greubel, Jan 06 2024

A051473 a(n) = A028321(n)/2.

Original entry on oeis.org

3, 4, 18, 5, 23, 6, 189, 102, 420, 291, 41, 7, 711, 48, 1551, 605, 8, 281, 4433, 2574, 72, 9, 7007, 1456, 81, 10, 39039, 27924, 15834, 7014, 2370, 588, 82654, 66963, 43758, 22848, 9384, 2958, 111, 11, 149617, 110721, 66606, 32232, 12342, 122, 314925
Offset: 0

Views

Author

James Sellers

Keywords

Links

Crossrefs

Cf. A028313, A028314, A028315, A028316, A028317, A028318, A028319.
Cf. A028320, A028321, A028322, A028323, A028324, A028325, A051472.

Programs

  • Magma
    T:= func< n, k | n le 1 select 1 else Binomial(n, k) + 3*Binomial(n-2, k-1) >; // T = A028323
b:=[T(n, k): k in [1+Floor(n/2)..n], n in [0..100]];
[b[n]/2: n in [1..150] | (b[n] mod 2) eq 0]; // G. C. Greubel, Jul 02 2024
  • Mathematica
    b:= Table[If[n<2, 1, Binomial[n,k] +3*Binomial[n-2,k-1]], {n,0,30}, {k, Floor[n/2]+1,n}]//Flatten;
Select[b, EvenQ]/2 (* G. C. Greubel, Jul 02 2024 *)
  • SageMath
    def A028323(n, k): return binomial(n, k) + 3*binomial(n-2, k-1) - 3*int(n==0)
b=flatten([[A028323(n, k) for k in range(1+(n//2),n+1)] for n in range(101)])
[b[n]/2 for n in (1..150) if b[n]%2==0] # G. C. Greubel, Jul 02 2024
Previous Showing 11-15 of 15 results.

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