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 1-7 of 7 results.

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)

A028315 Odd elements in the 5-Pascal triangle A028313.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 7, 7, 1, 1, 19, 19, 1, 1, 9, 27, 27, 9, 1, 1, 65, 65, 1, 1, 11, 101, 101, 11, 1, 1, 57, 147, 231, 231, 147, 57, 1, 1, 13, 69, 69, 13, 1, 1, 273, 273, 1, 1, 15, 355, 855, 855, 355, 15, 1, 1, 111, 451, 2277, 2277, 451, 111, 1, 1, 17, 127, 1661, 3487, 5379, 5379, 3487, 1661, 127, 17, 1
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Odd elements of A028313 as an irregular triangle:
  1;
  1,   1;
  1,   5,   1;
  1,   1;
  1,   7,   7,   1;
  1,  19,  19,   1;
  1,   9,  27,  27,   9,   1;
  1,  65,  65,   1;
  1,  11, 101, 101,  11,   1;
  1,  57, 147, 231, 231, 147, 57, 1;
  1,  13,  69,  69,  13,   1;
  1, 273, 273,   1;
  1,  15, 355, 855, 855, 355, 15, 1;
  ...

Links

Crossrefs

Cf. A028313, A028316, A028325.

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]: n in [1..150] | (a[n] mod 2) eq 1]; // 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;
a[n_]:= DeleteCases[{f[[n+1]]}, _?EvenQ];
Table[a[n], {n,0,150}]//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] for n in (0..150) if a[n]%2==1] # G. C. Greubel, Jan 06 2024

Extensions

More terms from James Sellers

A028316 Odd elements in the 5-Pascal triangle A028313 that are not 1.

Original entry on oeis.org

5, 7, 7, 19, 19, 9, 27, 27, 9, 65, 65, 11, 101, 101, 11, 57, 147, 231, 231, 147, 57, 13, 69, 69, 13, 273, 273, 15, 355, 855, 855, 355, 15, 111, 451, 2277, 2277, 451, 111, 17, 127, 1661, 3487, 5379, 5379, 3487, 1661, 127, 17, 689, 2223, 11583, 11583, 2223, 689, 19
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Comments

Odd elements of A028314. - G. C. Greubel, Jan 06 2024

Examples

			Odd elements of A028313 as an irregular triangle:
   5;
   7,   7;
  19,  19;
   9,  27,  27,   9;
  65,  65;
  11, 101, 101,  11;
  57, 147, 231, 231, 147, 57;
  ...

Links

Crossrefs

Cf. A028313, A028314, A028315, A028325.

Programs

  • Magma
    A028314:= func< n, k | Binomial(n+2, k+1) + 3*Binomial(n, k) >;
a:=[A028314(n, k): k in [0..n], n in [0..100]];
[a[n]: n in [1..150] | (a[n] mod 2) eq 1]; // G. C. Greubel, Jan 06 2024
  • Mathematica
    A028314[n_, k_]:= Binomial[n+2,k+1] +3*Binomial[n,k];
f= Table[A028314[n,k], {n,0,100}, {k,0,n}]//Flatten;
a[n_]:= DeleteCases[{f[[n+1]]}, _?EvenQ];
Table[a[n], {n,0,150}]//Flatten (* G. C. Greubel, Jan 06 2024 *)
  • SageMath
    def A028314(n, k): return binomial(n+2, k+1) + 3*binomial(n, k)
a=flatten([[A028314(n, k) for k in range(n+1)] for n in range(101)])
[a[n] for n in (0..150) if a[n]%2==1] # G. C. Greubel, Jan 06 2024

Extensions

More terms from James Sellers

A028319 Distinct odd elements in the 5-Pascal triangle A028313.

Original entry on oeis.org

1, 5, 7, 19, 9, 27, 65, 11, 101, 57, 147, 231, 13, 69, 273, 15, 355, 855, 111, 451, 2277, 17, 127, 1661, 3487, 5379, 689, 2223, 11583, 833, 7371, 20449, 181, 995, 3745, 10283, 21385, 34463, 43615, 21, 201, 1377, 23, 1599, 7293, 267, 1843, 31977, 25
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Links

Crossrefs

Cf. A028313, A028314, A028315, A028316, A028317, A028318, A028320.
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[OddQ] (* 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==1]
    return nd
a(b) # A028319 # G. C. Greubel, Jul 13 2024

Extensions

More terms from James Sellers, Dec 08 1999

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

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
Showing 1-7 of 7 results.

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