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 1-10 of 15 results. Next

A028323 Elements to the right of the central elements of the 5-Pascal triangle A028313.

Original entry on oeis.org

1, 1, 6, 1, 7, 1, 19, 8, 1, 27, 9, 1, 65, 36, 10, 1, 101, 46, 11, 1, 231, 147, 57, 12, 1, 378, 204, 69, 13, 1, 840, 582, 273, 82, 14, 1, 1422, 855, 355, 96, 15, 1, 3102, 2277, 1210, 451, 111, 16, 1, 5379, 3487, 1661, 562, 127, 17, 1, 11583, 8866, 5148, 2223, 689, 144, 18, 1
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			This sequence represents the following portion of A028313(n,k), with x being the elements of A028313(2*n,n):
  x,
  .,  1,
  .,  x,  1,
  .,  .,  6,   1,
  .,  .,  x,   7,   1,
  .,  ., ..,  19,   8,   1,
  .,  ., ..,   x,  27,   9,   1,
  ., .., ..,  ..,  65,  36,  10,  1,
  ., .., .., ...,   x, 101,  46, 11,  1,
  ., .., .., ..., ..., 231, 147, 57, 12, 1.
As an irregular triangle this sequence begins as:
     1;
     1;
     6,    1;
     7,    1;
    19,    8,    1;
    27,    9,    1;
    65,   36,   10,   1;
   101,   46,   11,   1;
   231,  147,   57,  12,   1;
   378,  204,   69,  13,   1;
   840,  582,  273,  82,  14,  1;
  1422,  855,  355,  96,  15,  1;
  3102, 2277, 1210, 451, 111, 16,  1;

Links

Crossrefs

Cf. A028313.

Programs

  • Magma
    A028323:= func< n,k | n eq 0 select 1 else Binomial(n+1, k + Floor((n+1)/2) + 1) + 3*Binomial(n-1, k + Floor((n+1)/2)) >;
[A028323(n,k): k in [0..Floor(n/2)], n in [0..16]]; // G. C. Greubel, Jan 05 2024
  • Mathematica
    T[n_, k_]:= Binomial[n+1, k +Floor[(n+1)/2] +1] + 3*Binomial[n-1, k+ Floor[(n+1)/2]] -3*Boole[n==0];
Table[T[n,k], {n,0,16}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Jan 05 2024 *)
  • SageMath
    def A028323(n,k): return binomial(n+1, k+1+(n+1)//2) + 3*binomial(n-1, k+((n+1)//2)) - 3*int(n==0)
flatten([[A028323(n,k) for k in range(1+(n//2))] for n in range(17)]) # G. C. Greubel, Jan 05 2024

Formula

From G. C. Greubel, Jan 05 2024: (Start)
a(n) = A028313(n, k), for 1 + floor(n/2) <= k <= n, n >= 0.
T(n, k) = binomial(n+1, k + floor((n+1)/2) + 1) + 3*binomial(n-1, k + floor((n+1)/2)) -3*[n=0], for 0 <= k <= floor(n/2), n >= 0. (End)

Extensions

More terms from James Sellers

A028322 Central elements in the 5-Pascal triangle A028313.

Original entry on oeis.org

1, 5, 12, 38, 130, 462, 1680, 6204, 23166, 87230, 330616, 1259700, 4820452, 18513068, 71318400, 275467320, 1066432950, 4136847390, 16075953960, 62570669700, 243882320220, 951797460900, 3718872587040, 14545727618760
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Links

Crossrefs

Cf. A000108, A000984, A028313.

Programs

  • Magma
    [(n+1)*(7*n-2)*Catalan(n)/(2*(2*n-1)): n in [0..40]]; // G. C. Greubel, Jan 05 2024
  • Mathematica
    Table[(n+1)*(7*n-2)*CatalanNumber[n]/(2*(2*n-1)), {n,0,40}] (* G. C. Greubel, Jan 05 2024 *)
  • SageMath
    [(7*n-2)*binomial(2*n,n)/(2*(2*n-1)) for n in range(41)] # G. C. Greubel, Jan 05 2024

Formula

G.f.: (1 + 3*x)/sqrt(1-4*x). - Vladeta Jovovic, Jan 08 2004
a(n) = A000984(n) + 3*A000984(n-1). - R. J. Mathar, Dec 15 2015
a(n) = (n+1)*(7*n-2)*A000108(n)/(2*(2*n-1)). - G. C. Greubel, Jan 05 2024

Extensions

More terms from James Sellers

A028325 Odd elements to the right of the central elements of the 5-Pascal triangle A028313.

Original entry on oeis.org

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

Views

Author

Mohammad K. Azarian

Keywords

Comments

Odd elements of A028323. - G. C. Greubel, Jan 06 2024

Links

Crossrefs

Cf. A028313, A028323.

Programs

  • Magma
    T:= func< n,k | Binomial(n+1, k+1+Floor((n+1)/2)) + 3*Binomial(n-1, k+Floor((n+1)/2)) >; // T = A028323, essentially
b:=[T(n, k): k in [0..Floor(n/2)], n in [0..100]];
[b[n]: n in [1..150] | (b[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,1+Floor[n/2],n}]//Flatten;
b[n_]:= DeleteCases[{f[[n+1]]}, _?EvenQ];
Table[b[n], {n,0,150}]//Flatten (* G. C. Greubel, Jan 06 2024 *)
  • SageMath
    def T(n, k): return binomial(n+1, k+1+(n+1)//2) + 3*binomial(n-1, k+((n+1)//2)) - 3*int(n==0) # T = A028323, essentially
b=flatten([[T(n, k) for k in range(1+(n//2))] for n in range(101)])
[b[n] for n in (1..150) if b[n]%2==1] # G. C. Greubel, Jan 06 2024

Extensions

More terms from James Sellers

A028314 Elements in the 5-Pascal triangle A028313 that are not 1.

Original entry on oeis.org

5, 6, 6, 7, 12, 7, 8, 19, 19, 8, 9, 27, 38, 27, 9, 10, 36, 65, 65, 36, 10, 11, 46, 101, 130, 101, 46, 11, 12, 57, 147, 231, 231, 147, 57, 12, 13, 69, 204, 378, 462, 378, 204, 69, 13, 14, 82, 273, 582, 840, 840, 582, 273, 82, 14, 15, 96, 355, 855, 1422, 1680, 1422, 855, 355, 96, 15
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Triangle begins as:
   5;
   6,  6;
   7, 12,   7;
   8, 19,  19,   8;
   9, 27,  38,  27,   9;
  10, 36,  65,  65,  36,  10;
  11, 46, 101, 130, 101,  46,  11;
  12, 57, 147, 231, 231, 147,  57,  12;
  13, 69, 204, 378, 462, 378, 204,  69,  13;

Links

Crossrefs

Cf. A000027, A010892, A022112, A028313, A028322,
Cf. A051936, A051937, A121262, A176448.

Programs

  • Magma
    A028314:= func< n,k | Binomial(n+2,k+1) + 3*Binomial(n,k) >;
[A028314(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 06 2024
  • Mathematica
    A028314[n_, k_]:= Binomial[n+2,k+1] + 3*Binomial[n,k];
Table[A028314[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 06 2024 *)
  • SageMath
    def A028314(n,k): return binomial(n+2,k+1) + 3*binomial(n,k)
flatten([[A028314(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 06 2024

Formula

From G. C. Greubel, Jan 06 2024: (Start)
T(n, k) = binomial(n+2, k+1) + 3*binomial(n, k).
T(n, n-k) = T(n, k).
T(n, 0) = T(n, n) = A000027(n+5).
T(n, 1) = T(n, n-1) = A051936(n+4).
T(n, 2) = T(n, n-2) = A051937(n+3).T(2*n, n) = A028322(n+1).
Sum_{k=0..n} T(n, k) = A176448(n).
Sum_{k=0..n} (-1)^k * T(n, k) = 1 + (-1)^n + 3*[n=0].
Sum_{k=0..n} T(n-k, k) = A022112(n+1) - (3-(-1)^n)/2.
Sum_{k=0..n} (-1)^k * T(n-k, k) = 4*A010892(n) - 2*A121262(n+1) - (3 - (-1)^n)/2. (End)
G.f.: (5 - 4*x - 4*x*y + 3*x^2*y)/((1 - x)*(1 - x*y)*(1 - x - x*y)). - Stefano Spezia, Dec 06 2024

Extensions

More terms from James Sellers

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

A028317 Even elements in the 5-Pascal triangle A028313.

Original entry on oeis.org

6, 6, 12, 8, 8, 38, 10, 36, 36, 10, 46, 130, 46, 12, 12, 204, 378, 462, 378, 204, 14, 82, 582, 840, 840, 582, 82, 14, 96, 1422, 1680, 1422, 96, 16, 1210, 3102, 3102, 1210, 16, 562, 6204, 562, 18, 144, 5148, 8866, 8866, 5148, 144, 18, 162, 2912, 14014, 23166
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			Even elements of A028313 as an irregular triangle:
   6,   6;
  12;
   8,   8;
  38;
  10,  36, 36, 10;
  46, 130, 46;
  12,  12;
  ...

Links

Crossrefs

Cf. A028313, A051472.

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..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];
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] for n in (0..200) if a[n]%2==0] # G. C. Greubel, Jan 06 2024

Formula

a(n) = 2*A051472(n). - G. C. Greubel, Jan 06 2024

Extensions

More terms from James Sellers

A028318 Distinct elements in the 5-Pascal triangle A028313.

Original entry on oeis.org

1, 5, 6, 7, 12, 8, 19, 9, 27, 38, 10, 36, 65, 11, 46, 101, 130, 57, 147, 231, 13, 69, 204, 378, 462, 14, 82, 273, 582, 840, 15, 96, 355, 855, 1422, 1680, 16, 111, 451, 1210, 2277, 3102, 17, 127, 562, 1661, 3487, 5379, 6204, 18, 144, 689, 2223, 5148, 8866
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Links

Programs

  • Mathematica
    DeleteDuplicates[Table[If[n<2, 1, Binomial[n,k] +3*Binomial[n-2,k-1]], {n,0,30}, {k,0,n}]//Flatten] (* G. C. Greubel, Jul 03 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)]
    return nd
a(b) # A028318 # G. C. Greubel, Jul 03 2024

Extensions

More terms from James Sellers, Dec 08 1999

A028324 Elements to the right of the central elements of the 5-Pascal triangle A028313 that are not 1.

Original entry on oeis.org

6, 7, 19, 8, 27, 9, 65, 36, 10, 101, 46, 11, 231, 147, 57, 12, 378, 204, 69, 13, 840, 582, 273, 82, 14, 1422, 855, 355, 96, 15, 3102, 2277, 1210, 451, 111, 16, 5379, 3487, 1661, 562, 127, 17, 11583, 8866, 5148, 2223, 689, 144, 18, 20449, 14014, 7371, 2912, 833
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Examples

			This sequence represents the following portion of A028313(n,k), with x being the elements of A028313(2*n,n):
  x;
  .,  .;
  .,  x,  .;
  .,  .,  6,   .;
  .,  .,  x,   7,   .;
  .,  .,  .,  19,   8,   .;
  .,  .,  .,   x,  27,   9,   .;
  .,  .,  .,   .,  65,  36,  10,   .;
  .,  .,  .,   .,   x, 101,  46,  11,  .;
  .,  .,  .,   .,   ., 231, 147,  57, 12,  .;
  .,  .,  .,   .,   .,   x, 378, 204, 69, 13,  .;
As an irregular triangle this sequence begins as:
      6;
      7;
     19,    8;
     27,    9;
     65,   36,   10;
    101,   46,   11;
    231,  147,   57,   12;
    378,  204,   69,   13;
    840,  582,  273,   82,  14;
   1422,  855,  355,   96,  15;
   3102, 2277, 1210,  451, 111,  16;
   5379, 3487, 1661,  562, 127,  17;
  11583, 8866, 5148, 2223, 689, 144, 18;

Links

Crossrefs

Cf. A028313, A028323.

Programs

  • Magma
    A028324:= func< n,k | Binomial(n+3, k+2+Floor((n+1)/2)) + 3*Binomial(n+1, k+1+Floor((n+1)/2)) >;
[A028324(n,k): k in [0..Floor(n/2)], n in [0..16]]; // G. C. Greubel, Jan 06 2024
  • Mathematica
    T[n_, k_]:= Binomial[n+3, k+2+Floor[(n+1)/2]] + 3*Binomial[n+1, k+1 + Floor[(n+1)/2]];
Table[T[n,k], {n,0,16}, {k,0,Floor[n/2]}]//Flatten (* G. C. Greubel, Jan 06 2024 *)
  • SageMath
    def A028324(n,k): return binomial(n+3, k+2+(n+1)//2) + 3*binomial(n+1, k+1+((n+1)//2))
flatten([[A028324(n,k) for k in range(1+(n//2))] for n in range(17)]) # G. C. Greubel, Jan 06 2024

Formula

T(n, k) = binomial(n+3, k + 2 + floor((n+1)/2)) + 3*binomial(n+1, k + 1 + floor((n+1)/2)), for 0 <= k <= floor(n/2), n >= 0. - 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
Showing 1-10 of 15 results. Next

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

Content is available under The OEIS End-User License Agreement.