A028321 -id:A028321 - cp's OEIS frontend

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

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

James Sellers

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A028313, A028314, A028315, A028316, A028317, A028318, A028319.

Cf. A028320, A028321, A028322, A028323, A028324, A028325, A051472.

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

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 *)

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

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

Mohammad K. Azarian

			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;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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.

[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

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 *)

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

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)

More terms from Sam Alexander (pink2001x(AT)hotmail.com)

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

Mohammad K. Azarian

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A028313, A028314, A028315, A028316, A028317, A028318, A028320.

Cf. A028321, A028322, A028323, A028324, A028325, A051472.

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 *)

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

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

Mohammad K. Azarian

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A028313, A028314, A028315, A028316, A028317, A028318, A028319.

Cf. A028321, A028322, A028323, A028324, A028325, A051472.

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 *)

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

More terms from James Sellers, Dec 08 1999

cp's OEIS Frontend

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

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

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

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

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Mathematica

SageMath

Extensions