A060747 -id:A060747 - cp's OEIS frontend

A204040 Triangle T(n,k), read by rows, given by (0, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, -4, 4, 6, 1, 0, -4, -8, 12, 8, 1, 0, 4, -24, -4, 24, 10, 1, 0, 12, -8, -60, 16, 40, 12, 1, 0, 4, 56, -84, -96, 60, 60, 14, 1, 0, -20, 88, 84, -272, -100, 136, 84, 16, 1, 0, -28, -40

Offset: 0

Philippe Deléham, Jan 27 2012

Antidiagonal sums : periodic sequence 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, ... (see A007877 or A098178).Riordan array (1, x*(1+x)/(1-x+2*x^2)) .

			Triangle begins :
1
0, 1
0, 2, 1
0, 0, 4, 1
0, -4, 4, 6, 1
0, -4, -8, 12, 8, 1
0, 4, -24, -4, 24, 10, 1
0, 12, -8, -60, 16, 40, 12, 1
0, 4, 56, -84, -96, 60, 60, 14, 1
0, -20, 88, 84, -272, -100, 136, 84, 16, 1

Cf. A005408.

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2, k-1) - 2*T(n-2,k).
G.f.: (1-x+2*x^2)/(1-(1+y)*x + (2-y)*x^2).
T(n,n) = n = A000012(n), T(n+1,n) = 2n = A005843(n), T(n+2,n) = A046092(n-1) for n>0, T(n+1,1) = A078050(n)*(-1)^n.
Sum_{k, 0<=k<=n} T(n,k) = A060747(n) = A005408(n-1).

A342797 Irregular triangle read by rows: T(n, k) is the k-th antidiagonal sum of the n X n matrices defined in A069480 and A078475.

Original entry on oeis.org

1, 1, 5, 4, 1, 5, 15, 15, 9, 1, 5, 15, 34, 36, 29, 16, 1, 5, 15, 34, 65, 70, 63, 47, 25, 1, 5, 15, 34, 65, 111, 120, 114, 96, 69, 36, 1, 5, 15, 34, 65, 111, 175, 189, 185, 166, 135, 95, 49, 1, 5, 15, 34, 65, 111, 175, 260, 280, 279, 260, 226, 180, 125, 64

Offset: 1

Stefano Spezia, Apr 25 2021

			The triangle T(n, k) begins:
1
1    5    4
1    5   15   15    9
1    5   15   34   36   29   16
1    5   15   34   65   70   63   47   25
...

Cf. A000290 (diagonal), A006003, A037270 (row sums), A060747 (row length), A069480, A078475.

T[n_,k_]:=If[k<=n,(k+k^3)/2,(k^3+2n-6k^2n-4n^3+k(10n^2-1))/2]; Flatten[Table[T[n,k],{n,8},{k,2n-1}]]

T(n, k) = A006003(k) for 1 <= k <= n.
T(n, k) = (k^3 + 2*n - 6*k^2*n - 4*n^3 + k*(10*n^2 - 1))/2 for n < k <= 2*n - 1.
T(n, 2*n-1) = A000290(n).

A346179 Number of nonisomorphic vertex-transitive tournaments of order 2n-1.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 16, 16, 30, 110, 94, 214, 694, 586, 1096, 3280, 5472, 7286, 25206, 26216, 49940, 196624, 182362, 407856, 907116

Offset: 1

Brendan McKay, Jul 09 2021

The circulant tournaments A049288 are included.

Up to 49 vertices, non-circulant vertex-transitive tournaments occur on 21, 25, 27, 39, 45 and 49 vertices.

Brendan McKay, vertex-transitive tournaments

If the automorphism group contains a full-length cycle, the tournament is circulant and is counted by A049288.

Cf. A060747.

A356480 a(n) is the minimal number of river crossings necessary to solve the missionaries and cannibals problem for n missionaries and n cannibals where the boat capacity is the minimum necessary to allow a solution.

Original entry on oeis.org

1, 5, 11, 9, 11, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135

Offset: 1

Sela Fried, Aug 09 2022

The problem is: n missionaries and n cannibals must cross the river using a boat. The missionaries must not be outnumbered by the cannibals at either river bank, or on the boat, and the boat cannot cross the river by itself.
It turns out that the necessary boat capacity is two people for n=1,2,3; three people for n=4,5; and four people for n > 5.
This problem is a generalization of the classical missionaries and cannibals problem, in which n = 3 and the boat capacity is two people. In this case the minimal number of crossings is a(3) = 11 (see example).

			Suppose n = 3 and that all the people must cross from the left river side to the right. Let m and c denote the number of missionaries and the number of the cannibals on the left bank of the river at any time. Let b=L if the boat is on the left bank, b=R if the boat is on the right bank. Then (m, c, b) fully captures the condition of the system. A solution of minimal length is then given by (3, 3, L)-->(2, 2, R)-->(3, 2, L)-->(3, 0, R)-->(3, 1, L)-->(1, 1, R)-->(2, 2, L)-->(0, 2, R)-->(0, 3, L)-->(0, 1, R)-->(1, 1, L)-->(0, 0, R).

P. Norvig and S. J. Russell, Artificial Intelligence: A Modern Approach, Third Edition, 2010. Exercise 3.9.

R. Fraley, K. L. Cooke, and P. Detrick, Graphical Solution of Difficult Crossing Puzzles, Mathematics Magazine, Vol. 39 (3), pp. 151-157, (1966).
Wikipedia, Missionaries and cannibals problem.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A060747, A167484.

G.f.: x*(4*x^6 - 4*x^5 + 4*x^4 - 8*x^3 + 2*x^2 + 3*x + 1)/(x-1)^2.

A360373 Triangular array T read by rows related to the multiplication table.

Original entry on oeis.org

1, 2, 4, 2, 3, 6, 9, 6, 3, 4, 8, 12, 16, 12, 8, 4, 5, 10, 15, 20, 25, 20, 15, 10, 5, 6, 12, 18, 24, 30, 36, 30, 24, 18, 12, 6, 7, 14, 21, 28, 35, 42, 49, 42, 35, 28, 21, 14, 7, 8, 16, 24, 32, 40, 48, 56, 64, 56, 48, 40, 32, 24, 16, 8, 9, 18, 27, 36, 45, 54, 63, 72, 81

Offset: 1

Philippe Deléham, Feb 04 2023

			Table T(n, k) , n>=1 , 1<=k<=2*n-1.
n = 1 : 1 ;
n = 2 : 2,  4,  2 ;
n = 3 : 3,  6,  9,  6,  3 ;
n = 4 : 4,  8, 12, 16, 12,  8,  4 ;
n = 5 : 5, 10, 15, 20, 25, 20, 15, 10,  5 ;
n = 6 : 6, 12, 18, 24, 30, 36, 30, 24, 18, 12,  6 ;
n = 7 : 7, 14, 21, 28, 35, 42, 49, 42, 35, 28, 21, 14,  7 ;
n = 8 : 8, 16, 24, 32, 40, 48, 56, 64, 56, 48, 40, 32, 24, 16,  8 ;
...

Cf. A000290 (central terms), A000578 (row sums), A060747 (row lengths).

Cf. A003991, A004737, A075362, A272125 .

T:= (n, k)-> n*min(k, 2*n-k):
seq(seq(T(n,k), k=1..2*n-1), n=1..10);  # Alois P. Heinz, Feb 04 2023

T(n, k) = T(n, 2*n-k) = n*k for 1<=k<=n .
Sum_{k=1..2*n-1} T(n, k) = n^3.
Sum_{k=1..2*n-1} T(n, k)^2 = n^3*(2*n^2 + 1)/3 = A272125(n).
T(n, k) = n * A004737(n,k).

cp's OEIS Frontend

A204040 Triangle T(n,k), read by rows, given by (0, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A342797 Irregular triangle read by rows: T(n, k) is the k-th antidiagonal sum of the n X n matrices defined in A069480 and A078475.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A346179 Number of nonisomorphic vertex-transitive tournaments of order 2n-1.

Views

Author

Keywords

Comments

Links

Crossrefs

A356480 a(n) is the minimal number of river crossings necessary to solve the missionaries and cannibals problem for n missionaries and n cannibals where the boat capacity is the minimum necessary to allow a solution.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A360373 Triangular array T read by rows related to the multiplication table.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula