A216241 -id:A216241 - cp's OEIS frontend

A068913 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than -n.

1, 0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 4, 4, 2, 1, 0, 4, 6, 4, 2, 1, 0, 8, 12, 8, 4, 2, 1, 0, 8, 18, 14, 8, 4, 2, 1, 0, 16, 36, 28, 16, 8, 4, 2, 1, 0, 16, 54, 48, 30, 16, 8, 4, 2, 1, 0, 32, 108, 96, 60, 32, 16, 8, 4, 2, 1, 0, 32, 162, 164, 110, 62, 32, 16, 8, 4, 2, 1, 0, 64, 324, 328, 220, 124, 64, 32, 16, 8, 4, 2, 1

Offset: 0

Henry Bottomley, Mar 06 2002

			Rows start:
  1,  0,  0,  0,  0, ...
  1,  2,  2,  4,  4, ...
  1,  2,  4,  6, 12, ...
  1,  2,  4,  8, 14, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Cf. early rows: A000007, A016116 (without initial term), A068911, A068912, A216212, A216241, A235701.

Central and lower diagonals are A000079, higher diagonals include A000918, A028399.

T[n_,0]=1; T[n_,k_]:=2^k/(n+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (n+1))]^k Cot[(Pi (1-2r))/(4 (n+1))],{r,1,n+1}]; Table[T[r,n-r],{n,0,20},{r,0,n}]//Round//Flatten (* Herbert Kociemba, Sep 23 2020 *)

Starting with T(n, 0) = 1, if (k-n) is negative or even then T(n, k) = 2*T(n, k-1), otherwise T(n, k) = 2*T(n, k-1) - A061897(n+1, (k-n-1)/2). So for n>=k, T(n, k) = 2^k. [Corrected by Sean A. Irvine, Mar 23 2024]

T(n,0) = 1, T(n,k) = (2^k/(n+1))*Sum_{r=1..n+1} (-1)^r*cos((Pi*(2*r-1))/(2*(n+1)))^k*cot((Pi*(1-2*r))/(4*(n+1))). - Herbert Kociemba, Sep 23 2020

A214846 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 6 or if k-n >= 6, T(k,0) = T(0,k) = 1 if 0 <= k <= 5, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 0, 0, 6, 21, 35, 35, 21, 6, 0, 0, 0, 27, 56, 70, 56, 27, 0, 0, 0, 0, 27, 83, 126, 126, 83, 27, 0, 0, 0, 0, 0, 110, 209, 252, 209, 110, 0, 0, 0, 0, 0, 0, 110, 319, 461, 461, 319, 110, 0, 0, 0, 0, 0, 0, 0, 429, 780, 922, 780, 429, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 16 2013

An arithmetic hexagon of E. Lucas.

			Square array begins:
  1, 1,  1,   1,   1,   1,    0,    0,    0,     0,     0, ...
  1, 2,  3,   4,   5,   6,    6,    0,    0,     0,     0, ...
  1, 3,  6,  10,  15,  21,   27,   27,    0,     0,     0, ...
  1, 4, 10,  20,  35,  56,   83,  110,  110,     0,     0, ...
  1, 5, 15,  35,  70, 126,  209,  319,  429,   429,     0, ...
  1, 6, 21,  56, 126, 252,  461,  780, 1209,  1638,  1638, ...
  0, 6, 27,  83, 209, 461,  922, 1702, 2911,  4549,  6187, ...
  0, 0, 27, 110, 319, 780, 1702, 3404, 6315, 10864, 17051, ...
  ...

E. Lucas, Théorie des nombres, Gauthier-Villars, Paris 1891, Tome 1, p. 89.

Cf. similar sequences: A000007, A216218, A216216, A216210, A216219.

T(n,n) = A087944(n).
T(n,n+1) = T(n+1,n) = A087946(n).
T(n+2,n) = T(n,n+2) = A001353(n+1).
T(n+3,n) = T(n,n+3) = A216271(n).
T(n+5,n) = T(n+4,n) = T(n,n+4) = T(n,n+5) = A216263(n).
Sum_{k=0..n} T(n-k,k) = A216241(n).

A235701 Number of n step walks (each step +/- 1 starting from 0) which are never more than 6 or less than -6.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 126, 252, 490, 980, 1890, 3780, 7252, 14504, 27734, 55468, 105840, 211680, 403368, 806736, 1535954, 3071908, 5845406, 11690812, 22238062, 44476124, 84582428, 169164856, 321661970, 643323940, 1223146232, 2446292464, 4650833040

Offset: 0

Geoffrey Critzer, Jan 14 2014

Index entries for linear recurrences with constant coefficients, signature (0,7,0,-14,0,7)

Cf. A068911, A068912, A216212, A216241.

Row n=6 of A068913.

nn=30;r=Solve[{s==1 + x a + x b, a==x s + x c, b==x s +x d, c==x a +x e, d==x b + x f, e==x c+x g, f==x d+x h,g==x e+x i, h==x f+x j,i==x g+x k,j==x h+x l,k==x i,l==x j, z==x k + x l }, {s,a,b,c,d,e,f,g,h,i,j,k,l,z}]; CoefficientList[Series[s+a+b+c+d+e+f+g+h+i+j+k+l/.r,{x,0,nn}],x]

G.f.: (1 + x - 2*x^2 - x^3)^2/(1 - 7*x^2 + 14*x^4 - 7*x^6).

cp's OEIS Frontend

A068913 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than -n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A214846 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 6 or if k-n >= 6, T(k,0) = T(0,k) = 1 if 0 <= k <= 5, T(n,k) = T(n-1,k) + T(n,k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A235701 Number of n step walks (each step +/- 1 starting from 0) which are never more than 6 or less than -6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula