A216263 -id:A216263 - cp's OEIS frontend

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).

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).

A216271 Expansion of (1-x)/((1-2x)(1-4x+x^2)).

Original entry on oeis.org

1, 5, 21, 83, 319, 1209, 4549, 17051, 63783, 238337, 890077, 3322995, 12403951, 46296905, 172791861, 644886923, 2406788599, 8982333009, 33522674509, 125108627171, 466912358463, 1742541855257, 6503257159717, 24270490977915, 90578715140551, 338044386361505, 1261598863859901

Offset: 0

Philippe Deléham, Mar 16 2013

Partial sums are in A216263.

Diagonal of square array A214846.

Index entries for linear recurrences with constant coefficients, signature (6,-9,2).

Cf. A001353, A087946, A214846, A216263.

CoefficientList[Series[(1-x)/((1-2x)(1-4x+x^2)),{x,0,30}],x] (* Harvey P. Dale, Oct 05 2019 *)

a(n) = A001353(n+2) - A087946(n+1).
G.f.: (1-x)/(1-6x+9x^2-2x^3).
a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3), a(0) = 1, a(1) = 5, a(2) = 21.
Sum_{k=0..n} a(k) = A216263(n).

cp's OEIS Frontend

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).

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