A094790 -id:A094790 - cp's OEIS frontend

A216054 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 6, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 0, 5, 9, 5, 0, 0, 0, 0, 5, 14, 14, 0, 0, 0, 0, 0, 0, 19, 28, 14, 0, 0, 0, 0, 0, 0, 19, 47, 42, 0, 0, 0, 0, 0, 0, 0, 0, 66, 89, 42, 0, 0, 0, 0, 0, 0, 0, 0, 66, 155, 131, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 221, 286, 131, 0, 0

Offset: 0

Philippe Deléham, Mar 16 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=0
0, 1, 2, 3, 4, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=1
0, 0, 2, 5, 9, 14, 19, 19, 0, 0, 0, 0, 0, 0, 0, ... row n=2
0, 0, 0, 5, 14, 28, 47, 66, 66, 0, 0, 0, 0, 0, 0, ... row n=3
0, 0, 0, 0, 14, 42, 89, 155, 221, 221, 0, 0, 0, 0, ... row n=4
0, 0, 0, 0, 0, 0, 42, 131, 286, 507, 728, 728, 0, 0, ... row n=5
0, 0, 0, 0, 0, 0, 131, 417, 924, 1652, 2380, 2380, 0, ... row n=6
...

E. Lucas, Théorie des nombres, A.Blanchard, Paris, 1958, Tome 1, p.89

Cf. A006053, A052547, A096976, A187066,

Cf. Similar sequences A216230, A216228, A216226, A216238

Clear[t]; t[0, k_ /; k <= 5] = 1; t[n_, k_] /; k < n || k > n+5 = 0; t[n_, k_] := t[n, k] = t[n-1, k] + t[n, k-1]; Table[t[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 18 2013 *)

T(n,n) = A080937(n).
T(n,n+1) = A080937(n+1).
T(n,n+2) = A094790(n+1).
T(n,n+3) = A094789(n+1).
T(n,n4) = T(n,n+5) = A005021(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A028495(n).

A216235 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 2 or if k-n >= 5, T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 2, 0, 1, 4, 5, 0, 0, 0, 5, 9, 5, 0, 0, 0, 5, 14, 14, 0, 0, 0, 0, 0, 19, 28, 14, 0, 0, 0, 0, 0, 19, 47, 42, 0, 0, 0, 0, 0, 0, 0, 66, 89, 42, 0, 0, 0, 0, 0, 0, 0, 66, 155, 131, 0, 0, 0, 0, 0, 0, 0, 0, 0, 221, 286, 131, 0, 0, 0, 0, 0, 0, 0, 0, 0, 221, 507, 417, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 14 2013

Arithmetic hexagon of E. Lucas.

			Square array begins:
  1, 1, 1,  1,  1,   0,   0,   0,   0,   0, ... row n=0
  1, 2, 3,  4,  5,   5,   0,   0,   0,   0, ... row n=1
  0, 2, 5,  9, 14,  19,  19,   0,   0,   0, ... row n=2
  0, 0, 5, 14, 28,  47,  66,  66,   0,   0, ... row n=3
  0, 0, 0, 14, 42,  89, 155, 221, 221,   0, ... row n=4
  0, 0, 0,  0, 42, 131, 286, 507, 728, 728, ... row n=5
  ...

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

Cf. A005021, A080937, A094789, A094790.

Similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232.

T(n,n) = T(n+1,n) = A080937(n+1).
T(n,n+1) = A094790(n+1).
T(n,n+2) = A094789(n+1).
T(n,n+3) = T(n,n+4) = A005021(n).
Sum_{k=0..n} T(n-k,k) = A028495(n+1). - Philippe Deléham, Mar 23 2013

A099918 A Chebyshev transform related to the 7th cyclotomic polynomial.

Original entry on oeis.org

1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1

Offset: 0

Paul Barry, Oct 30 2004

The g.f. is a Chebyshev transform of 1/(1+x-2x^2-x^3) under the Chebyshev mapping g(x)->(1/(1+x^2))g(x/(1+x^2)). The denominator is the 7th cyclotomic polynomial. The inverse of the 7 cyclotomic polynomial A014016 is given by sum{k=0..n, A099918(n-k)(k/2+1)(-1)^(k/2)(1+(-1)^k)/2}.

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1).

Cf. A099860.

LinearRecurrence[{-1,-1,-1,-1,-1,-1},{1,-1,2,-2,1,-1},90] (* Harvey P. Dale, May 25 2019 *)

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

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)^(-1)^k*b(n-2k), where b(n)=A094790(n/2+1)(1+(-1)^n)/2+A094789((n+1)/2)(1-(-1)^n)/2=(-1)^n*A006053(n+2).

cp's OEIS Frontend

A216054 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 6, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A216235 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 2 or if k-n >= 5, T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A099918 A Chebyshev transform related to the 7th cyclotomic polynomial.

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Programs

Mathematica

Formula