A216219 -id:A216219 - cp's OEIS frontend

A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000, 40957275390625, 148184814453125

Offset: 0

Wolfdieter Lang

Number of (s(0), s(1), ..., s(2n+4)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+4, s(0) = 1, s(2n+4) = 5. - Herbert Kociemba, Jun 14 2004
Binomial transform of A002878. - Philippe Deléham, Oct 04 2005
Diagonal of square array A216219. - Philippe Deléham, Mar 15 2013
Lim_{n->oo} a(n+1)/a(n) = 2 + phi = A296184, where phi = A001622. - Wolfdieter Lang, Nov 16 2023~

			G.f. = 1 + 5*x + 20*x^2 + 75*x^3 + 275*x^4 + 1000*x^5 + 3625*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Santiago Alzate, Oscar Correa, and Rigoberto Flórez, Fibonacci identities from Jordan Identities, arXiv:2009.02639 [math.NT], 2020.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0 ,b=1; p=5, q=-5.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=5.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (5,-5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000032, A000045, A001622.

Cf. A002878, A039717, A081567, A109466, A216219.

a:=[1,5];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 28 2019

I:=[1,5]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 28 2019

seq(coeff(series(1/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 28 2019

Table[MatrixPower[{{2,1},{1,3}},n][[1]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
a[ n_]:= (((5 + Sqrt[5])/2)^(n + 1) - ((5 - Sqrt[5])/2)^(n + 1)) / Sqrt[5] // Expand; (* Michael Somos, Aug 27 2015 *)
Table[If[EvenQ[n], 5^(n/2)*LucasL[n+1], 5^((n+1)/2)*Fibonacci[n+1]], {n,0,30}] (* G. C. Greubel, Dec 28 2019 *)

{a(n) = imag((quadgen(5) + 2)^(n+1))}; /* Michael Somos, Aug 27 2015 */

[lucas_number1(n,5,5) for n in range(1, 22)] # Zerinvary Lajos, Apr 22 2009

a(n) = (sqrt(5))^n*U(n, sqrt(5)/2).
G.f.: 1/(1 - 5*x + 5*x^2).
a(2*k+1) = 5^(k+1)*Fibonacci(2*k+2).
a(2*k) = 5^k*Lucas(2*k+1).
a(n-1) = Sum_{k=0..n} C(n, k)*Fibonacci(2*k). - Benoit Cloitre, Jun 21 2003
a(n) = 5*a(n-1) - 5*a(n-2). - Benoit Cloitre, Oct 23 2003
a(n-1) = (((5+sqrt(5))/2)^n - ((5-sqrt(5))/2)^n)/sqrt(5) is the 2nd binomial transform of Fibonacci(n), the first binomial transform of Fibonacci(2n) and its n-th term is the n-th term of the third binomial transform of Fibonacci(3n) divided by 2^n. - Paul Barry, Mar 23 2004
a(n) = Sum_{k-0..n} 5^k*A109466(n,k). - Philippe Deléham, Nov 28 2006
a(n) = 5*A039717(n), n>0. - Philippe Deléham, Mar 12 2013
a(n) = A216219(n,n+3) = A216219(n,n+4) = A216219(n+3,n) = A216219(n+4,n). - Philippe Deléham, Mar 15 2013
G.f.: 1/(1-5*x/(1+x/(1-x))). - Philippe Deléham, Mar 15 2013
a(n) = -a(-2-n) * 5^(n+1) for all n in Z. - Michael Somos, Aug 27 2015
E.g.f.: exp((5-sqrt(5))*x/2)*((5 + sqrt(5))*exp(sqrt(5)*x) - 5 + sqrt(5))/(2*sqrt(5)). - Stefano Spezia, Dec 29 2019
a(n) = Sum_{k=0..n} A081567(n-k)*2^k. - Philippe Deléham, Mar 10 2023

A216226 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=4, T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 0, 3, 2, 0, 0, 0, 3, 5, 0, 0, 0, 0, 0, 8, 5, 0, 0, 0, 0, 0, 8, 13, 0, 0, 0, 0, 0, 0, 0, 21, 13, 0, 0, 0, 0, 0, 0, 0, 21, 34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 34, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 13 2013

			Square array begins:
1, 1, 1, 1,  0,  0,  0,   0,   0, 0, ... row n=0
0, 1, 2, 3,  3,  0,  0,   0,   0, 0, ... row n=1
0, 0, 2, 5,  8,  8,  0,   0,   0, 0, ... row n=2
0, 0, 0, 5, 13, 21, 21,   0,   0, 0, ... row n=3
0, 0, 0, 0, 13, 34, 55,  55,   0, 0, ... row n=4
0, 0, 0, 0,  0, 34, 89, 144, 144, 0, ... row n=5
...

Cf. A000045 (Fibonacci numbers), A000285, A001519, A001906, A068914

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

T(n,n) = A000045(2*n-1) = A001519(n).
T(n,n+1) = A000045(2*n+1) = A001519(n+1).
T(n,n+2) = T(n,n+3) = A000045(2*n+2) = A001906(n+1).
Sum_{k, 0<=k<=n} T(n-k,k) = A000045(n+1).
Sum_{k, k>=0} T(n,k) = A000285(2*n+1).
Sum_{n, n>=0} T(n,k) = A000285(2*k-2), k>=2.

A216230 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=2, T(0,0) = T(0,1) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 14 2013

			Square array begins:
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, ...
...

Cf. A000012, A068914

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

T(n,n) = T(n,n+1) = 1.

Sum_{k, 0<=k<=n} T(n-k, k) = 1.

A223968 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 5 or if k-n >= 6, T(4,0) = T(3,0) = T(2,0) = T(1,0) = 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).

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, 0, 0, 6, 15, 20, 15, 5, 0, 0, 6, 21, 35, 35, 20, 0, 0, 0, 0, 27, 56, 70, 55, 20, 0, 0, 0, 0, 27, 83, 126, 125, 75, 0, 0, 0, 0, 0, 0, 110, 209, 251, 200, 75, 0, 0, 0, 0, 0, 0, 110, 319, 460, 451, 275, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 30 2013

			Square array begins:
1....1....1....1....1....1....0....0....0....0....0....0
1....2....3....4....5....6....6....0....0....0....0....0
1....3....6...10...15...21...27...27....0....0....0....0
1....4...10...20...35...56...83..110..110....0....0....0
1....5...15...35...70..126..209..319..429..429....0....0
0....5...20...55..125..251..460..779.1208.1637.1637....0
0....0...20...75..200..451..911.1690.2898.4535.6172.6172
...
Square array, read by diagonals, with 0 omitted:
1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, ...
1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, ...
1, 4, 15, 55, 200, 726, 2638, 9604, 35037, ...
1, 3, 10, 35, 125, 451, 1637, 5965, 21794, ...
1, 2, 6, 20, 70, 251, 911, 3327, 12190, 44744, ...
1, 3, 10, 35, 126, 460, 1690, 6225, 22950, ...
1, 4, 15, 56, 209, 779, 2898, 10760, 39882, ...
1, 5, 21, 83, 319, 1208, 4535, 16932, 62986, ...
1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, ...
1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, ...

sum(T(n-k,k), 0<=k<=n) = A223940(n).
T(n,n+5) = T(n,n+4) = A221863(n).
T(n,n+3) = A221862(n).
T(n,n+2) = A221859(n).
T(n,n+1) = A216710(n).
T(n,n) = A224514(n).
T(n+1,n) = A224509(n).
T(n+2,n) = A220948(n).
T(n+3,n) = T(n+4,n) = A224422(n). - Philippe Deléham, Apr 13 2013

A216232 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 3 or if k-n >= 5, T(2,0) = 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, 1, 1, 3, 3, 0, 1, 4, 6, 3, 0, 0, 5, 10, 9, 0, 0, 0, 5, 15, 19, 9, 0, 0, 0, 0, 20, 34, 28, 0, 0, 0, 0, 0, 20, 54, 62, 28, 0, 0, 0, 0, 0, 0, 74, 116, 90, 0, 0, 0, 0, 0, 0, 0, 74, 190, 206, 90, 0, 0, 0, 0, 0, 0, 0, 0, 264, 396, 296, 0, 0, 0, 0, 0, 0, 0, 0, 0, 264, 660, 692, 296, 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, 0, ... row n=0
  1, 2, 3,  4,  5,   5,   0,   0,   0,   0, 0, ... row n=1
  1, 3, 6, 10, 15,  20,  20,   0,   0,   0, 0, ... row n=2
  0, 3, 9, 19, 34,  54,  74,  74,   0,   0, 0, ... row n=3
  0, 0, 9, 28, 62, 116, 190, 264, 264,   0, 0, ... row n=4
  0, 0, 0, 28, 90, 206, 396, 660, 924, 924, 0, ... row n=5
  ...
Array, read by rows, with 0 omitted:
   1,   1,   1,   1,    1
   1,   2,   3,   4,    5,    5
   1,   3,   6,  10,   15,   20,   20
        3,   9,  19,   34,   54,   74,   74
             9,  28,   62,  116,  190,  264,  264
                 28,   90,  206,  396,  660,  924,  924
                       90,  296,  692, 1352, 2276, 3200, 3200
  ...

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

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

Cf. A007052, A094803, A094806, A094817, A094821

Cf. similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230.

T(n,n) = A094817(n), for n > 0.
T(n+1,n) = T(n+2,n) = A094803(n).
T(n,n+1) = A007052(n).
T(n,n+2) = A094821(n+1).
T(n,n+3) = T(n,n+4) = A094806(n).
Sum_{k=0..n} T(n-k,k) = A217730(n). - Philippe Deléham, Mar 22 2013

A217770 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=4 or if k-n >= 6, T(3,0) = T(2,0) = T(1,0) = 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).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 0, 1, 5, 10, 10, 4, 0, 0, 6, 15, 20, 14, 0, 0, 0, 6, 21, 35, 34, 14, 0, 0, 0, 0, 27, 56, 69, 48, 0, 0, 0, 0, 0, 27, 83, 125, 117, 48, 0, 0, 0, 0, 0, 0, 110, 208, 242, 165, 0, 0, 0, 0, 0, 0, 0, 110, 318, 450, 407, 165

Offset: 0

Philippe Deléham, Mar 24 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
n=0: 1, 1,  1,  1,   1,   1,   0,   0,    0,    0,    0, 0, ...
n=1: 1, 2,  3,  4,   5,   6,   6,   0,    0,    0,    0, 0, ...
n=2: 1, 3,  6, 10,  15,  21,  27,  27,    0,    0,    0, 0, ...
n=3: 1, 4, 10, 20,  35,  56,  83, 110,  110,    0,    0, 0, ...
n=4: 0, 4, 14, 34,  69, 125, 208, 318,  428,  428,    0, 0, ...
n=5: 0, 0, 14, 48, 117, 242, 450, 768, 1196, 1624, 1624, 0, ...
...
Square array, read by rows, with 0 omitted:
...1,    1,     1,     1,     1,      1
...1,    2,     3,     4,     5,      6,      6
...1,    3,     6,    10,    15,     21,     27,     27
...1,    4,    10,    20,    35,     56,     83,    110,    110
...4,   14,    34,    69,   125,    208,    318,    428,    428
..14,   48,   117,   242,   450,    768,   1196,   1624,   1624
..48,  165,   407,   857,  1625,   2821,   4445,   6069,   6069
.165,  572,  1429,  3054,  5875,  10320,  16389,  22458,  22458
.572, 2001,  5055, 10930, 21250,  37639,  60097,  82555,  82555
2001, 7056, 17986, 39236, 76875, 136972, 219527, 302082, 302082
...
Triangle begins:
1
1, 1
1, 2,  1
1, 3,  3,  1
1, 4,  6,  4,  0
1, 5, 10, 10,  4,  0
0, 6, 15, 20, 14,  0, 0
0, 6, 21, 35, 34, 14, 0, 0
...

Cf. Similar sequences: A214846, A216054, A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235, A216236, A216238, A217257, A217315, A217593, A217765.

T(n,n+4) = T(n,n+5) = A094788(n+2).
T(n,n+3) = A217783(n).
T(n,n+2) = A217779(n).
T(n,n+1) = A081567(n).
T(n,n)   = A217782(n).
T(n+1,n) = A217778(n).
T(n+3,n) = T(n+2,n) = A094667(n+1).
Sum(T(n-k,k), k=0..n) = A217777(n).

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).

A216238 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=5, 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, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 0, 4, 5, 0, 0, 0, 0, 4, 9, 5, 0, 0, 0, 0, 0, 13, 14, 0, 0, 0, 0, 0, 0, 13, 27, 14, 0, 0, 0, 0, 0, 0, 0, 40, 41, 0, 0, 0, 0, 0, 0, 0, 0, 40, 81, 41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121, 122, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 14 2013

Hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1,  1,  0,   0,   0,   0,    0,    0, ... row n=0
0, 1, 2, 3,  4,  4,   0,   0,   0,    0,    0, ... row n=1
0, 0, 2, 5,  9, 13,  13,   0,   0,    0,    0, ... row n=2
0, 0, 0, 5, 14, 27,  40,  40,   0,    0,    0, ... row n=3
0, 0, 0, 0, 14, 41,  81, 121, 121,    0,    0, ... row n=4
0, 0, 0, 0,  0, 41, 122, 243, 364,  364,    0, ... row n=5
0, 0, 0, 0,  0,  0, 122, 365, 729, 1093, 1093, ... row n=6
...

E. Lucas, Théorie des nombres, Albert Blanchard, Paris, 1958, Tome1, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Cf. A000244, A003462, A124302, A182522.

Similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235, A216236.

T(n,n) = A124302(n).
T(n,n+1) = A124302(n+1).
T(n,n+2) = 3^n = A000244(n).
T(n,n+3) = T(n,n+4) = A003462(n+1).
Sum_{k, 0<=k<=n} T(n-k,k) = A182522(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

A216236 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=4 or if k-n>=5, T(3,0) = T(2,0) = 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, 1, 1, 3, 3, 1, 1, 4, 6, 4, 0, 0, 5, 10, 10, 4, 0, 0, 5, 15, 20, 14, 0, 0, 0, 0, 20, 35, 34, 14, 0, 0, 0, 0, 20, 55, 69, 48, 0, 0, 0, 0, 0, 0, 75, 124, 117, 48, 0, 0, 0, 0, 0, 0, 75, 199, 241, 165, 0, 0, 0, 0, 0, 0, 0, 0, 274, 440, 406, 165, 0, 0, 0, 0, 0, 0, 0, 0, 274, 714, 846, 571, 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, 0, ...
1, 2, 3, 4, 5, 5, 0, 0, 0, 0, 0, ...
1, 3, 6, 10, 15, 20, 20, 0, 0, 0, ...
1, 4, 10, 20, 35, 55, 75, 75, 0, 0, 0, ...
0, 4, 14, 34, 69, 124, 199, 274, 274, 0, 0, ...
0, 0, 14, 48, 117, 241, 440, 714, 988, 988, 0, ...
...

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

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235.

T(n+3,n) = T(n+2,n) = A094827(n).
T(n+1,n) = A094832(n).
T(n,n) = A094854(n).
T(n,n+1) = A094855(n).
T(n,n+2) = A094833(n+1).
T(n,n+3) = T(n,n+4) = A094828(n).
Sum( T(n-k,k), 0<=k<=n ) = A217733(n). - Philippe Deléham, Mar 22 2013

cp's OEIS Frontend

A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

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A216226 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=4, T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A216230 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=2, T(0,0) = T(0,1) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A223968 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 5 or if k-n >= 6, T(4,0) = T(3,0) = T(2,0) = T(1,0) = 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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A216232 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 3 or if k-n >= 5, T(2,0) = 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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A217770 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=4 or if k-n >= 6, T(3,0) = T(2,0) = T(1,0) = 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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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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A216238 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=5, 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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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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