A216232 -id:A216232 - cp's OEIS frontend

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).

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

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).

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

A094817 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.

Original entry on oeis.org

2, 6, 19, 62, 206, 692, 2340, 7944, 27032, 92112, 314128, 1071776, 3657824, 12485696, 42623040, 145512576, 496787840, 1696093440, 5790732544, 19770612224, 67500721664, 230461137920, 786842059776, 2686443866112

Offset: 1

Herbert Kociemba, Jun 12 2004

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.

Michael De Vlieger, Table of n, a(n) for n = 1..1875
Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

Cf. A007052, A216232.

Rest@ CoefficientList[Series[-x (2 - 6 x + 3 x^2)/((2 x - 1) (2 x^2 - 4 x + 1)), {x, 0, 24}], x] (* Michael De Vlieger, Feb 12 2022 *)

a(n) = (1/4) * Sum_{r=1..7} sin(3*r*Pi/8)^2*(2*cos(r*Pi/8))^(2*n).
a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4.
G.f.: -x*(2-6*x+3*x^2) / ( (2*x-1)*(2*x^2-4*x+1) ).
a(n) = A216232(n,n), for n >= 1. - Philippe Deléham, Mar 21 2013
4*a(n) = 2*A007052(n) + 2^n. - R. J. Mathar, Nov 14 2019

A217730 Expansion of (1+2x-x^3)/(1-4x^2+2*x^4).

Original entry on oeis.org

1, 2, 4, 7, 14, 24, 48, 82, 164, 280, 560, 956, 1912, 3264, 6528, 11144, 22288, 38048, 76096, 129904, 259808, 443520, 887040, 1514272, 3028544, 5170048, 10340096, 17651648, 35303296, 60266496, 120532992, 205762688, 411525376, 702517760, 1405035520, 2398545664, 4797091328, 8189147136, 16378294272, 27959497216

Offset: 0

Philippe Deléham, Mar 22 2013

In general, a(n,j,m) = Sum_{r=1..m} (2^n*(1-(-1)^r)*cos(Pi*r/(m+1))^n*cot(Pi*r/(2*(m+1)))*sin(j*Pi*r/(m+1)))/(m+1) gives the number of paths of length n starting at the j-th node on the path graph P_m. Here we have the case m=7 and j=3. - Herbert Kociemba, Sep 17 2020

Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Math. 332 (2014), 45--54. MR3227977. See Fig. 5. - _N. J. A. Sloane_, Aug 04 2014
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Cf. A007070, A077957, A204089, A216232.

First differences are in A062113.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x-x^3)/(1-4*x^2+2*x^4))); // Bruno Berselli, Mar 22 2013

CoefficientList[Series[(1 + 2 x - x^3)/(1 - 4 x^2 + 2 x^4), {x, 0, 40}], x] (* Bruno Berselli, Mar 22 2013 *)
a[n_,j_,m_]:=Sum[(2^(n+1)Cos[Pi r/(m+1)]^n Cot[Pi r/(2(m+1))] Sin[j Pi r/(m+1)])/(m+1),{r,1,m,2}]
Table[a[n,3,7],{n,0,40}]//Round (* Herbert Kociemba, Sep 17 2020 *)

makelist(coeff(taylor((1+2*x-x^3)/(1-4*x^2+2*x^4), x, 0, n), x, n), n, 0, 40); /* Bruno Berselli, Mar 22 2013 */

G.f.: (1+x)*(1+x-x^2)/(1-4*x^2+2*x^4).
a(n) = Sum_{k=0..n} A216232(n-k,k).
a(n) = 4*a(n-2) - 2*a(n-4) for n>=4, a(0)=1, a(1)=2, a(2)=4, a(3)=7.
a(2*n) = A007070(n), a(2*n-1) = a(2*n)/2 = A007070(n)/2.
a(n)*a(n+1)-a(n-1)*a(n+2) = (1-(-1)^n)*2^floor(n/2-1) for n>0. - Bruno Berselli, Mar 22 2013
a(n) = Sum_{r=1..7} (2^n*(1-(-1)^r)*cos(Pi*r/8)^n*cot(Pi*r/16)*sin(3*Pi*r/8))/8. - Herbert Kociemba, Sep 17 2020

cp's OEIS Frontend

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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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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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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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).

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A094817 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.

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A217730 Expansion of (1+2*x-x^3)/(1-4*x^2+2*x^4).

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A217730 Expansion of (1+2x-x^3)/(1-4x^2+2*x^4).