A069429 -id:A069429 - cp's OEIS frontend

A069382 Number of n X 7 binary arrays with a path of adjacent 1's from top row to bottom row.

127, 14197, 1461797, 144769425, 14142942975, 1374273318721, 133267613878665, 12913642307413193, 1251002814214706415, 121180225103268487525, 11738064445690843712709, 1137001292953873957167377, 110135446775389136829629647, 10668293121854243285020490369

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069383 Number of n X 8 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

255, 58975, 12519345, 2541013617, 506544513343, 100193001007135, 19752116290408897, 3888492110032895921, 765068584835726823447, 150495460332124702700743, 29601436900513534956486417, 5822276945638780019957186657, 1145175890285973091225709069559

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069384 Number of n X 9 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

511, 242461, 105683341, 43843180113, 17792504911231, 7146486319870849, 2856738227918860537, 1139431807812280823353, 454016084146597129480639, 180827014727915133583355805, 72007238386097441302774324845, 28672084199736350940864453233905

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069385 Number of n X 10 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1023, 989527, 882516857, 746691527217, 615793150236223, 501394148790682463, 405671453894948801169, 327195886636045773364289, 263496356226918378172858791, 212041997127527144580159558415, 170577316159214009818521435822905, 137200626549758017615769211506956897

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069386 Number of n X 11 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

2047, 4017157, 7308428597, 12588144461329, 21067276157958271, 34728307983514473409, 56797093221062281961353, 92506527736225809457497129, 150343681421865609328377897039, 244072929303780515911965529991989, 396017759826921556057184566557279157

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..100

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A069387 Number of n X 12 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

4095, 16245775, 60131384705, 210502738714097, 714097521397778495, 2380826437768134741023, 7862987107940658832635809, 25834416752721407468468718385, 84641600245355780352751022513463, 276889755885138056467643815501003447, 905061607197623234620822097612564300097

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..100

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

More terms from Sean A. Irvine, Apr 30 2024

A337129 Triangular array read by rows: T(n,0) = 2^n, T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j) for k > 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 16, 8, 12, 32, 84, 16, 24, 64, 168, 440, 32, 48, 128, 336, 880, 2304, 64, 96, 256, 672, 1760, 4608, 12064, 128, 192, 512, 1344, 3520, 9216, 24128, 63168, 256, 384, 1024, 2688, 7040, 18432, 48256, 126336, 330752, 512, 768, 2048, 5376, 14080, 36864, 96512, 252672, 661504, 1731840

Offset: 0

Oboifeng Dira, Sep 14 2020

			The triangle  T(n,k) begins:
   n\k  0    1    2    3    4    5
   0:   1
   1:   2    3
   2:   4    6    16
   3:   8    12   32  84
   4:   16   24   64  168  440
   5:   32   48   128 336  880  2304
   ...
T(3,2) = ((3+sqrt(5))^3-(3-sqrt(5))^3)*(2)/(4*sqrt(5)) = (64*sqrt(5))/(2*sqrt(5)) = 32.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Cf. A000079 (1st column), A069429 (diagonal), A018903 (row sums), A001906, A004171.

T := proc (n, k) if k = 0 and 0 <= n then 2^n elif 1 <= k and k <= n then round((((3+sqrt(5))^(k+1)-(3-sqrt(5))^(k+1))*(2^(n-k))/(4*sqrt(5)))) else 0 end if end proc:seq(print(seq(T(n, k), k=0..n)), n=0..9);

T[n_, 0] := 2^n;
T[n_, n_] := 2^(n-1) Fibonacci[2n+2];
T[n_, k_] /; 0Jean-François Alcover, Nov 13 2020 *)

T(n,k) = if (k == 0, 2^n, my(w=quadgen(5, 'w)); ((2*w+2)^(k+1)-(4-2*w)^(k+1))*(2^(n-k))/(4*(2*w-1))); \\ Michel Marcus, Sep 14 2020

Row(n)={Vecrev(polcoef((1-x*y)*(1-2*x*y)/((1-6*x*y+4*x^2*y^2)*(1-2*x)) + O(x*x^n), n))} \\ Andrew Howroyd, Sep 23 2020

T(n,0) = 2^n.
T(n,k) = ((3+sqrt(5))^(k+1)-(3-sqrt(5))^(k+1))*(2^(n-k))/(4*sqrt(5)) for 1<=k<=n.
T(n+1,n) = 2*T(n,n).
T(n+m,n) = 2^m*T(n,n), for m>=1.
T(n,n) = A069429(n) = 2^(n-1)*A001906(n+1) for n>=1.
T(2*n,n) = (1/2)*A099157(n+1) = A004171(n-1)*A001906(n+1) for n>=1.
G.f.: (1 - x*y)*(1 - 2*x*y)/((1 - 6*x*y + 4*x^2*y^2)*(1 - 2*x)). - Andrew Howroyd, Sep 23 2020

A069364 Number of 6 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 239, 29089, 2764991, 232824241, 18287614751, 1374273318721, 100193001007135, 7146486319870849, 501394148790682463, 34728307983514473409, 2380826437768134741023, 161857825276476548906497

Offset: 1

R. H. Hardin, Mar 22 2002

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

a(9)-a(13) from Sean A. Irvine, Apr 29 2024

A069365 Number of 7 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 577, 153769, 30856705, 5388274121, 868972410929, 133267613878665, 19752116290408897, 2856738227918860537, 405671453894948801169, 56797093221062281961353, 7862987107940658832635809, 1078636993747832299068001753

Offset: 1

R. H. Hardin, Mar 22 2002

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

a(7)-a(13) from Sean A. Irvine, Apr 29 2024

A069366 Number of 8 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 1393, 812849, 344356289, 124693133113, 41278350729313, 12913642307413193, 3888492110032895921, 1139431807812280823353, 327195886636045773364289, 92506527736225809457497129, 25834416752721407468468718385, 7143459701634079128687197933529

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

a(7)-a(13) from Sean A. Irvine, Apr 29 2024

cp's OEIS Frontend

A069382 Number of n X 7 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069383 Number of n X 8 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069384 Number of n X 9 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069385 Number of n X 10 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069386 Number of n X 11 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069387 Number of n X 12 binary arrays with a path of adjacent 1's from top row to bottom row.

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A337129 Triangular array read by rows: T(n,0) = 2^n, T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j) for k > 0.

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A069364 Number of 6 X n binary arrays with a path of adjacent 1's from top row to bottom row.

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Keywords

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Extensions

A069365 Number of 7 X n binary arrays with a path of adjacent 1's from top row to bottom row.

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Author

Keywords

Crossrefs

Extensions

A069366 Number of 8 X n binary arrays with a path of adjacent 1's from top row to bottom row.

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Author

Keywords

Crossrefs

Extensions