A069395 -id:A069395 - cp's OEIS frontend

A069362 Number of 4 X n binary arrays with a path of adjacent 1's from top row to bottom row.

1, 41, 1041, 22193, 433809, 8057905, 144769425, 2541013617, 43843180113, 746691527217, 12588144461329, 210502738714097, 3497001564166609, 57781030561348017, 950437243856526737, 15574913193760097649, 254416775893204873553, 4144677558181255455025

Offset: 1

R. H. Hardin, Mar 22 2002

Colin Barker, Table of n, a(n) for n = 1..800
Index entries for linear recurrences with constant coefficients, signature (35,-378,1264,-1272,-128).

Row 4 of A359576.

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.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+6*x-16*x^2-8*x^3)/((1-16*x)*(1-19*x+ 74*x^2 -80*x^3-8*x^4)))); // G. C. Greubel, Apr 22 2018

Rest[CoefficientList[Series[x*(1+6*x-16*x^2-8*x^3)/((1-16*x)*(1-19*x+ 74*x^2 -80*x^3-8*x^4)), {x,0,50}],x]] (* G. C. Greubel, Apr 22 2018 *)
LinearRecurrence[{35,-378,1264,-1272,-128},{1,41,1041,22193,433809},20] (* Harvey P. Dale, Jan 01 2019 *)

Vec(x*(1 + 6*x - 16*x^2 - 8*x^3) / ((1 - 16*x)*(1 - 19*x + 74*x^2 - 80*x^3 - 8*x^4)) + O(x^30)) \\ Colin Barker, Oct 12 2017

G.f.: x*(1 +6*x -16*x^2 -8*x^3)/((1 -16*x)*(1 -19*x +74*x^2 -80*x^3 - 8*x^4)).

A069403 a(n) = 2Fibonacci(2n+1) - 1.

Original entry on oeis.org

1, 3, 9, 25, 67, 177, 465, 1219, 3193, 8361, 21891, 57313, 150049, 392835, 1028457, 2692537, 7049155, 18454929, 48315633, 126491971, 331160281, 866988873, 2269806339, 5942430145, 15557484097, 40730022147, 106632582345, 279167724889, 730870592323, 1913444052081

Offset: 0

R. H. Hardin, Mar 22 2002

Half the number of n X 3 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Indices of A017245 = 9*n + 7 = 7, 16, 25, 34, for submitted A153819 = 16, 34, 88,. A153819(n) = 9*a(n) + 7 = 18*F(2*n+1) -2; F(n) = Fibonacci = A000045, 2's = A007395. Other recurrence: a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - Paul Curtz, Jan 02 2009

Colin Barker, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 41, 56.
J. Hietarinta and C.-M. Viallet, Singularity confinement and chaos in discrete systems, Physical Review Letters 81 (1998), pp. 326-328.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A000045, A084707.
Cf. 1 X n A000225, 2 X n A016269, 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.
Equals A052995 - 1.
Bisection of A001595, A062114, A066983.

List([0..30], n-> 2*Fibonacci(2*n+1)-1); # G. C. Greubel, Jul 11 2019

[2*Fibonacci(2*n+1)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

a[n_]:= a[n] = 3a[n-1] - 3a[n-3] + a[n-4]; a[0] = 1; a[1] = 3; a[2] = 9; a[3] = 25; Table[ a[n], {n, 0, 30}]
Table[2*Fibonacci[2*n+1]-1, {n,0,30}] (* G. C. Greubel, Apr 22 2018 *)
LinearRecurrence[{4,-4,1},{1,3,9},30] (* Harvey P. Dale, Sep 22 2020 *)

a(n) = 2*fibonacci(2*n+1)-1 \\ Charles R Greathouse IV, Jun 11 2015

Vec((1-x+x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016

[2*fibonacci(2*n+1)-1 for n in (0..30)] # G. C. Greubel, Jul 11 2019

a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25; a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
a(n) = 3*a(n-1) - a(n-2) + 1 for n>1, a(1) = 3, a(0) = 0. - Reinhard Zumkeller, May 02 2006
From R. J. Mathar, Feb 23 2009: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
G.f.: (1-x+x^2)/((1-x)*(1-3*x+x^2)). (End)
a(n) = 1 + 2*Sum_{k=0..n} Fibonacci(2*k) = 1+2*A027941(n). - Gary Detlefs, Dec 07 2010
a(n) = (2^(-n)*(-5*2^n -(3-sqrt(5))^n*(-5+sqrt(5)) +(3+sqrt(5))^n*(5+sqrt(5))))/5. - Colin Barker, Nov 02 2016

Simpler definition from Vladeta Jovovic, Mar 19 2003

A069378 Number of n X 3 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

7, 37, 197, 1041, 5503, 29089, 153769, 812849, 4296863, 22713981, 120070149, 634712209, 3355201895, 17736195433, 93756691401, 495614587553, 2619907077991, 13849295944501, 73209847696773

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Column 3 of A359576.

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.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4+2*x^5))); // G. C. Greubel, Apr 22 2018

Rest[CoefficientList[Series[x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4 +2*x^5), {x,0,50}], x]] (* G. C. Greubel, Apr 22 2018 *)

x='x+O('x^30); Vec(x*(7-12*x+x^2+2*x^3-2*x^4)/(1 -7*x+9*x^2 +x^3- 4*x^4+2*x^5)) \\ G. C. Greubel, Apr 22 2018

G.f.: x*(7-12*x+x^2+2*x^3-2*x^4)/(1-7*x+9*x^2+x^3-4*x^4+2*x^5). - Vladeta Jovovic, Jul 02 2003

A069379 Number of n X 4 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

15, 175, 1985, 22193, 247759, 2764991, 30856705, 344356289, 3842988975, 42887577455, 478623939553, 5341429762353, 59610217019311, 665248512113343, 7424156719466465, 82853403589520257, 924641917817567951

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

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

Column 4 of A359576.

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.

A069380 Number of n X 5 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

31, 781, 18621, 433809, 10056959, 232824241, 5388274121, 124693133113, 2885579381831, 66776768695477, 1545323639404349, 35761396310047985, 827579980089997079, 19151628770974955241, 443201843190147840905

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

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

Column 5 of A359576.

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.

A069404 Half the number of n X 4 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

7, 55, 377, 2427, 15253, 94847, 587031, 3625675, 22372413, 137993145, 850987067, 5247512077, 32357022035, 199515609775, 1230218484787, 7585536760417, 46772417567873, 288398549126971, 1778263916566525, 10964764644841043, 67608669872179151, 416874624972396255

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..250
Sean A. Irvine, Java program (github)

Cf. 1 X n A000225, 2 X n A016269, 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, Aug 18 2024

A069405 Half the number of n X 5 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

15, 285, 4541, 66579, 944157, 13182673, 182702967, 2522968803, 34777826197, 478971480223, 6593672923115, 90751017499077, 1248904863846397, 17186379018703213, 236498372271010941, 3254365880309197587, 44781833050605593997, 616220311429717310963

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..250

Cf. 1 X n A000225, 2 X n A016269, 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, Aug 19 2024

A069363 Number of 5 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 99, 5503, 247759, 10056959, 384479935, 14142942975, 506544513343, 17792504911231, 615793150236223, 21067276157958271, 714097521397778495, 24022705580163949439, 803089367467759614015, 26706726258154287563903

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Row 5 of A359576.

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.

A069377 Number of 19 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 22619537, 73209847696773, 115159568055775538305, 127111602733664216603859933, 114600698023505978867449552531361, 90979848541738331379871952593270363301, 66310152669631463041584664319631353072678161, 45499186393097406209322583222635994035907090539853

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(5)-a(9) from Sean A. Irvine, Apr 30 2024

A069381 Number of n X 6 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

63, 3367, 167337, 8057905, 384479935, 18287614751, 868972410929, 41278350729313, 1960665141991079, 93127506982471999, 4423369428678533705, 210101996822111263265, 9979493366382754619551, 474010149850018604630031, 22514756623847166766088601

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

cp's OEIS Frontend

A069362 Number of 4 X n binary arrays with a path of adjacent 1's from top row to bottom row.

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A069403 a(n) = 2*Fibonacci(2*n+1) - 1.

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A069378 Number of n X 3 binary arrays with a path of adjacent 1's from top row to bottom row.

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Magma

Mathematica

PARI

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A069379 Number of n X 4 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069380 Number of n X 5 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069404 Half the number of n X 4 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

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A069405 Half the number of n X 5 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Views

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Keywords

Links

Crossrefs

Extensions

A069363 Number of 5 X n binary arrays with a path of adjacent 1's from top row to bottom row.

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Author

Keywords

Crossrefs

A069377 Number of 19 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

A069381 Number of n X 6 binary arrays with a path of adjacent 1's from top row to bottom row.

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A069403 a(n) = 2Fibonacci(2n+1) - 1.