cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A069403 a(n) = 2*Fibonacci(2*n+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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • GAP
    List([0..30], n-> 2*Fibonacci(2*n+1)-1); # G. C. Greubel, Jul 11 2019
  • Magma
    [2*Fibonacci(2*n+1)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
  • Mathematica
    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 *)
  • PARI
    a(n) = 2*fibonacci(2*n+1)-1 \\ Charles R Greathouse IV, Jun 11 2015
  • PARI
    Vec((1-x+x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016
  • Sage
    [2*fibonacci(2*n+1)-1 for n in (0..30)] # G. C. Greubel, Jul 11 2019

Formula

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

Extensions

Simpler definition from Vladeta Jovovic, Mar 19 2003

A121231 Number of n X n binary matrices M (that is, real matrices with entries 0 and 1) such that M^2 is also a binary matrix.

Original entry on oeis.org

1, 2, 11, 172, 6327, 474286, 67147431, 17080038508
Offset: 0

Views

Author

Dan Dima, Aug 21 2006

Keywords

Comments

Comments from Brendan McKay, Aug 21 2006: Equivalently, directed graphs (simple but loops allowed) without a few small forbidden subgraphs (those allowing 2 distinct paths of length 2 from vertex x to vertex y for some x,y; I think there are 6 possibilities). One can also consider isomorphism classes of those digraphs.
Comment from Rob Pratt, Aug 03 2008: A121294 provides a lower bound on the maximum number of 1's in such a matrix M. There are cases where a higher number is reached; the following 5 X 5 matrix has 11 ones and its square is binary:
0 0 1 0 0
0 0 0 0 1
1 1 0 0 1
1 1 0 1 0
1 1 0 1 0.
The optimal values seem to match A070214, verified for n <= 7.
Term (5,1) of the n-th power of the 5 X 5 matrix shown is A001045(n), the Jacobsthal sequence. - Gary W. Adamson, Oct 03 2008
a(n) >= A226321(n).

Links

Crossrefs

Cf. A226321, A225371, A055084, A052264, A051589, A069452, A053304, A001045, A121294, A070214.

Extensions

Edited by R. J. Mathar, Oct 01 2008
a(7) from R. H. Hardin, Jun 19 2012. This makes it clear that the old A122527 was really a badly-described version of this sequence, and that a(7) was earlier found by Balakrishnan (bvarada2(AT)jhu.edu), Sep 17 2006. - N. J. A. Sloane, Jun 19 2012
Entry revised by N. J. A. Sloane, Jun 19 2012

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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Links

Crossrefs

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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Links

Crossrefs

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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Links

Crossrefs

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.

Extensions

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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Links

Crossrefs

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.

Extensions

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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Crossrefs

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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Crossrefs

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.

Extensions

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

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Links

Crossrefs

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.

Extensions

More terms from Sean A. Irvine, Apr 30 2024
Previous Showing 11-20 of 92 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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