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

A082761 Trinomial transform of the Fibonacci numbers (A000045).

Original entry on oeis.org

1, 4, 20, 104, 544, 2848, 14912, 78080, 408832, 2140672, 11208704, 58689536, 307302400, 1609056256, 8425127936, 44114542592, 230986743808, 1209462292480, 6332826779648, 33159111507968, 173623361929216, 909103725543424, 4760128905543680, 24924358531088384, 130505635564355584
Offset: 0

Views

Author

Emanuele Munarini, May 21 2003

Keywords

Comments

Hankel transform of Sum_{k=0..n} (-1)^k*C(2k, k) (see A054108). - Paul Barry, Jan 13 2009
Hankel transform of A046748. - Paul Barry, Apr 14 2010
For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(2)'s along the three central diagonals. - John M. Campbell, Jul 12 2011
The limiting ratio is: Lim_{n -> oo} a(n)/a(n-1) = 1 + phi^3. - Bob Selcoe, Mar 18 2014
Invert transform of A052984. Invert transform is A083066. Binomial transform of A033887. Binomial transform is A163073. - Michael Somos, May 26 2014

Examples

			a(5) = 2848 = 5*(544) + 4 + 20 + 104.
G.f. = 1 + 4*x + 20*x^2 + 104*x^3 + 544*x^4 + 2848*x^5 + 14912*x^6 + ...

Links

Crossrefs

Cf. A000045, A001519, A027907, A033887, A046748, A052984, A054108, A069429, A083066, A147703, A163073.

Programs

  • Magma
    [2^n * Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jul 15 2011
  • Mathematica
    a[ n_] := 2^n Fibonacci[ 2 n + 1]; (* Michael Somos, May 26 2014 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ (2 - x) / (4 - 6 x + x^2), {x, 0, -1 - n}], SeriesCoefficient[ (1 - 2 x) / (1 - 6 x + 4 x^2), {x, 0, n}]]; (* Michael Somos, Oct 22 2017 *)
LinearRecurrence[{6,-4},{1,4},30] (* Harvey P. Dale, Jul 11 2014 *)
  • PARI
    a(n)=fibonacci(2*n+1)<Charles R Greathouse IV, Jul 15 2011
  • PARI
    {a(n) = if( n<0, n = -1 - n; 2^(-1-2*n), 1) * polcoeff( (1 - 2*x) / (1 - 6*x + 4*x^2) + x * O(x^n), n)}; /* Michael Somos, Oct 22 2017 */
  • SageMath
    [2^n*fibonacci(2*n+1) for n in range(41)] # G. C. Greubel, Jul 28 2024

Formula

a(n) = Sum_{k=0..2*n} A027907(n, k)*A000045(k+1).
From Paul Barry, Jul 16 2003: (Start)
Third binomial transform of (1, 1, 5, 5, 25, 25, ....).
a(n) = ((1+sqrt(5))(3+sqrt(5))^n-(1-sqrt(5))*(3-sqrt(5))^n)/(2*sqrt(5)). (End)
From R. J. Mathar, Nov 04 2008: (Start)
G.f.: (1-2*x)/(1-6*x+4*x^2).
a(n) = 6*a(n-1) - 4*a(n-2). (End)
a(n) = Sum_{k=0..n} A147703(n,k)*3^k. - Philippe Deléham, Nov 14 2008
For n>=2: a(n) = 5*a(n-1) + Sum_{i=1..n-2} a(i). - Bob Selcoe, Mar 18 2014
a(n) = a(-1-n) * 2^(2*n+1) for all n in Z. - Michael Somos, Mar 18 2014
a(n) = 2^n*Fibonacci(2*n+1), or 2^n*A001519(n+1). - Bob Selcoe, May 25 2014
From Michael Somos, May 26 2014: (Start)
a(n) - a(n-1) = A069429(n).
a(n+1) * a(n-1) - a(n)^2 = 4^n.
G.f.: 1 / (1 - 4*x / (1 - x / (1 - x))). (End)
E.g.f.: exp(3*x)*(5*cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, May 24 2024

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
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