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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A051633 a(n) = 5*2^n - 2.

Original entry on oeis.org

3, 8, 18, 38, 78, 158, 318, 638, 1278, 2558, 5118, 10238, 20478, 40958, 81918, 163838, 327678, 655358, 1310718, 2621438, 5242878, 10485758, 20971518, 41943038, 83886078, 167772158, 335544318, 671088638, 1342177278, 2684354558, 5368709118, 10737418238, 21474836478
Offset: 0

Views

Author

Asher Auel

Keywords

Examples

			a(5) = 5*2^4 - 2 = 80 - 2 = 78.

Links

Crossrefs

Cf. A000079, A020714, A118654.
Cf. A123208, A048487, A131051, A153894.

Programs

  • Mathematica
    LinearRecurrence[{3, -2},{3, 8},30] (* Ray Chandler, Jul 18 2020 *)

Formula

a(n) = A118654(n, 5).
a(n) = A000079(n)*5 - 2 = A020714(n) - 2. - Omar E. Pol, Dec 23 2008
a(n) = 2*(a(n-1)+1) with a(0)=3. - Vincenzo Librandi, Aug 06 2010
a(n) = A123208(2*n+1) = A048487(n)+2 = A131051(n+2) = A153894(n)-1. - Philippe Deléham, Apr 15 2013
G.f.: ( 3-x ) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Mar 23 2023
E.g.f.: exp(x)*(5*exp(x) - 2). - Stefano Spezia, Oct 03 2023

A022105 Fibonacci sequence beginning 1, 15.

Original entry on oeis.org

1, 15, 16, 31, 47, 78, 125, 203, 328, 531, 859, 1390, 2249, 3639, 5888, 9527, 15415, 24942, 40357, 65299, 105656, 170955, 276611, 447566, 724177, 1171743, 1895920, 3067663, 4963583, 8031246, 12994829
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n-1)=sum(P(15;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=14. These are the SW-NE diagonals in P(15;n,k), the (15,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Links

Crossrefs

a(n) = A109754(14, n+1).
a(k) = A118654(4, k).

Programs

  • Magma
    a0:=1; a1:=15; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013
  • Mathematica
    a={};b=1;c=15;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,12,1}];a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
LinearRecurrence[{1,1},{1,15},40] (* Harvey P. Dale, Oct 11 2015 *)

Formula

a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=15. a(-1):=14.
G.f.: (1+14*x)/(1-x-x^2).
a(n) = A101220(14,0,n+1). - Ross La Haye, May 02 2006

A022401 Fibonacci sequence beginning 1, 31.

Original entry on oeis.org

1, 31, 32, 63, 95, 158, 253, 411, 664, 1075, 1739, 2814, 4553, 7367, 11920, 19287, 31207, 50494, 81701, 132195, 213896, 346091, 559987, 906078, 1466065, 2372143, 3838208, 6210351, 10048559, 16258910
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Programs

  • Magma
    I:=[1, 31]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 30 2014
  • Mathematica
    CoefficientList[Series[(1 + 30 x)/(1 - x - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 30 2014 *)
Table[Fibonacci[n + 2] + 29*Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Mar 01 2018 *)
  • PARI
    for(n=0, 40, print1(fibonacci(n+2) + 29*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 01 2018

Formula

a(n) = A118654(5, n).
G.f.: (1+30*x)/(1-x-x^2). - Philippe Deléham, Nov 20 2008
a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-61+sqrt(5)) + (1+sqrt(5))^n*(61+sqrt(5)))) / sqrt(5). - Colin Barker, Mar 02 2018

A119587 2^n + 1 - 2*Fibonacci(n+1).

Original entry on oeis.org

0, 1, 1, 3, 7, 17, 39, 87, 189, 403, 847, 1761, 3631, 7439, 15165, 30795, 62343, 125905, 253783, 510759, 1026685, 2061731, 4136991, 8295873, 16627167, 33311647, 66716029, 133582107, 267406999, 535206833, 1071049287
Offset: 0

Views

Author

Ross La Haye, May 31 2006, Jun 27 2007

Keywords

Links

Crossrefs

Cf. A000032, A000045, A000051, A000071, A006355, A008466, A027934, A101220, A118654.

Programs

  • Mathematica
    Table[2^n + 1 - 2 Fibonacci[n + 1], {n, 0, 30}]

Formula

a(n) = 2^n + 1 - 2*Fibonacci(n+1) = 2^n + 1 + Fibonacci(n) - Fibonacci(n+3) = 2^n + 1 - Fibonacci(n) - Lucas(n). a(n) = 2(2^(n-1) - Fibonacci(n+1)) + 1, for n > 0. a(n) = A000051(n) - A006355(n+2) = A000051(n) - A000045(n) - A000032(n). a(n) = A101220(2,2,n-1) - A101220(1,1,n-3), for n > 2. a(n) = A008466(n) - A000071(n-1), for n > 0. a(n) = 2*A008466(n-1) + 1, for n > 0.
a(n) = 2*A101220(2,2,n-2) + 1, for n > 1. a(n) = Sum[2^(n-k)Fibonacci(k) - Fibonacci(k-2),{k,0,n}] = antidiagonal sums of A118654. a(n+1) - a(n) = 2(2^(n-1) - Fibonacci(n)), for n > 0. a(n+1) - a(n) = 2*A027934(n-2), for n > 1. a(n+1) - a(n) = 2*A101220(1,2,n-1), for n > 0. a(0) = 0; a(1) = 1; a(n) = a(n-1) + a(n-2) + 2^(n-2) - 1, for n > 1. a(0) = 0; a(1) = 1; a(2) = 1; a(3) = 3; a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4), for n > 3.
O.g.f. = x(1-3x+3x^2)/((1-x)(1-2x)(1-x-x^2)).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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