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.

Showing 1-5 of 5 results.

A004277 1 together with positive even numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Also number of non-attacking bishops on n X n board. - Koksal Karakus (karakusk(AT)hotmail.com), May 27 2002
Engel expansion of e^(1/2) (see A006784 for definition) [when offset by 1]. - Henry Bottomley, Dec 18 2000
Numbers n such that a 2n-group (i.e., a group of order 2n) has subgroup C_2. - Lekraj Beedassy, Oct 14 2004
Image of 1/(1-2x) under the mapping g(x)->g(x/(1+x^2)). - Paul Barry, Jan 16 2005
Position of n in A113322: A113322(a(n-1)) = n for n>0. - Reinhard Zumkeller, Oct 26 2005
Incrementally largest terms in the continued fraction for e. - Nick Hobson, Jan 11 2007
Conjecturally, the differences of two consecutive primes (without repetition). - Juri-Stepan Gerasimov, Nov 09 2009
Equals (1, 2, 2, 2, ...) convolved with (1, 0, 2, 0, 2, 0, 2, ...). - Gary W. Adamson, Mar 03 2010
a(n) is the number of 0-dimensional elements (vertices) in an n-cross polytope. - Patrick J. McNab, Jul 06 2015
Numbers k such that in the symmetric representation of sigma(k) there is no pair bars as its ends (Cf. A237593). - Omar E. Pol, Sep 28 2018
Also, the coordination sequence of the L-lattice (see A332419). - Sean A. Irvine, Jul 29 2020

Links

Crossrefs

Cf. A004275, A008486, A030978, A097134.
INVERT transformation yields A098182 without A098182(0). - R. J. Mathar, Sep 11 2008

Programs

Formula

G.f.: (1+x^2)/(1-x)^2. - Paul Barry, Feb 28 2003
Inverse binomial transform of Cullen numbers A002064. a(n)=2n+0^n. - Paul Barry, Jun 12 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1)*(-1)^k*2^(n-2k). - Paul Barry, Jan 16 2005
Equals binomial transform of [1, 1, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Jul 15 2008
E.g.f.: 1+x*sinh(x) (aerated sequence). - Paul Barry, Oct 11 2009
a(n) = 0^n + 2*n = A000007(n) + A005843(n). - Reinhard Zumkeller, Jan 11 2012

Extensions

Corrected by Charles R Greathouse IV, Mar 18 2010

A264080 a(n) = 6*F(n)*F(n+1) + (-1)^n, where F = A000045.

Original entry on oeis.org

1, 5, 13, 35, 91, 239, 625, 1637, 4285, 11219, 29371, 76895, 201313, 527045, 1379821, 3612419, 9457435, 24759887, 64822225, 169706789, 444298141, 1163187635, 3045264763, 7972606655, 20872555201, 54645058949, 143062621645, 374542805987, 980565796315
Offset: 0

Views

Author

Bruno Berselli, Nov 03 2015

Keywords

Comments

a(n) is prime for n = 1, 2, 5, 7, 14, 15, 29, 40, 49, 57, 70, 87, 105, 127, 175, 279, 362, 647, 727, ...

Links

Crossrefs

Cf. A000032, A000045; A001654, A002878.
Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n: A226205 (k=1); A236428 (k=2); A014742 (k=3); A061647 (k=4); A002878 (k=5).

Programs

  • Magma
    [6*Fibonacci(n)*Fibonacci(n+1)+(-1)^n: n in [0..30]];
  • Maple
    a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,5,13>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016
  • Mathematica
    Table[6 Fibonacci[n] Fibonacci[n + 1] + (-1)^n, {n, 0, 30}]
LinearRecurrence[{2,2,-1},{1,5,13},30] (* Harvey P. Dale, Jul 12 2019 *)
  • Maxima
    makelist(6*fib(n)*fib(n+1)+(-1)^n, n, 0, 30);
  • PARI
    for(n=0, 30, print1(6*fibonacci(n)*fibonacci(n+1)+(-1)^n", "));
  • PARI
    a(n) = round((2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ Colin Barker, Sep 28 2016
  • PARI
    Vec((1+3*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 28 2016
  • Sage
    [6*fibonacci(n)*fibonacci(n+1)+(-1)^n for n in (0..30)]

Formula

G.f.: (1+3*x+x^2) / ((1+x)*(1-3*x+x^2)). - Corrected by Colin Barker, Sep 28 2016
a(n) = -a(-n-1) = 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z.
a(n) = L(2*n+1) + F(n)*F(n+1) = A002878(n) + A001654(n). See similar identity for A061647.
a(n) = A001654(n+1) + 3*A001654(n) + A001654(n-1).
a(n) - a(n-1) = 2*A099016(n) with a(-1)=-1.
a(n) + a(n-1) = 2*A097134(n) for n>0.
Sum_{i>=0} 1/a(i) = 1.3232560865206157372628688449331...
a(n) = (2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Sep 28 2016
E.g.f.: (1/5)*exp(-x)*(-1 + 6*exp(5*x/2)*(cosh((sqrt(5)*x)/2) + sqrt(5)*sinh((sqrt(5)*x)/2))). - Stefano Spezia, Dec 09 2019

A097135 a(0) = 1; for n>0, a(n) = 3*Fibonacci(n).

Original entry on oeis.org

1, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, 699, 1131, 1830, 2961, 4791, 7752, 12543, 20295, 32838, 53133, 85971, 139104, 225075, 364179, 589254, 953433, 1542687, 2496120, 4038807, 6534927, 10573734, 17108661, 27682395, 44791056, 72473451, 117264507
Offset: 0

Views

Author

Paul Barry, Jul 26 2004

Keywords

Comments

Binomial transform is A097136.

Links

Crossrefs

Cf. A000032, A000045.
Essentially the same as A022086.

Programs

Formula

G.f. : (1+2*x-x^2)/(1-x-x^2).
a(n) = a(n-1)+a(n-2) for n>2.
a(2n) = A097134(n); a(2n+1) = 3*F(2n+1).

Extensions

Definition rewritten by N. J. A. Sloane, Jan 24 2010

A097133 a(n) = 3*Fibonacci(n) + (-1)^n.

Original entry on oeis.org

1, 2, 4, 5, 10, 14, 25, 38, 64, 101, 166, 266, 433, 698, 1132, 1829, 2962, 4790, 7753, 12542, 20296, 32837, 53134, 85970, 139105, 225074, 364180, 589253, 953434, 1542686, 2496121, 4038806, 6534928, 10573733, 17108662, 27682394, 44791057, 72473450, 117264508
Offset: 0

Views

Author

Paul Barry, Jul 26 2004

Keywords

Comments

Binomial transform is A097134.

Links

Crossrefs

Cf. A000045, A097134, A097136.

Programs

  • Haskell
    a097133 n = a097133_list !! n
a097133_list = 1 : 2 : 4 : zipWith (+)
               (map (* 2) $ tail a097133_list) a097133_list
-- Reinhard Zumkeller, Feb 24 2015
  • Mathematica
    CoefficientList[Series[(1+2x+2x^2)/((1+x)(1-x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,1},{1,2,4},40] (* Harvey P. Dale, May 07 2011 *)

Formula

G.f.: (1+2*x+2*x^2)/((1+x)*(1-x-x^2));
a(n) = 2*a(n-2)+a(n-3);
a(2*n) = 3*F(2*n)+1 = A097136(n).

A099256 Expansion of g.f. (3-x)*(1+3*x+x^2)/((1-x-x^2)*(1+x-x^2)).

Original entry on oeis.org

3, 8, 9, 23, 24, 61, 63, 160, 165, 419, 432, 1097, 1131, 2872, 2961, 7519, 7752, 19685, 20295, 51536, 53133, 134923, 139104, 353233, 364179, 924776, 953433, 2421095, 2496120, 6338509, 6534927, 16594432, 17108661, 43444787, 44791056, 113739929, 117264507, 297775000, 307002465, 779585071
Offset: 0

Views

Author

Creighton Dement, Oct 18 2004

Keywords

Comments

One of two sequences involving the Lucas/Fibonacci numbers. This sequence consists of pairs of numbers more or less close to each other with "jumps" in between pairs.
a(n+3) + a(n) - a(n+2) appears to be mysteriously connected with a(n+1).
Both this sequence and A099255 were created using "Floretion dynamical symmetries" (see link for further details).

Links

Crossrefs

Cf. A000045, A099255, A000032, A055273 (bisection), A097134 (bisection).

Programs

  • Mathematica
    LinearRecurrence[{0,3,0,-1},{3,8,9,23},40] (* Harvey P. Dale, Apr 22 2012 *)

Formula

a(2n+2) - a(2n+1) = Fibonacci(2n-1).
A099255(n)/2 - a(n)/2 = (-1)^n*A000032(n)
a(0) = 3, a(1) = 8, a(2) = 9, a(3) = 23, a(n+4) = 3a(n+2) - a(n).
a(2n) = A022086(2n+2), a(2n+1) = A022097(2n+2).
a(n) = A013655(n+2)-A061084(n+1).

Extensions

Definition corrected, extended. - R. J. Mathar, Nov 13 2008
Showing 1-5 of 5 results.

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

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