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 1-3 of 3 results.

A163403 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152
Offset: 1

Views

Author

Klaus Brockhaus, Jul 26 2009

Keywords

Comments

a(n+1) is the number of palindromic words of length n using a two-letter alphabet. - Michael Somos, Mar 20 2011

Examples

			x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + 16*x^8 + 16*x^9 + 32*x^10 + ...

Links

Crossrefs

Equals A016116 without initial 1. Unsigned version of A152166.
Partial sums are in A136252.
Binomial transform is A078057, second binomial transform is A007070, third binomial transform is A102285, fourth binomial transform is A163350, fifth binomial transform is A163346.
Cf. A000079 (powers of 2), A009116, A009545, A051032.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

Programs

  • Magma
    [ n le 2 select n else 2*Self(n-2): n in [1..43] ];
  • Mathematica
    LinearRecurrence[{0, 2}, {1, 2}, 50] (* Paolo Xausa, Feb 02 2024 *)
  • PARI
    {a(n) = if( n<1, 0, 2^(n\2))} /* Michael Somos, Mar 20 2011 */
  • Sage
    def A163403():
    x, y = 1, 1
    while True:
        yield x
        x, y = x + y, x - y
a = A163403(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

Formula

a(n) = 2^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 2*x)/(1 - 2*x^2).
a(n) = A051032(n) - 1.
G.f.: x / (1 - 2*x / (1 + x / (1 + x))) = x * (1 + 2*x / (1 - x / (1 - x / (1 + 2*x)))). - Michael Somos, Jan 03 2013
From R. J. Mathar, Aug 06 2009: (Start)
a(n) = A131572(n).
a(n) = A060546(n-1), n > 1. (End)
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = |A009116(n-1)| + |A009545(n-1)|. - Bruno Berselli, May 30 2011
E.g.f.: cosh(sqrt(2)*x) + sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, Feb 05 2023

A102285 G.f. (1-x)/(7*x^2-6*x+1).

Original entry on oeis.org

1, 5, 23, 103, 457, 2021, 8927, 39415, 174001, 768101, 3390599, 14966887, 66067129, 291634565, 1287337487, 5682582967, 25084135393, 110726731589, 488771441783, 2157541529575, 9523849084969, 42040303802789
Offset: 0

Views

Author

Creighton Dement, Feb 19 2005

Keywords

Comments

A floretion-generated sequence relating to the second binomial transform of Pell numbers A000129.
Floretion Algebra Multiplication Program, FAMP Code: (a(n)) = jesforseq[ + .5'i + .5i' + 2'jj' + .5'ij' + .5'ji' ]; A000004 = vesforseq.

Links

Crossrefs

Cf. A086351, A027649, A007070 (inverse binomial transform), A081179, A163350 (binomial transform).

Programs

  • Magma
    [Floor(((1+Sqrt(2))*(3+Sqrt(2))^n+(1-Sqrt(2))*(3-Sqrt(2))^n)/2): n in [0..30]]; // Vincenzo Librandi, Oct 12 2011
  • Mathematica
    CoefficientList[Series[(1-x)/(7x^2-6x+1),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{1,5},30] (* Harvey P. Dale, Dec 10 2017 *)

Formula

a(n) = A086351(n+1) - 3*A086351(n) (FAMP result); Inversion gives A027649 (SuperSeeker result); Inverse binomial transform of A007070 (SuperSeeker result);
From Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009: (Start)
a(n) = ((1+sqrt(2))*(3+sqrt(2))^n + (1-sqrt(2))*(3-sqrt(2))^n)/2 offset 0.
Third binomial transform of 1,2,2,4,4. (End)
a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0)=1, a(1)=5. - Philippe Deléham, Sep 19 2009
a(n) = A081179(n) + A086351(n). - Joseph M. Shunia, Sep 09 2019
a(n) = A081179(n+1)-A081179(n). - R. J. Mathar, Sep 11 2019

A163346 a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 47, 309, 2009, 12983, 83623, 537621, 3452881, 22163527, 142219007, 912428949, 5853252329, 37546657463, 240841771063, 1544844588981, 9909085155361, 63559426007047, 407685301497167, 2614986216809589, 16773100233661049, 107586319349989943
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

Keywords

Comments

Binomial transform of A163350. Fifth binomial transform of A163403.

Links

Crossrefs

Cf. A163350, A163403.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+r)*(5+r)^n+(1-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009
  • Mathematica
    CoefficientList[Series[(1 - 3 x)/(1 - 10 x + 23 x^2), {x, 0, 21}], x] (* Michael De Vlieger, Jun 30 2016 *)
LinearRecurrence[{10,-23}, {1, 7}, 50] (* G. C. Greubel, Dec 19 2016 *)
  • PARI
    Vec((1-3*x)/(1-10*x+23*x^2) + O(x^99)) \\ Altug Alkan, Jul 05 2016

Formula

a(n) = 10*a(n-1)-23*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
a(n) = ((1+sqrt(2))*(5+sqrt(2))^n + (1-sqrt(2))*(5-sqrt(2))^n)/2.
G.f.: (1-3*x)/(1-10*x+23*x^2).
E.g.f.: (sqrt(2)*sinh(sqrt(2)*x) + cosh(sqrt(2)*x))*exp(5*x). - Ilya Gutkovskiy, Jun 30 2016

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009
New name from G. C. Greubel, Dec 19 2016
Showing 1-3 of 3 results.

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

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