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 1-7 of 7 results.

A081655 2*5^n-1.

Original entry on oeis.org

1, 9, 49, 249, 1249, 6249, 31249, 156249, 781249, 3906249, 19531249, 97656249, 488281249, 2441406249, 12207031249, 61035156249, 305175781249, 1525878906249, 7629394531249, 38146972656249, 190734863281249
Offset: 0

Views

Author

Paul Barry, Mar 26 2003

Keywords

Comments

Binomial transform of A081654. Inverse binomial transform of A081656.

Links

Programs

Formula

G.f.: (1+3*x)/((1-5*x)*(1-x)).
E.g.f. 2*exp(5*x)-exp(x).
a(n) = 5*a(n-1)+4, with a(0)=1. - Vincenzo Librandi, Aug 01 2010

A303537 Expansion of ((1 + 4*x)/(1 - 4*x))^(1/4).

Original entry on oeis.org

1, 2, 2, 12, 22, 124, 276, 1496, 3686, 19436, 51068, 263720, 724860, 3681880, 10466920, 52450992, 153093254, 758495564, 2261603564, 11096526344, 33676743956, 163842737928, 504738342808, 2437418983888, 7605947276508, 36487283224952, 115140704639576
Offset: 0

Views

Author

Seiichi Manyama, Apr 25 2018

Keywords

Comments

Let ((1 + k*x)/(1 - k*x))^(1/k) = a(0) + a(1)*x + a(2)*x^2 + ...
Then n*a(n) = 2*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Links

Crossrefs

Cf. A063886, A081654, A151403, A303538.

Programs

  • Mathematica
    CoefficientList[Series[Surd[(1+4x)/(1-4x),4],{x,0,40}],x] (* Harvey P. Dale, Jul 25 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(1/4))

Formula

a(n) ~ 2^(2*n + 1/4) / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 26 2018
n*a(n) = 2*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.
G.f.: A(x)=F(x*G(x^2)), where F(x) is the g.f. for A063886, and G(x) is the g.f. for A151403. - Alexander Burstein, Nov 13 2023

A304941 Expansion of ((1 + 4*x)/(1 - 4*x))^(3/4).

Original entry on oeis.org

1, 6, 18, 68, 246, 948, 3572, 13896, 53286, 208452, 807132, 3169080, 12346300, 48602760, 190150440, 750018448, 2943363078, 11627329764, 45736940364, 180897649368, 712881236052, 2822389182104, 11138924119512, 44137230865392, 174405194802524, 691557285091176
Offset: 0

Views

Author

Seiichi Manyama, May 22 2018

Keywords

Comments

Let ((1 + k*x)/(1 - k*x))^(m/k) = a(0) + a(1)*x + a(2)*x^2 + ... then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Links

Crossrefs

((1 + 4*x)/(1 - 4*x))^(m/4): A303537 (m=1), A304940 (m=2), this sequence (m=3), A081654 (m=4).

Programs

  • Magma
    [n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 8*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // G. C. Greubel, Jun 07 2023
  • Mathematica
    CoefficientList[Series[((1+4x)/(1-4x))^(3/4),{x,0,30}],x] (* Harvey P. Dale, Oct 24 2020 *)
  • PARI
    N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(3/4))
  • SageMath
    @CachedFunction
def a(n): # a = A304941
    if n<2: return 6^n
    else: return 2*(3*a(n-1) + 8*(n-2)*a(n-2))//n
[a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

n*a(n) = 6*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.
a(n) ~ 2^(2*n + 3/4) / (Gamma(3/4) * n^(1/4)). - Vaclav Kotesovec, May 28 2018

A081632 2*3^n-(-1)^n.

Original entry on oeis.org

1, 7, 17, 55, 161, 487, 1457, 4375, 13121, 39367, 118097, 354295, 1062881, 3188647, 9565937, 28697815, 86093441, 258280327, 774840977, 2324522935, 6973568801, 20920706407, 62762119217, 188286357655, 564859072961, 1694577218887
Offset: 0

Views

Author

Paul Barry, Mar 26 2003

Keywords

Comments

Binomial transform of A081631. Inverse binomial transform of A081654.

Links

Programs

  • Magma
    [2*3^n-(-1)^n: n in [0..40]]; // Vincenzo Librandi, Aug 10 2013
  • Mathematica
    Total/@Partition[Riffle[2*3^Range[0,30],{-1,1}],2] (* or *) LinearRecurrence[{2,3},{1,7},30] (* Harvey P. Dale, May 24 2011 *)
CoefficientList[Series[(1 + 5 x) / ((1 - 3 x) (1 + x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)

Formula

G.f.: (1+5*x)/((1-3*x)(1+x)).
E.g.f. 2*exp(3*x)-exp(-x).
a(0)=1, a(1)=7, a(n)=2*a(n-1)+3*a(n-2) [ Harvey P. Dale, May 24 2011]

A187093 a(0)=0, a(1)=a(2)=1; thereafter, a(n+1) = n^2 - a(n-1).

Original entry on oeis.org

0, 1, 1, 3, 8, 13, 17, 23, 32, 41, 49, 59, 72, 85, 97, 111, 128, 145, 161, 179, 200, 221, 241, 263, 288, 313, 337, 363, 392, 421, 449, 479, 512, 545, 577, 611, 648, 685, 721, 759, 800, 841, 881, 923, 968, 1013, 1057, 1103, 1152, 1201, 1249, 1299, 1352, 1405, 1457
Offset: 0

Views

Author

Benjamin Coinsin, Mar 04 2011

Keywords

Comments

The original definition was equivalent to: Let S(n) = sum_{i=0..n} a(i), then n^2+a(n)-S(n+1) = S(n-2). This in turn simplifies to the present definition.

Links

Crossrefs

Cf. A194274, A081654.

Programs

  • Maple
    A000034 := proc(n) op(1+(n mod 2),[1,2]) ; end proc:
A187093 := proc(n) (n^2-1+(-1)^floor(n/2)*A000034(n))/2 ;end proc: # R. J. Mathar
  • Mathematica
    LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 1, 3, 8}, 60] (* Jean-François Alcover, Mar 30 2020 *)
Join[{0},RecurrenceTable[{a[1]==a[2]==1,a[n+1]==n^2-a[n-1]},a,{n,60}]] (* Harvey P. Dale, Jan 05 2023 *)
  • PARI
    a(n) = (n^2-1+(-1)^(n\2)*(1 + (n % 2)))/2; \\ Michel Marcus, Sep 11 2016
  • Python
    print(0, end=',')       # a(-1)=0
prpr = prev = 1         # a(0)=a(1)=1
for n in range(2, 77):
    print(prpr, end=',')
    curr = n*n - prpr   # a(n) = n^2 - a(n-2)
    prpr = prev
    prev = curr
# from Alex Ratushnyak, Aug 05 2012

Formula

a(n) = (n^2 - 1 + (-1)^floor(n/2) * A000034(n))/2.
G.f.: x*(-1+2*x+x^3-4*x^2) / ( (x^2+1)*(x-1)^3 ).
a(2^(n+1)) = A081654(n). - Anton Zakharov, Sep 13 2016

Extensions

Edited by N. J. A. Sloane, Mar 09 2011
More terms from Alex Ratushnyak, Aug 05 2012

A304940 Expansion of ((1 + 4*x)/(1 - 4*x))^(1/2).

Original entry on oeis.org

1, 4, 8, 32, 96, 384, 1280, 5120, 17920, 71680, 258048, 1032192, 3784704, 15138816, 56229888, 224919552, 843448320, 3373793280, 12745441280, 50981765120, 193730707456, 774922829824, 2958796259328, 11835185037312, 45368209309696, 181472837238784
Offset: 0

Views

Author

Seiichi Manyama, May 22 2018

Keywords

Comments

Let ((1 + k*x)/(1 - k*x))^(m/k) = a(0) + a(1)*x + a(2)*x^2 + ...
Then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Links

Crossrefs

((1 + 4*x)/(1 - 4*x))^(m/4): A303537 (m=1), this sequence (m=2), A304941 (m=3), A081654 (m=4).
Cf. A063886.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(1/2))

Formula

n*a(n) = 4*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.
a(n) = 2^n * A063886(n).

A236967 Expansion of (1+3*x)^2/(1-3*x)^2.

Original entry on oeis.org

1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300
Offset: 0

Views

Author

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 22 2014

Keywords

Links

Crossrefs

Cf. Expansion of (1 + k*x)^m/(1 - k*x)^m where the values of k,m are:
......|..m = 1..|..m = 2..|..m = 3..|..m = 4..|..m = 5..|..m = 6..|
k = 1 | A040000 | A008574 | A005899 | A008412 | A008413 | A008414 |
k = 2 | A151821 | A241204 | | | | |
k = 3 | A099856 | A236967 | | | | |
k = 4 | A081654 | | | | | |
-------------------------------------------------------------------

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)^2/(1-3*x)^2));

Formula

For n >= 1, a(n) = 4*n*3^n. - Robert Israel, May 08 2014

Extensions

Edited by Wolfdieter Lang, May 07 2014
Showing 1-7 of 7 results.

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

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