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.

A061981 a(n) = 3^n - 2*n - 1.

Original entry on oeis.org

0, 0, 4, 20, 72, 232, 716, 2172, 6544, 19664, 59028, 177124, 531416, 1594296, 4782940, 14348876, 43046688, 129140128, 387420452, 1162261428, 3486784360, 10460353160, 31381059564, 94143178780, 282429536432, 847288609392
Offset: 0

Views

Author

Henry Bottomley, May 24 2001

Keywords

Links

Crossrefs

Column of A061980.
Cf. A000340, A186948.

Programs

  • Mathematica
    Table[3^n-2n-1,{n,0,30}] (* or *) LinearRecurrence[{5,-7,3},{0,0,4},30] (* Harvey P. Dale, Mar 30 2018 *)
  • PARI
    a(n) = { 3^n - 2*n - 1 } \\ Harry J. Smith, Jul 29 2009
  • SageMath
    [3^n -(2*n+1) for n in (0..40)] # G. C. Greubel, Jun 13 2022

Formula

From Bruno Berselli, Jan 31 2012: (Start)
G.f.: 4*x^2/((1-3*x)*(1-x)^2).
a(n) = A186948(n) - 1.
a(n+2) = 4*A000340(n). (End)
From Deisy J. Camacho, Feb 26 2021 (Start)
a(n) = Sum_{j=2..n} Sum_{i=0..j} n!/((j-i)!*i!*(n-j)!).
a(n) = 4 + 4*a(n-1) - 3*a(n-2). (End)
E.g.f.: exp(3*x) - (2*x + 1)*exp(x). - G. C. Greubel, Jun 13 2022

A186947 a(n) = 4^n - n*2^n.

Original entry on oeis.org

1, 2, 8, 40, 192, 864, 3712, 15488, 63488, 257536, 1038336, 4171776, 16728064, 67002368, 268206080, 1073250304, 4293918720, 17177640960, 68714758144, 274867945472, 1099490656256, 4398002470912, 17592093769728, 70368551239680, 281474574057472, 1125899067981824
Offset: 0

Views

Author

Paul Barry, Mar 01 2011

Keywords

Comments

Binomial transform of A186948.
Second binomial transform of A186949.

Examples

			G.f. = 1 + 2*x + 8*x^2 + 40*x^3 + 192*x^4 + 864*x^5 + 3712*x^6 + ... - _Michael Somos_, Jul 18 2018

Links

Crossrefs

Cf. A000325, A002064, A186948, A186949.

Programs

  • Magma
    [4^n - n*2^n: n in [0..30]]; // G. C. Greubel, Aug 14 2018
  • Mathematica
    Table[4^n-n 2^n,{n,0,30}] (* or *) LinearRecurrence[{8,-20,16},{1,2,8},30] (* Harvey P. Dale, Apr 23 2017 *)
  • PARI
    {a(n) = 2^n * (2^n - n)}; /* Michael Somos, Jul 18 2018 */

Formula

G.f.: (1 - 6*x + 12*x^2)/((1 - 2*x)^2*(1 - 4*x)).
a(n) = 4*a(n-1) + 2^n*(n-2), n >= 1. - Vincenzo Librandi, Mar 13 2011
a(n) = 2^n*A000325(n) = 4^n*A002064(-n) for all n in Z. - Michael Somos, Jul 18 2018
From Elmo R. Oliveira, Sep 15 2024: (Start)
E.g.f.: exp(2*x)*(exp(2*x) - 2*x).
a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3) for n > 2. (End)

A186949 a(n) = 2^n - 2*(binomial(1,n) - binomial(0,n)).

Original entry on oeis.org

1, 0, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824
Offset: 0

Views

Author

Paul Barry, Mar 01 2011

Keywords

Comments

Binomial transform is A186948.
Second binomial transform is A186947.
Inverse binomial transform is (-1)^n * A168277(n).
Essentially the same as A000079, A151821, A155559, A171449, and A171559.

Links

Programs

  • GAP
    Concatenation([1,0], List([2..30], n-> 2^n )); # G. C. Greubel, Dec 01 2019
  • Magma
    [n lt 2 select 1-n else 2^n: n in [0..30]]; // G. C. Greubel, Dec 01 2019
  • Maple
    seq( `if`(n<2, 1-n, 2^n), n=0..30); # G. C. Greubel, Dec 01 2019
  • Mathematica
    Table[If[n<2, 1-n, 2^n], {n, 0, 30}] (* G. C. Greubel, Dec 01 2019 *)
  • PARI
    vector(31, n, if(n<3, 2-n, 2^(n-1))) \\ G. C. Greubel, Dec 01 2019
  • Sage
    [1,0]+[2^n for n in (2..30)] # G. C. Greubel, Dec 01 2019

Formula

G.f.: (1 - 2*x + 4*x^2)/(1-2*x).
a(n) = Sum_{k=0..n} binomial(n,k)*(3^k - 2*k).
E.g.f.: exp(2*x) - 2*x. - G. C. Greubel, Dec 01 2019
Showing 1-3 of 3 results.

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