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.

Previous Showing 11-20 of 20 results.

A199301 a(n) = (2n+1)*8^n.

Original entry on oeis.org

1, 24, 320, 3584, 36864, 360448, 3407872, 31457280, 285212672, 2550136832, 22548578304, 197568495616, 1717986918400, 14843406974976, 127543348822016, 1090715534753792, 9288674231451648, 78812993478983680, 666532744850833408, 5620492334958379008, 47269781688880726016
Offset: 0

Views

Author

Philippe Deléham, Nov 04 2011

Keywords

Links

Crossrefs

Cf. A001018 (Powers of 8), A005408 (2n+1).
Cf. A014480, A058962, A124647, A155988, A171220, A199299, A199300.

Programs

Formula

a(n) = 16*a(n-1)-64*a(n-2).
G.f.: (1+8*x)/(1-8*x)^2.
a(n) = 8*(a(n-1)+2^(3*n-2)). - Vincenzo Librandi, Nov 05 2011
a(n) = A005408(n) * A001018(n). - Wesley Ivan Hurt, Oct 30 2014
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(8)*arccoth(sqrt(8)).
Sum_{n>=0} (-1)^n/a(n) = sqrt(8)*arccot(sqrt(8)). (End)
E.g.f.: exp(8*x)*(1 + 16*x). - Stefano Spezia, May 09 2023

Extensions

a(18) corrected by Vincenzo Librandi, Nov 05 2011

A157327 Egyptian fraction expansion for Pi/4 = arctan(1/2) + arctan(1/3) (Hutton 1776).

Original entry on oeis.org

2, 3, -24, -81, 160, 1215, -896, -15309, 4608, 177147, -22528, -1948617, 106496, 20726199, -491520, -215233605, 2228224, 2195382771, -9961472, -22082967873, 44040192, 219667417263, -192937984, -2165293113021, 838860800
Offset: 0

Views

Author

Jaume Oliver Lafont, Feb 27 2009

Keywords

Comments

Sum_{n>=0} 1/a(n) = Pi/4.

Links

Crossrefs

Cf. A157142, A155988, A058962.

Programs

  • Mathematica
    CoefficientList[Series[2 (1 - 4 x^2)/(1 + 4 x^2)^2 + 3 x (1 - 9 x^2)/(1 + 9 x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 12 2012 *)

Formula

G.f.: 2*(1-4*x^2)/(1+4*x^2)^2 + 3*x*(1-9*x^2)/(1+9*x^2)^2.

A165283 a(n) = (2*n + 1)*16^n.

Original entry on oeis.org

1, 48, 1280, 28672, 589824, 11534336, 218103808, 4026531840, 73014444032, 1305670057984, 23089744183296, 404620279021568, 7036874417766400, 121597189939003392, 2089670227099910144, 35740566642812256256, 608742554432415203328, 10330176681277348904960
Offset: 0

Views

Author

Jaume Oliver Lafont, Sep 13 2009

Keywords

Links

Crossrefs

Cf. A058962 ((2n+1)4^n), A155988 ((2n+1)9^n).

Programs

  • Magma
    [(2*n+1)*16^n: n in [0..20]]; // Vincenzo Librandi, Jun 08 2011
  • PARI
    a(n)=(2*n+1)*16^n

Formula

G.f.: (1+16*x)/(1-16*x)^2.
Sum_{n>=0} 1/a(n) = 2*log(5/3).
Sum_{n>=0} (-1)^n/a(n) = 4 * arctan(1/4). - Amiram Eldar, Jul 12 2020
E.g.f.: exp(16*x)*(1 + 32*x). - Stefano Spezia, May 09 2023

A166725 a(n) = (2*n+1)*25^n.

Original entry on oeis.org

1, 75, 3125, 109375, 3515625, 107421875, 3173828125, 91552734375, 2593994140625, 72479248046875, 2002716064453125, 54836273193359375, 1490116119384765625, 40233135223388671875, 1080334186553955078125, 28870999813079833984375, 768341124057769775390625
Offset: 0

Views

Author

Jaume Oliver Lafont, Oct 20 2009

Keywords

Links

Crossrefs

Cf. A058962 ((2n+1)*4^n), A155988 ((2n+1)*9^n), A016578 (log(3/2)).

Programs

  • Magma
    [(2*n+1)*25^n: n in [0..20]]; // Vincenzo Librandi, Jun 08 2011
  • Mathematica
    Table[5^(2*n) *(2*n + 1), {n,0,10}] (* G. C. Greubel, May 24 2016 *)
LinearRecurrence[{50,-625},{1,75},30] (* Harvey P. Dale, Mar 02 2018 *)
  • PARI
    a(n)=(2*n+1)*25^n

Formula

G.f.: (1+25*x)/(1-25*x)^2.
Sum_{k>=0} 1/a(k) = (5/2)*log(3/2).
E.g.f.: (50*x + 1)*exp(25*x). - G. C. Greubel, May 24 2016
Sum_{n>=0} (-1)^n/a(n) = 5*arctan(1/5). - Amiram Eldar, Feb 26 2022

A157024 a(0)=1, a(n) = (3n-1)*3n*(3n+1)/2 for n>0.

Original entry on oeis.org

1, 12, 105, 360, 858, 1680, 2907, 4620, 6900, 9828, 13485, 17952, 23310, 29640, 37023, 45540, 55272, 66300, 78705, 92568, 107970, 124992, 143715, 164220, 186588, 210900, 237237, 265680, 296310, 329208, 364455, 402132, 442320, 485100, 530553, 578760, 629802
Offset: 0

Views

Author

Jaume Oliver Lafont, Feb 21 2009

Keywords

Crossrefs

Cf. A002391, A027480, A058962, A097321, A154920.

Programs

  • Mathematica
    nxt[{n_,a_}]:={n+1,((3n)(3n-1)(3n+1))/2}; NestList[nxt,{0,1},40][[All,2]]/.(0->Nothing) (* Harvey P. Dale, Sep 24 2016 *)

Formula

Sum_{n>=0} 1/a(n) = log(3).
G.f.: (1+8x+63x^2+8x^3+x^4)/(1-x)^4.
a(n) = A027480(3n-1), n>0. - R. J. Mathar, Apr 07 2009
Sum_{n>=0} (-1)^n/a(n) = 4*log(2)/3. - Amiram Eldar, Feb 27 2022

A236538 Triangle read by rows: T(n,k) = (n+1)*2^(n-2)+(k-1)*2^(n-1) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 28, 36, 44, 48, 64, 80, 96, 112, 112, 144, 176, 208, 240, 272, 256, 320, 384, 448, 512, 576, 640, 576, 704, 832, 960, 1088, 1216, 1344, 1472, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 2816, 3328, 3840, 4352, 4864, 5376
Offset: 1

Views

Author

Fedor Igumnov, Jan 28 2014

Keywords

Comments

1, 9, 45, 161, 497, 1409, ... is the sequence of perimeters (sum of border elements) of the triangle.
1, 5, 80, 3520, 394240, 107233280, 68629299200, ... is the sequence of determinants of the triangle.
Only the first three terms are odd.

Examples

			Triangle begins:
================================================
\k |    1     2     3     4     5     6     7
n\ |
================================================
1  |    1;
2  |    3,    5;
3  |    8,   12,   16;
4  |   20,   28,   36,   44;
5  |   48,   64,   80,   96,  112;
6  |  112,  144,  176,  208,  240,  272;
7  |  256,  320,  384,  448,  512,  576,  640;
...

Links

Crossrefs

Cf. A001792 (column 1), A053220 (right border). Also:
A014477, row sums;
A036826, partial sums;
A058962, central elements in odd rows;
A045623, second column;
A045891, third column;
A034007, fourth column;
A167667, subdiagonal;
A130129, second subdiagonal.

Programs

  • C
    int a(int n, int k) {return (n+1)*pow(2,n-2)+(k-1)*pow(2,n-1);}
  • Magma
    /* As triangle: */ [[(n+1)*2^(n-2)+(k-1)*2^(n-1): k in [1..n]]: n in [1..10]]; // Bruno Berselli, Jan 28 2014
  • Mathematica
    t[n_, k_] := (n + 1)*2^(n - 2) + (k - 1)*2^(n - 1); Table[t[n, k], {n, 10}, {k, n}] // Flatten (* Bruno Berselli, Jan 28 2014 *)

Formula

T(n,k) = T(n-1,k) + T(n-1,k+1).
Sum_{k=1..n} T(n,k) = n^2*2^(n-1) = A014477(n-1).

Extensions

More terms from Bruno Berselli, Jan 28 2014

A362885 Array read by ascending antidiagonals: A(n, k) = (1 + 2*n)*k^n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 5, 6, 1, 0, 7, 20, 9, 1, 0, 9, 56, 45, 12, 1, 0, 11, 144, 189, 80, 15, 1, 0, 13, 352, 729, 448, 125, 18, 1, 0, 15, 832, 2673, 2304, 875, 180, 21, 1, 0, 17, 1920, 9477, 11264, 5625, 1512, 245, 24, 1, 0, 19, 4352, 32805, 53248, 34375, 11664, 2401, 320, 27, 1
Offset: 0

Views

Author

Stefano Spezia, May 08 2023

Keywords

Examples

			The array begins:
    1,  1,   1,    1,     1,     1, ...
    0,  3,   6,    9,    12,    15, ...
    0,  5,  20,   45,    80,   125, ...
    0,  7,  56,  189,   448,   875, ...
    0,  9, 144,  729,  2304,  5625, ...
    0, 11, 352, 2673, 11264, 34375, ...
    ...

Crossrefs

Cf. A000007 (k=0), A000012 (n=0), A004248, A005408 (k=1), A008585 (n=1), A014480 (k=2), A033429 (n=2), A058962 (k=4), A124647 (k=3), A155988 (k=9), A171220 (k=5), A176043, A199299 (k=6), A199300 (k=7), A199301 (k=8), A244727 (n=3), A362886 (antidiagonal sums).

Programs

  • Mathematica
    A[n_,k_]:=(1+2n)k^n; Join[{1}, Table[A[n-k,k],{n,10},{k,0,n}]]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1+k*x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+2k*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

A(n, k) = A005408(n)*A004248(n, k).
O.g.f. of column k: (1 + k*x)/(1 - k*x)^2.
E.g.f. of column k: exp(k*x)*(1 + 2*k*x).
A(n, n) = A176043(n+1).

A157718 Greedy Egyptian fraction expansion of log(3).

Original entry on oeis.org

1, 11, 130, 91827, 42593758221, 2068726045016880942060, 20697114911379630588051784011292634933847536, 832769470129253476302780470023395858447487389073547955500158020204885523374048803963217
Offset: 0

Views

Author

Jaume Oliver Lafont, Mar 04 2009

Keywords

Examples

			log(3) = Sum_{n>=0} 1/a(n) = 1/1 + 1/11 + 1/130 + 1/91827 + 1/42593758221 + ...

Links

Crossrefs

Cf. A058962, A154920, A157024, A002391, A118324.

Programs

  • PARI
    x=log(3); for (k=1, 8, d=ceil(1/x); x=x-1/d; print(d,","))

A266491 a(n) = n*A130658(n).

Original entry on oeis.org

0, 1, 4, 6, 4, 5, 12, 14, 8, 9, 20, 22, 12, 13, 28, 30, 16, 17, 36, 38, 20, 21, 44, 46, 24, 25, 52, 54, 28, 29, 60, 62, 32, 33, 68, 70, 36, 37, 76, 78, 40, 41, 84, 86, 44, 45, 92, 94, 48, 49, 100, 102, 52, 53, 108, 110, 56, 57, 116, 118, 60, 61, 124, 126, 64
Offset: 0

Views

Author

Paul Curtz, Dec 30 2015

Keywords

Comments

Successive differences:
r(0): 0, 1, 4, 6, 4, 5, 12, 14, ...
r(1): 1, 3, 2, -2, 1, 7, 2, -6, ...
r(2): 2, -1, -4, 3, 6, -5, -8, 7, ... (see A103889)
r(3): -3, -3, 7, 3, -11, -3, 15, 3, ...
r(4): 0, 10, -4, -14, 8, 18, -12, -22, ...
r(5): 10, -14, -10, 22, 10, -30, -10, 38, ...
r(6): -24, 4, 32, -12, -40, 20, 48, -28, ...
r(7): 28, 28, -44, -28, 60, 28, -76, -28, ...
r(8): 0, -72, 16, 88, -32, -104, 48, 120, ...
r(9): -72, 88, 72, -120, -72, 152, 72, -184, ...
r(10): 160, -16, -192, 48, 224, -80, -256, 112, ...
etc.
Let b(n) = 1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 16, 16, ..., with n>=0, which is formed from the terms of A011782 repeated twice.
Conjecture: all terms of the row r(i) are divisible by b(i).
Conjecture: the terms of the first column divided by b(n) provide 0, 1, 2, -3, 0, 5, -6, 7, 0, -9, 10, -11, ..., the absolute values of which are listed in A190621.

Crossrefs

Cf. A005408, A011782, A042963, A058962, A103889, A128135, A130658, A190621, A227380.

Programs

  • Magma
    [n*(3-(-1)^((n-1)*n div 2))/2: n in [0..70]]; // Vincenzo Librandi, Jan 08 2016
  • Mathematica
    Table[n (3 - (-1)^((n - 1) n/2))/2, {n, 0, 55}]
Table[n (Boole@ OddQ@ Floor[n/2] + 1), {n, 0, 55}] (* or *) Table[SeriesCoefficient[x (3/(1 - x)^2 + 2 x/(1 + x^2)^2 - (1 - x^2)/(1 + x^2)^2)/2, {x, 0, n}], {n, 0, 55}] (* Michael De Vlieger, Jan 04 2016 *)
  • PARI
    vector(60, n, n--; n*(3-(-1)^((n-1)*n/2))/2) \\ Altug Alkan, Jan 04 2016

Formula

a(n) = n*(3 - (-1)^((n-1)*n/2))/2.
a(n) = a(n-4) + 4*A130658(n) for n>3.
a(n) = 2*a(n-1) -3*a(n-2) +4*a(n-3) -3*(n-4) +2*a(n-5) -a(n-6) for n>5.
G.f.: x*(3/(1 - x)^2 + 2*x/(1 + x^2)^2 - (1 - x^2)/(1 + x^2)^2)/2. - Michael De Vlieger, Jan 04 2016

Extensions

Edited by Bruno Berselli, Jan 07 2016

A288443 a(n) = (2n + 1)*2^(2n + 1); numbers k such that v(k)*2^v(k) = k, where v(n) = A007814(n) is 2-adic valuation of n.

Original entry on oeis.org

2, 24, 160, 896, 4608, 22528, 106496, 491520, 2228224, 9961472, 44040192, 192937984, 838860800, 3623878656, 15569256448, 66571993088, 283467841536, 1202590842880, 5085241278464, 21440476741632, 90159953477632, 378231999954944, 1583296743997440, 6614661952700416, 27584547717644288, 114841790497947648
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Jun 24 2017

Keywords

Crossrefs

Cf. A007814, A058962, A098713, A286683.
Odd bisection of A036289.
Cf. A073000, A156057.

Programs

Formula

a(n) = (2n + 1)*2^(2n + 1).
a(n) = A036289(2n + 1).
a(n) = A098713(n) + 1.
a(n) = 2*A058962(n). - Joerg Arndt, Jun 25 2017
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=0} 1/a(n) = arctanh(1/2) = log(3)/2 (A156057).
Sum_{n>=0} (-1)^n/a(n) = arctan(1/2) (A073000). (End)
Previous Showing 11-20 of 20 results.

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