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.

A142705 Numerator of 1/4 - 1/(2n)^2.

Original entry on oeis.org

0, 3, 2, 15, 6, 35, 12, 63, 20, 99, 30, 143, 42, 195, 56, 255, 72, 323, 90, 399, 110, 483, 132, 575, 156, 675, 182, 783, 210, 899, 240, 1023, 272, 1155, 306, 1295, 342, 1443, 380, 1599, 420, 1763, 462, 1935, 506, 2115, 552, 2303, 600, 2499, 650, 2703, 702
Offset: 1

Views

Author

Paul Curtz, Sep 24 2008

Keywords

Comments

Read modulo 10 (the last digits), a sequence with period length 10 results: 0, 3, 2, 5, 6, 5, 2, 3, 0, 9. Read modulo 9, a sequence with period length 18 results.
Denominators are in A154615.
a(n) is the numerator of (n-1)*(n+1)/4. - Altug Alkan, Apr 19 2018

Links

Crossrefs

Essentially the same as A070260. Cf. A078371 (second bisection of A061037), A142888 (first differences), A154615 (denominators), A225948.

Programs

  • Magma
    [-(3/4)*(-1)^n*n-(3/8)*(-1)^n*n^2+(5/8)*n^2+(5/4)*n: n in [0..60]]; // Vincenzo Librandi, Jul 02 2011
  • Mathematica
    Numerator[Table[(1/4)*(1 - 1/n^2), {n,1,50}]] (* G. C. Greubel, Jul 20 2017 *)
  • PARI
    for(n=1, 50, print1(numerator((1/4)*(1 - 1/n^2)), ", ")) \\ G. C. Greubel, Jul 20 2017
  • PARI
    a(n) = if(n%2,(n^2-1)/4,n^2-1); \\ Altug Alkan, Apr 19 2018

Formula

a(n) = A061037(2*n).
a(n) = A070260(n-1), n>1.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
a(2^(n-1)) = a(1+A000225(n-1)) = 4^(n-1)-1 = A024036(n-1).
G.f.: x^2*(3+2x+6x^2-x^4)/(1-x^2)^3. - R. J. Mathar, Oct 24 2008
E.g.f.: 1 + (1/4)*((4*x^2 + x - 4)*cosh(x) + (x^2 + 4*x -1)*sinh(x)). - G. C. Greubel, Jul 20 2017
Sum_{n>=2} 1/a(n) = 3/2. - Amiram Eldar, Aug 11 2022

Extensions

Edited by R. J. Mathar, Oct 24 2008

A226008 a(0) = 0; for n>0, a(n) = denominator(1/4 - 4/n^2).

Original entry on oeis.org

0, 4, 4, 36, 1, 100, 36, 196, 16, 324, 100, 484, 9, 676, 196, 900, 64, 1156, 324, 1444, 25, 1764, 484, 2116, 144, 2500, 676, 2916, 49, 3364, 900, 3844, 256, 4356, 1156, 4900, 81, 5476, 1444, 6084, 400, 6724, 1764, 7396, 121, 8100
Offset: 0

Views

Author

Paul Curtz, May 22 2013

Keywords

Comments

Numerators are in A225948.
Repeated terms of A016826 are in the positions 1, 2, 3, 6, 5, 10, ... (A043547).

Examples

			a(0) = (-1+1)^2 = 0, a(1) = (-3+5)^2 = 4, a(2) = (-1+3)^2 = 4.

Crossrefs

Cf. A016754, A016802, A016826, A017090, A017138, A154615, A225948 (numerators).
Cf. A225975 (associated square roots).

Programs

  • Magma
    [0] cat [Denominator(1/4-4/n^2): n in [1..50]]; // Bruno Berselli, May 23 2013
  • Mathematica
    Join[{0},Table[Denominator[1/4 - 4/n^2], {n, 49}]] (* Alonso del Arte, May 22 2013 *)

Formula

a(n) = 3*a(n-8) -3*a(n-16) +a(n-24).
a(8n) = A016802(n), a(8n+4) = A016754(n).
a(4n) = A154615(n).
a(4n+1) = A017090(n).
a(4n+2) = a(2n+1) = A016826(n); a(2n) = A061038(n).
a(4n+3) = A017138(n).
From Bruno Berselli, May 23 2013: (Start)
G.f.: x*(4 +4*x +36*x^2 +x^3 +100*x^4 +36*x^5 +196*x^6 +16*x^7 +312*x^8 +88*x^9 +376*x^10 +6*x^11 +376*x^12 +88*x^13 +312*x^14 +16*x^15 +196*x^16 +36*x^17 +100*x^18 +x^19 +36*x^20 +4*x^21 +4*x^22)/(1-x^8)^3.
a(n) = n^2*(6*cos(3*Pi*n/4)+6*cos(Pi*n/4)-54*cos(Pi*n/2)-219*(-1)^n+293)/128.
a(n+9) = a(n+1)*((n+9)/(n+1))^2. (End)
Sum_{n>=1} 1/a(n) = 19*Pi^2/96. - Amiram Eldar, Aug 14 2022

Extensions

Edited by Bruno Berselli, May 23 2013

A168068 Array T(n,k) read by antidiagonals: T(n,2k+1) = 2k+1. T(n,2k) = 2^n*k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 2, 0, 1, 8, 3, 4, 5, 0, 1, 16, 3, 8, 5, 3, 0, 1, 32, 3, 16, 5, 6, 7, 0, 1, 64, 3, 32, 5, 12, 7, 4, 0, 1, 128, 3, 64, 5, 24, 7, 8, 9, 0, 1, 256, 3, 128, 5, 48, 7, 16, 9, 5, 0, 1, 512, 3, 256, 5, 96, 7, 32, 9, 10, 11, 0, 1, 1024, 3, 512, 5, 192, 7, 64, 9, 20, 11, 6, 0, 1, 2048, 3, 1024, 5
Offset: 0

Views

Author

Paul Curtz, Nov 18 2009

Keywords

Comments

The array is constructed multiplying the even-indexed A026741(k) by 2^n, and keeping the odd-indexed A026471(k) as they are.
Connections to the hydrogen spectrum: The squares of the second row are T(1,k)^2 = A001477(k)^2 = A000290(k) which are the denominators of the Lyman lines (see A171522). The squares of the row T(2,k) are in A154615, denominators of the Balmer series. Row T(3,k) is related to A106833 and A061038.

Examples

			The array starts in row n=0 with columns k>=0 as:
0,1,1,3,2,5,3,7,4, A026741
0,1,2,3,4,5,6,7,8, A001477
0,1,4,3,8,5,12,7,16, A022998
0,1,8,3,16,5,24,7,32, A144433
0,1,16,3,32,5,48,7,64,
0,1,32,3,64,5,96,7,128,

Programs

  • Maple
    A168068 := proc(n,k) if type(k,'odd') then k; else 2^(n-1)*k ; end if; end proc: # R. J. Mathar, Jan 22 2011

A226044 Period of length 8: 1, 64, 16, 64, 4, 64, 16, 64.

Original entry on oeis.org

1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64
Offset: 0

Views

Author

Paul Curtz, May 24 2013

Keywords

Comments

A002378(n)/A016754(n) gives 0/1, 2/9, 6/25, 12/49, 20/81, 30/121, 42/169, 56/225,..., where A016754(n) = 4*A002378(n) + 1;
A142705(n)/A154615(n+1) gives 0/1, 3/16, 2/9, 15/64, 6/25, 35/144, 12/49, 63/256,..., where A142705(n) = 4*A154615(n+1) + A010685(n);
A061037(n)/A061038(n) gives 0/1, 5/36, 3/16, 21/100, 2/9, 45/196, 15/64, 77/324,..., where A061038(n) = 4*A061037(n) + A177499(n);
A225948(n)/A226008(n) gives 0/1, 9/100, 5/36, 33/196, 3/16, 65/324, 21/100, 105/484,..., where A226008(n) = 4*A225948(n) + a(n).
See also the triangle in Example lines.

Examples

			Triangle in which the terms of each line are repeated:
A000012: 1,   ...
A010685: 1,   4,  ...
A177499: 1,  16,  4,  16,  ...
A226044: 1,  64, 16,  64,  4,  64, 16,  64, ...
         1, 256, 64, 256, 16, 256, 64, 256, 4, 256, 64, 256, 16, 256, 64, 256, ...

Links

Crossrefs

Cf. A010685, A177499, A205383, A225948.

Programs

Formula

a(n) = A205383(n+7)^2.
G.f.: (1+64*x+16*x^2+64*x^3+4*x^4+64*x^5+16*x^6+64*x^7)/((1-x)*(1+x)*(1+x^2)*(1+x^4)). [Bruno Berselli, May 25 2013]

A168251 a(n) = n^2 if n is odd, n^2*2^(n-2) if n is even.

Original entry on oeis.org

0, 1, 4, 9, 64, 25, 576, 49, 4096, 81, 25600, 121, 147456, 169, 802816, 225, 4194304, 289, 21233664, 361, 104857600, 441, 507510784, 529, 2415919104, 625, 11341398016, 729, 52613349376, 841, 241591910400, 961, 1099511627776, 1089, 4964982194176, 1225
Offset: 0

Views

Author

Paul Curtz, Nov 21 2009

Keywords

Comments

This is the main diagonal of the following array defined by T(n,2k+1) = A168077(k) for odd column indices and T(n,2k) = A168077(2k)*2^n for even column indices:
0, 1, 1, 9, 4, 25, ... A168077
0, 1, 2, 9, 8, 25, ... A129194
0, 1, 4, 9, 16,25, ... A000290
0, 1, 8, 9, 32,25, ...
0, 1, 16,9, 64,25, ... A154615

Links

Programs

  • Magma
    [(n^2)*2^((n-2)*(1+(-1)^n) div 2): n in [0..40]]; // Vincenzo Librandi, Jul 17 2016
  • Maple
    A168251 := proc(n)
        if type(n,'even') then
                n^2*2^n/4 ;
        else
                n^2 ;
        end if;
end proc: # R. J. Mathar, Sep 20 2011
  • Mathematica
    Table[(n^2)*2^((n - 2)*(1 + (-1)^n)/2), {n, 0, 50}] (* G. C. Greubel, Jul 16 2016 *)
Table[If[OddQ[n],n^2,n^2 2^(n-2)],{n,0,50}] (* or *) LinearRecurrence[{0,15,0,-87,0,245,0,-348,0,240,0,-64},{0,1,4,9,64,25,576,49,4096,81,25600,121},41] (* Harvey P. Dale, May 14 2022 *)

Formula

a(2n) = A128782(n). a(2n+1) = A016754(n).
a(n) = +15*a(n-2) -87*a(n-4) +245*a(n-6) -348*a(n-8) +240*a(n-10) - 64*a(n-12).
G.f.: x*(1 + 4*x - 6*x^2 + 4*x^3 - 23*x^4 - 36*x^5 + 212*x^6 + 44*x^7 - 336*x^8 - 16*x^9 - 64*x^10) / ( (1-x)^3*(2*x+1)^3*(1-2*x)^3*(1+x)^3 ). - R. J. Mathar, Sep 20 2011
a(n) = (n^2)*2^((n-2)*(1+(-1)^n)/2). - Luce ETIENNE, Feb 03 2015
Showing 1-5 of 5 results.

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