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.

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A194767 Denominator of the fourth increasing diagonal of the autosequence of second kind from (-1)^n / (n+1).

Original entry on oeis.org

2, 2, 12, 20, 10, 42, 56, 24, 90, 110, 44, 156, 182, 70, 240, 272, 102, 342, 380, 140, 462, 506, 184, 600, 650, 234, 756, 812, 290, 930, 992, 352, 1122, 1190, 420, 1332, 1406, 494, 1560, 1640, 574, 1806, 1892, 660, 2070, 2162, 752, 2352, 2450, 850, 2652, 2756, 954, 2970, 3080, 1064, 3306, 3422, 1180, 3660
Offset: 0

Views

Author

Paul Curtz, Sep 02 2011

Keywords

Comments

The autosequence of first kind from (-1)^n/(n+1) is A189733.
For the second kind (the second increasing diagonal is (-1)^n/(n+1), half of the main one):
2, 1, 0, -1/2, -1/3, 1/6, 1/2, 5/12,
-1, -1, -1/2, 1/6, 1/2, 1/3, -1/12, -7/20,
0, 1/2, 2/3, 1/3, -1/6, -5/12, -4/15, 1/12,
1/2, 1/6, -1/3, -1/2, -1/4, 3/20, 7/20, 13/60,
-1/3, -1/2, -1/6, 1/4, 2/5, 1/5, -2/15, -3/10,
-1/6, 1/3, 5/12, 3/20, -1/5, -1/3, -1/6, 5/42,
1/2, 1/12, -4/15, -7/20, -2/15, 1/6, 2/7, 1/7,
-5/12, -7/20, -1/12, 13/60, 3/10, 5/42, -1/7, -1/4.
Main diagonal: (period 2:repeat 2, -1)/A026741(n+1).
Second (increasing) diagonal: (-1)^n / (n+1).
Third (increasing) diagonal: (-1)^(n+1)*A026741(n) / A045896(n).
Fourth (increasing) diagonal: (-1)^(n+1)*A146535(n)/ a(n).

Links

Crossrefs

Cf. A001504 = 2*A060544, A049450 = 2*A000326, A045945 = 6*A005449.
Cf. A026741, A045896, A051176, A146535, A189733, A306368.

Programs

  • Mathematica
    c = Table[1/9 (7 n + 7 n^2 + 2 n Cos[2 n *Pi/3] + 2 n^2 Cos[2 n *Pi/3] + 2 Sqrt[3] n Sin[2 n *Pi/3] + 2 Sqrt[3] n^2 Sin[2 n *Pi/3]), {n, 1, 50}] (* Roger Bagula, Mar 25 2012 *)
a[n_] := (n+1) * Numerator[(n+2)/3]; Array[a, 60, 0] (* Amiram Eldar, Sep 17 2023 *)
LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{2,2,12,20,10,42,56,24,90},60] (* Harvey P. Dale, May 15 2025 *)

Formula

a(3*n) = (3*n+1)*(3*n+2), a(3*n+1) = (n+1)*(3*n+2), a(3*n+2) = 3*(n+1)*(3*n+4).
G.f.: 2*(1+x+6*x^2+7*x^3+2*x^4+3*x^5+x^6)/(1-x^3)^3. - Jean-François Alcover, Nov 11 2016
a(n+2) = 2 * A306368(n) for n >= 0. - Joerg Arndt, Aug 25 2023
a(n) = (n+1) * A051176(n+2) for n >= 0. - Paul Curtz, Sep 13 2023
Sum_{n>=0} 1/a(n) = 1 + log(3) - Pi/(3*sqrt(3)). - Amiram Eldar, Sep 17 2023

A194531 Numerator of row 4 in A051714(n) or row 3 in A176672(n).

Original entry on oeis.org

0, 1, 1, 2, 5, 5, 7, 28, 3, 15, 55, 22, 13, 91, 35, 40, 34, 51, 57, 190, 35, 77, 253, 92, 25, 325, 117, 126, 203, 145, 155, 496, 44, 187, 595, 210, 111, 703, 247, 260, 205, 287, 301, 946, 165, 345, 1081, 376, 98, 1225, 425
Offset: 0

Views

Author

Paul Curtz, Aug 28 2011

Keywords

Comments

Akiyama-Tanigawa algorithm from 1/n leads to Bernoulli A164555(n)/A027642(n):
1, 1/2, 1/3, 1/4,
1/2, 1/3, 1/4, 1/5,
1/6, 1/6, 3/20, 2/15, =A026741(n+1)/A045896(n+1),
0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495 =a(n)/b(n).

Crossrefs

Cf. A193220 (denominators).

Programs

  • Mathematica
    a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); Table[a[3, k], {k, 0, 50}] // Numerator (* Jean-François Alcover, Sep 19 2012 *)

A100652 Denominator of 1 - Sum_{i=1..n} |Bernoulli(i)|.

Original entry on oeis.org

1, 2, 3, 3, 10, 10, 105, 105, 70, 70, 1155, 1155, 1430, 1430, 2145, 2145, 24310, 24310, 4849845, 4849845, 58786, 58786, 2028117, 2028117, 965770, 965770, 1448655, 1448655, 28007330, 28007330, 100280245065, 100280245065, 66853496710, 66853496710, 100280245065
Offset: 1

Views

Author

N. J. A. Sloane, Dec 05 2004

Keywords

Comments

Contribution from Paul Curtz, Aug 07 2012 (Start):
Take a(0)=1. Then instead of the Akiyama-Tanigawa algorithm we create the extended (or prolonged) Akiyama-Tanigawa algorithm using A028310(n)=1,1,2,3,4,5,... instead of A000027(n)=1,2,3,4,5,.. .
Hence the array (A051714 with an additional column)
2, 1, 1/2, 1/3, 1/4,
1, 1/2, 1/3, 1/4, 1/5,
1/2, 1/6, 1/6, 3/20, 2/15, A026741(n+1)/A045896(n+1)
1/3, 0, 1/30, 1/20, 2/35, A194531(n)/A193220(n)
1/3, -1/30, -1/30, -3/140, -1/105. A051722(n)/A051723(n).
a(n) is the denominator of the (first) column before the Akiyama-Tanigawa algorithm leading to the second Bernoulli numbers A164555(n)/A027642(n). See A176672(n).
(End)

Examples

			1, 1/2, 1/3, 1/3, 3/10, 3/10, 29/105, 29/105, 17/70, 17/70, 193/1155, 193/1155, -123/1430, -123/1430, -2687/2145, -2687/2145, -202863/24310, -202863/24310, -307072861/4849845, ... = A100651/A100652.

Links

Programs

  • Mathematica
    Denominator[1-(Accumulate[Abs[BernoulliB[Range[0,40]]]])] (* Harvey P. Dale, Jan 28 2013 *)

A230262 Numerators of Akiyama-Tanigawa algorithm applied to harmonic numbers, written by antidiagonals.

Original entry on oeis.org

1, 3, -1, 11, -2, 1, 25, -3, 1, 0, 137, -4, 3, 1, -1, 49, -5, 2, 1, -1, 0, 363, -6, 5, 2, -3, -1, 1, 761, -7, 3, 5, -1, -1, 1, 0, 7129, -8, 7, 5, 0, -4, 1, 1, -1, 7381, -9, 4, 7, 1, -1, -1, 1, -1, 0, 83711, -10, 9, 28, 49, -29, -5, 8, 1, -5, 5
Offset: 0

Views

Author

Paul Curtz, Nov 09 2013

Keywords

Comments

Leading column gives the Bernoulli numbers: A027641(n)/A027642(n). In A051714, A164555 must be written instead of A027641.

Examples

			Numerators of
1,    3/2, 11/6, 25/12,...
-1/2, -2/3, -3/4,  -4/5,...
1/6,   1/6, 3/20,  2/15,... =A026741(n+1)/A045896(n+1)
0,    1/30, 1/20,  2/35,... =A194531/A193220.

Links

Crossrefs

Cf. A051714/A051715, A001008/A002805.

Programs

  • Mathematica
    t[1, k_] := HarmonicNumber[k]; t[n_, k_] := t[n, k] = k*(t[n-1, k] - t[n-1, k+1]); Table[t[n-k+1, k] // Numerator, {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 15 2013 *)

Extensions

More terms from Jean-François Alcover, Nov 15 2013
Previous Showing 11-14 of 14 results.

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