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 1-4 of 4 results.

A118391 Numerator of sum of reciprocals of first n tetrahedral numbers A000292.

Original entry on oeis.org

1, 5, 27, 7, 10, 81, 35, 22, 81, 65, 77, 135, 52, 119, 405, 76, 85, 567, 209, 115, 378, 275, 299, 486, 175, 377, 1215, 217, 232, 1485, 527, 280, 891, 629, 665, 1053, 370, 779, 2457, 430, 451, 2835, 989, 517, 1620, 1127, 1175, 1836, 637, 1325, 4131, 715, 742
Offset: 1

Views

Author

Jonathan Vos Post, Apr 27 2006

Keywords

Comments

Denominators are A118392. Fractions are: 1/1, 5/4, 27/20, 7/5, 10/7, 81/56, 35/24, 22/15, 81/55, 65/44, 77/52, 135/91, 52/35, 119/80, 405/272, 76/51, 85/57, 567/380, 209/140, 115/77, 378/253, 275/184, 299/200, 486/325, 175/117, 377/252, 1215/812, 217/145, 232/155, 1485/992.
2n+3 divides a(2n). 2n-1 divides a(2n-1). p divides a(p) for prime p>2. The only primes in a(n) are a(2) = 5 and a(4) = 7. - Alexander Adamchuk, May 08 2007

Examples

			a(1) = 1 = numerator of 1/1.
a(2) = 5 = numerator of 5/4 = 1/1 + 1/4.
a(3) = 27 = numerator of 27/20 = 1/1 + 1/4 + 1/10.
a(4) = 7 = numerator of 7/5 = 1/1 + 1/4 + 1/10 + 1/20.
a(5) = 10 = numerator of 10/7 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35.
a(20) = 115 = numerator of 115/77 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35 + 1/56 + 1/84 + 1/120 + 1/165 + 1/220 + 1/286 + 1/364 + 1/455 + 1/560 + 1/680 + 1/816 + 1/969 + 1/1140 + 1/1330 + 1/1540.

Links

Crossrefs

Cf. A000292, A022998, A118392.

Programs

  • Magma
    [Numerator(3*n*(n+3)/(2*(n+1)*(n+2))): n in [1..60]]; // G. C. Greubel, Feb 18 2021
  • Maple
    A118391:= n-> numer(3*n*(n+3)/(2*(n+1)*(n+2))); seq(A118391(n), n=1..60) # G. C. Greubel, Feb 18 2021
  • Mathematica
    Table[ Numerator[3n(n+3)/(2(n+1)(n+2))], {n,1,100} ] (* Alexander Adamchuk, May 08 2007 *)
Accumulate[1/Binomial[Range[60]+2,3]]//Numerator (* Harvey P. Dale, Aug 31 2023 *)
  • PARI
    s=0;for(i=3,50,s+=1/binomial(i,3);print(numerator(s))) /* Phil Carmody, Mar 27 2012 */
  • Sage
    [numerator(3*n*(n+3)/(2*(n+1)*(n+2))) for n in (1..60)] # G. C. Greubel, Feb 18 2021

Formula

A118391(n)/A118392(n) = Sum_{i=1..n} 1/A000292(n).
A118391(n)/A118392(n) = Sum_{i=1..n} 1/C(n+2,3).
A118391(n)/A118392(n) = Sum_{i=1..n} 6/(n*(n+1)*(n+2)).
a(n) = Numerator( 3*n*(n+3)/(2*(n+1)*(n+2)) ). - Alexander Adamchuk, May 08 2007

Extensions

More terms from Alexander Adamchuk, May 08 2007

A118411 Numerator of sum of reciprocals of first n pentatope numbers A000332.

Original entry on oeis.org

1, 6, 19, 136, 83, 119, 656, 73, 190, 121, 1816, 559, 679, 815, 3872, 1139, 886, 513, 2360, 2023, 2299, 2599, 11696, 3275, 7306, 1353, 5992, 1653, 5455, 5983, 26176, 7139, 15538, 8435, 12184, 3293, 3553, 11479, 49360
Offset: 1

Views

Author

Jonathan Vos Post, Apr 27 2006

Keywords

Comments

Denominators are A118412. Fractions are: 1/1, 6/5, 19/15, 136/105, 83/63, 119/90, 656/495, 73/55, 190/143, 121/91, 1816/1365, 559/420, 679/510, 815/612, 3872/2907, 1139/855, 886/665, 513/385, 2360/1771, 2023/1518, 2299/1725, 2599/1950, 11696/8775, 3275/2457, 7306/5481, 1353/1015, 5992/4495, 1653/1240, 5455/4092, 5983/4488, 26176/19635, 7139/5355, 15538/11655, 8435/6327, 12184/9139, 3293/2470, 3553/2665, 11479/8610, 49360/37023. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392.

Examples

			a(1) = 1 = numerator of 1/1.
a(2) = 6 = numerator of 6/5 = 1/1 + 1/5.
a(3) = 19 = numerator of 19/15 = 1/1 + 1/5 + 1/15.
a(4) = 136 = numerator of 136/105 = 1/1 + 1/5 + 1/15 + 1/35.
a(5) = 55 = numerator of 55/42 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70.
a(10) = 190 = numerator of 190/143 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715.
a(20) = 2360 = numerator of 2360/1771 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715 + 1/1001 + 1/1365 + 1/1820 + 1/2380 + 1/3060 + 1/3876 + 1/4845 + 1/5985 + 1/7315 + 1/8855.

Crossrefs

Cf. A000332, A022998, A026741, A118391, A118391, A118412.

Programs

  • PARI
    s=0;for(i=4,50,s+=1/binomial(i,4);print(numerator(s))) /* Phil Carmody, Mar 27 2012 */

Formula

A118411(n)/A118412(n) = SUM[i=1..n] (1/A000332(n)). A118411(n)/A118412(n) = SUM[i=1..n] (1/C(n+2,4)). A118411(n)/A118412(n) = SUM[i=1..n] (1/(n*(n+1)*(n+2)*(n+3)/24)).

A118412 Denominator of sum of reciprocals of first n pentatope numbers A000332.

Original entry on oeis.org

1, 5, 15, 105, 42, 63, 90, 495, 55, 143, 91, 1365, 420, 510, 612, 2907, 855, 665, 385, 1771, 1518, 1725, 1950, 8775, 2457, 5481, 1015, 4495, 1240, 4092, 4488, 19635, 5355, 11655, 6327, 9139, 2470, 2665, 8610, 37023
Offset: 1

Views

Author

Jonathan Vos Post, Apr 27 2006

Keywords

Comments

Numerators are A118411. Fractions are: 1/1, 6/5, 19/15, 136/105, 83/63, 119/90, 656/495, 73/55, 190/143, 121/91, 1816/1365, 559/420, 679/510, 815/612, 3872/2907, 1139/855, 886/665, 513/385, 2360/1771, 2023/1518, 2299/1725, 2599/1950, 11696/8775, 3275/2457, 7306/5481, 1353/1015, 5992/4495, 1653/1240, 5455/4092, 5983/4488, 26176/19635, 7139/5355, 15538/11655, 8435/6327, 12184/9139, 3293/2470, 3553/2665, 11479/8610, 49360/37023. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392.

Examples

			a(1) = 1 = denominator of 1/1.
a(2) = 5 = denominator of 6/5 = 1/1 + 1/5.
a(3) = 15 = denominator of 19/15 = 1/1 + 1/5 + 1/15.
a(4) = 105 = denominator of 136/105 = 1/1 + 1/5 + 1/15 + 1/35.
a(5) = 42 = denominator of 55/42 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70.
a(10) = 143 = denominator of 190/143 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715.
a(20) = 1771 = denominator of 2360/1771 = 1/1 + 1/5 + 1/15 + 1/35 + 1/70 + 1/126 + 1/210 + 1/330 + 1/495 + 1/715 + 1/1001 + 1/1365 + 1/1820 + 1/2380 + 1/3060 + 1/3876 + 1/4845 + 1/5985 + 1/7315 + 1/8855.

Crossrefs

Cf. A000332, A022998, A026741, A118391, A118391, A118411.

Programs

  • PARI
    s=0;for(i=4,50,s+=1/binomial(i,4);print(denominator(s))) /* Phil Carmody, Mar 27 2012 */

Formula

A118411(n)/A118412(n) = SUM[i=1..n] (1/A000332(n)). A118411(n)/A118412(n) = SUM[i=1..n] (1/C(n+2,4)). A118411(n)/A118412(n) = SUM[i=1..n] (1/(n*(n+1)*(n+2)*(n+3)/24)).

A118432 Denominator of sum of reciprocals of first n 5-simplex numbers A000389.

Original entry on oeis.org

1, 6, 14, 56, 504, 168, 264, 198, 286, 1001, 273, 1456, 1904, 2448, 15504, 969, 1197, 2926, 3542, 42504, 10120, 11960, 14040, 8190, 47502, 5481, 6293, 28768, 32736, 185504, 41888, 11781, 13209, 29526, 164502, 73112, 81016
Offset: 1

Views

Author

Jonathan Vos Post, Apr 28 2006

Keywords

Comments

Numerators are A118431. Fractions are: 1/1, 7/6, 17/14, 69/56, 625/504, 209/168, 329/264, 247/198, 357/286, 1250/1001, 341/273, 1819/1456, 2379/1904, 3059/2448, 19375/15504, 1211/969, 1496/1197, 3657/2926, 4427/3542, 53125/42504, 12649/10120, 14949/11960, 17549/14040, 10237/8190, 59375/47502, 6851/5481, 7866/6293, 35959/28768, 40919/32736, 231875/185504, 52359/41888, 14726/11781, 16511/13209, 36907/29526, 205625/164502, 91389/73112, 101269/81016. The denominator of sum of reciprocals of first n triangular numbers is A026741. The denominator of sum of reciprocals of first n tetrahedral numbers is A118392. The denominator of sum of reciprocals of first n pentatope numbers is A118412.

Examples

			a(1) = 1 = denominator of 1/1.
a(2) = 6 = denominator of 7/6 = 1/1 + 1/6.
a(3) = 14 = denominator of 17/14 = 1/1 + 1/6 + 1/21.
a(4) = 56 = denominator of 69/56 = 1/1 + 1/6 + 1/21 + 1/56.
a(5) = 42 = denominator of 55/42 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126.
a(10) = 1001 = denominator of 1250/1001 = 1/1+ 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002.
a(20) = 42504 = denominator of 53125/42504 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002 + 1/3003 + 1/4368 + 1/6188 + 1/8568 + 1/11628 + 1/15504 + 1/20349 + 1/26334 + 1/33649 + 1/42504.

Links

Crossrefs

Cf. A000332, A000389, A022998, A026741, A118391, A118391, A118411, A118412, A118431.

Programs

  • Mathematica
    Denominator[Accumulate[1/Binomial[Range[5,50],5]]] (* Harvey P. Dale, Jul 17 2016 *)

Formula

A118411(n)/A118412(n) = Sum_{i=1..n} (1/A000389(n)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/C(n,5)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/(n*(n+1)*(n+2)*(n+3)*(n+4)/120)).
Showing 1-4 of 4 results.

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