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 1-10 of 15 results. Next

A120071 a(n) = n*(n+20).

Original entry on oeis.org

0, 21, 44, 69, 96, 125, 156, 189, 224, 261, 300, 341, 384, 429, 476, 525, 576, 629, 684, 741, 800, 861, 924, 989, 1056, 1125, 1196, 1269, 1344, 1421, 1500, 1581, 1664, 1749, 1836, 1925, 2016, 2109, 2204, 2301, 2400, 2501, 2604, 2709, 2816, 2925, 3036, 3149, 3264
Offset: 0

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Links

Crossrefs

a(n-10), n >= 11, tenth column (used for the n=10 series of the hydrogen atom) of triangle A120070.
For n*(n+18) see A098850.
Cf. A001008, A001477, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A102928, A120061, A132759, A132760, A132761, A132762, A132765.

Programs

Formula

a(n) = (n+10)^2 - 10^2 = n*(n+20), n >= 0.
G.f.: x*(21-19*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 19 (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(20)/20 = A001008(20)/A102928(20) = 11167027/62078016, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 155685007/4655851200. (End)
From Elmo R. Oliveira, Jan 12 2025: (Start)
E.g.f.: exp(x)*x*(21 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A132765 a(n) = n*(n + 23).

Original entry on oeis.org

0, 24, 50, 78, 108, 140, 174, 210, 248, 288, 330, 374, 420, 468, 518, 570, 624, 680, 738, 798, 860, 924, 990, 1058, 1128, 1200, 1274, 1350, 1428, 1508, 1590, 1674, 1760, 1848, 1938, 2030, 2124, 2220, 2318, 2418, 2520, 2624, 2730, 2838, 2948, 3060, 3174, 3290, 3408
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A001477, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A056126, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766.

Programs

Formula

a(n) = n*(n + 23).
a(n) = 2*n + a(n-1) + 22 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(12 - 11*x)/(1-x)^3. (End)
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(23)/23 = A001008(23)/A102928(23) = 444316699/2736605872, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/23 - 3825136961/123147264240. (End)
E.g.f.: x*(24 + x)*exp(x). - G. C. Greubel, Mar 14 2022

A056119 a(n) = n*(n+13)/2.

Original entry on oeis.org

0, 7, 15, 24, 34, 45, 57, 70, 84, 99, 115, 132, 150, 169, 189, 210, 232, 255, 279, 304, 330, 357, 385, 414, 444, 475, 507, 540, 574, 609, 645, 682, 720, 759, 799, 840, 882, 925, 969, 1014, 1060, 1107, 1155, 1204, 1254, 1305, 1357, 1410, 1464, 1519, 1575
Offset: 0

Views

Author

Barry E. Williams, Jul 04 2000

Keywords

Links

Crossrefs

Cf. A000096, A001477, A056000, A056115, A056121.

Programs

Formula

G.f.: x*(7-6*x)/(1-x)^3.
a(n) = A126890(n,6) for n > 5. - Reinhard Zumkeller, Dec 30 2006
a(n) = A000096(n) + 5*A001477(n) = A056115(n) + A001477(n) = A056121(n) - A001477(n). - Zerinvary Lajos, Feb 22 2008
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n) = -f(n,n-1,7), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = n + a(n-1) + 6 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
Sum_{n>=1} 1/a(n) = 1145993/2342340 via A132759. - R. J. Mathar, Jul 14 2012
a(n) = 7*n - floor(n/2) + floor(n/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: x*(14 + x)*exp(x)/2. - G. C. Greubel, Jan 18 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/13 - 263111/2342340. - Amiram Eldar, Jan 10 2021

Extensions

More terms from James Sellers, Jul 05 2000

A132762 a(n) = n*(n + 19).

Original entry on oeis.org

0, 20, 42, 66, 92, 120, 150, 182, 216, 252, 290, 330, 372, 416, 462, 510, 560, 612, 666, 722, 780, 840, 902, 966, 1032, 1100, 1170, 1242, 1316, 1392, 1470, 1550, 1632, 1716, 1802, 1890, 1980, 2072, 2166, 2262, 2360, 2460, 2562, 2666, 2772, 2880, 2990, 3102, 3216
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A051942, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132763, A132764, A132765, A132766, A132767.

Programs

Formula

a(n) = 2*n + a(n-1) + 18 for n > 0, a(0) = 0. - Vincenzo Librandi, Aug 03 2010
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(10 - 9*x)/(1-x)^3. (End)
a(n) = 2*A051942(n+9). - R. J. Mathar, Sep 05 2018
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(19)/19 = A001008(19)/A102928(19) = 275295799/1474352880, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/19 - 33464927/884611728. (End)
E.g.f.: x*(20 + x)*exp(x). - G. C. Greubel, Mar 14 2022

A132764 a(n) = n*(n+22).

Original entry on oeis.org

0, 23, 48, 75, 104, 135, 168, 203, 240, 279, 320, 363, 408, 455, 504, 555, 608, 663, 720, 779, 840, 903, 968, 1035, 1104, 1175, 1248, 1323, 1400, 1479, 1560, 1643, 1728, 1815, 1904, 1995, 2088, 2183, 2280, 2379, 2480, 2583, 2688, 2795, 2904, 3015, 3128, 3243, 3360
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Examples

			a(1)=2*1+0+21=23; a(2)=2*2+23+21=48; a(3)=2*3+48+21=75. - _Vincenzo Librandi_, Aug 03 2010

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132765, A132766, A132767.

Programs

Formula

a(n) = n*(n + 22).
a(n) = 2*n + a(n-1) + 21 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=23, a(2)=48, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 02 2012
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(22)/22 = A001008(22)/A102928(22) = 19093197/113809696, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 156188887/5121436320. (End)
From G. C. Greubel, Mar 14 2022: (Start)
G.f.: x*(23 - 21*x)/(1-x)^3.
E.g.f.: x*(23 + x)*exp(x). (End)

A132763 a(n) = n*(n+21).

Original entry on oeis.org

0, 22, 46, 72, 100, 130, 162, 196, 232, 270, 310, 352, 396, 442, 490, 540, 592, 646, 702, 760, 820, 882, 946, 1012, 1080, 1150, 1222, 1296, 1372, 1450, 1530, 1612, 1696, 1782, 1870, 1960, 2052, 2146, 2242, 2340, 2440, 2542, 2646, 2752, 2860, 2970, 3082, 3196, 3312
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132764, A132765, A132766, A132767.

Programs

Formula

a(n) = n*(n + 21).
a(n) = 2*n + a(n-1) + 20 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=22, a(2)=46, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 25 2014
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(21)/21 = A001008(21)/A102928(21) = 18858053/108636528, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/21 - 166770367/4888643760. (End)
From Stefano Spezia, Jan 30 2021: (Start)
O.g.f.: 2*x*(11 - 10*x)/(1 - x)^3.
E.g.f.: x*(22 + x)*exp(x). (End)

A132766 a(n) = n*(n+24).

Original entry on oeis.org

0, 25, 52, 81, 112, 145, 180, 217, 256, 297, 340, 385, 432, 481, 532, 585, 640, 697, 756, 817, 880, 945, 1012, 1081, 1152, 1225, 1300, 1377, 1456, 1537, 1620, 1705, 1792, 1881, 1972, 2065, 2160, 2257, 2356, 2457, 2560, 2665, 2772, 2881, 2992, 3105, 3220, 3337
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098849, A098850, A098847, A098848, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765.

Programs

  • Mathematica
    Table[n (n + 24), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 25, 52}, 50] (* Harvey P. Dale, Feb 11 2016 *)
  • PARI
    a(n)=n*(n+24) \\ Charles R Greathouse IV, Jun 17 2017
  • Sage
    [n*(n+24) for n in (0..50)] # G. C. Greubel, Mar 14 2022

Formula

a(n) = n*(n + 24).
a(n) = 2*n + a(n-1) + 23 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=25, a(2)=52; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 11 2016
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(24)/24 = A001008(24)/A102928(24) = 1347822955/8566766208, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3602044091/128501493120. (End)
From G. C. Greubel, Mar 14 2022: (Start)
G.f.: 2*x*(13 - 12*x)/(1-x)^3.
E.g.f.: x*(26 + x)*exp(x). (End)

A132767 a(n) = n*(n + 25).

Original entry on oeis.org

0, 26, 54, 84, 116, 150, 186, 224, 264, 306, 350, 396, 444, 494, 546, 600, 656, 714, 774, 836, 900, 966, 1034, 1104, 1176, 1250, 1326, 1404, 1484, 1566, 1650, 1736, 1824, 1914, 2006, 2100, 2196, 2294, 2394, 2496, 2600, 2706, 2814, 2924, 3036, 3150, 3266, 3384
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Comments

a(n) is the Zagreb 1 index of the Mycielskian of the cycle graph C[n]. See p. 205 of the D. B. West reference. - Emeric Deutsch, Nov 04 2016

References

  • Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766.

Programs

Formula

a(n) = 2*n + a(n-1) + 24 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(n) = n^2 + 25*n. - Omar E. Pol, Nov 04 2016
From Chai Wah Wu, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 2*x*(13 - 12*x)/(1-x)^3. (End)
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(25)/25 = A001008(25)/A102928(25) = 34052522467/223092870000, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/25 - 19081066231/669278610000. (End)
E.g.f.: x*(26 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A132768 a(n) = n*(n + 26).

Original entry on oeis.org

0, 27, 56, 87, 120, 155, 192, 231, 272, 315, 360, 407, 456, 507, 560, 615, 672, 731, 792, 855, 920, 987, 1056, 1127, 1200, 1275, 1352, 1431, 1512, 1595, 1680, 1767, 1856, 1947, 2040, 2135, 2232, 2331, 2432, 2535, 2640, 2747, 2856, 2967, 3080, 3195, 3312, 3431
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759, A132760, A132761.
Cf. A132762, A132763, A132764, A132765, A132766, A132767.

Programs

Formula

a(n) = n*(n + 26).
a(n) = 2*n + a(n-1) + 25, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(26)/26 = A001008(26)/A102928(26) = 34395742267/232016584800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 18051406831/696049754400. (End)
From G. C. Greubel, Mar 13 2022: (Start)
G.f.: x*(27 - 25*x)/(1-x)^3.
E.g.f.: x*(27 + x)*exp(x). (End)

A132769 a(n) = n*(n + 27).

Original entry on oeis.org

0, 28, 58, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 520, 574, 630, 688, 748, 810, 874, 940, 1008, 1078, 1150, 1224, 1300, 1378, 1458, 1540, 1624, 1710, 1798, 1888, 1980, 2074, 2170, 2268, 2368, 2470, 2574, 2680, 2788, 2898, 3010, 3124, 3240, 3358, 3478
Offset: 0

Views

Author

Omar E. Pol, Aug 28 2007

Keywords

Links

Crossrefs

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759.
Cf. A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768.
Cf. A132756.

Programs

Formula

a(n) = 2*n + a(n-1) + 26, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=28, a(2)=58; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Oct 14 2012
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(27)/27 = A001008(27)/A102928(27) = 312536252003/2168462696400, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/27 - 57128792093/2168462696400. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 2*x*(14 - 13*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(28 + x).
a(n) = 2*A132756(n). (End)
Showing 1-10 of 15 results. Next

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