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 1-6 of 6 results.

A177071 a(n) = (7*n + 3)*(7*n + 4).

Original entry on oeis.org

12, 110, 306, 600, 992, 1482, 2070, 2756, 3540, 4422, 5402, 6480, 7656, 8930, 10302, 11772, 13340, 15006, 16770, 18632, 20592, 22650, 24806, 27060, 29412, 31862, 34410, 37056, 39800, 42642, 45582, 48620, 51756, 54990, 58322, 61752, 65280, 68906, 72630, 76452
Offset: 0

Views

Author

Vincenzo Librandi, May 31 2010

Keywords

Comments

Cf. Zumkeller's contribution in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2, therefore a(n) = 49*A002061(n+1) - 37. - Bruno Berselli, Aug 24 2010

Links

Crossrefs

Cf. A002061, A017017, A017029, A061792, A177059.

Programs

  • Mathematica
    Table[(7n+3)(7n+4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{12,110,306},40] (* Harvey P. Dale, Oct 09 2011 *)
  • PARI
    a(n)=2*binomial(7*n+4,2) \\ Charles R Greathouse IV, Jan 11 2012

Formula

a(n) = 98*n + a(n-1) with n > 0, a(0)=12.
From Harvey P. Dale, Oct 09 2011: (Start)
a(0)=12, a(1)=110, a(2)=306, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -((2*(x+6)*(6*x+1))/(x-1)^3). (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017017(n)*A017029(n).
Sum_{n>=0} 1/a(n) = tan(Pi/14)*Pi/7.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/14)*cos(sqrt(5)*Pi/14).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/14)*cosh(sqrt(3)*Pi/14). (End)
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: exp(x)*(12 + 49*x*(2 + x)).
a(n) = 2*A061792(n). (End)

Extensions

Edited by N. J. A. Sloane, Jun 22 2010

A001545 a(n) = (5*n+1)*(5*n+4).

Original entry on oeis.org

4, 54, 154, 304, 504, 754, 1054, 1404, 1804, 2254, 2754, 3304, 3904, 4554, 5254, 6004, 6804, 7654, 8554, 9504, 10504, 11554, 12654, 13804, 15004, 16254, 17554, 18904, 20304, 21754, 23254, 24804, 26404, 28054, 29754, 31504, 33304, 35154, 37054, 39004, 41004, 43054
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000217, A001622, A016861, A016897, A177059, A200135, A200138.

Programs

  • Mathematica
    Table[(5n+1)(5n+4),{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{4,54,154},60] (* Harvey P. Dale, Mar 17 2019 *)
  • PARI
    a(n)=(5*n+1)*(5*n+4) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 50*A000217(n) + 4.
a(n) = 50*n + a(n-1) with a(0)=4. - Vincenzo Librandi, Jan 20 2011
From Amiram Eldar, Jan 23 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(1 + 2/sqrt(5))*Pi/15 = 0.2882687....
Sum_{n>=0} (-1)^n/a(n) = 2*log(phi)/(3*sqrt(5)) + 2*log(2)/15, where phi is the golden ratio (A001622).
Product_{n>=0} (1 - 1/a(n)) = sqrt(2 + 2/sqrt(5)) * cos(sqrt(13)*Pi/10).
Product_{n>=0} (1 + 1/a(n)) = sqrt(2 + 2/sqrt(5)) * cos(Pi/(2*sqrt(5))).
Product_{n>=0} (1 + 2/a(n)) = phi. (End)
G.f.: 2*(2+21*x+2*x^2)/(1-x)^3. - R. J. Mathar, May 30 2022
Sum_{n>=0} 1/a(n) = (Psi(4/5) - Psi(1/5))/15. See A200135, A200138. - R. J. Mathar, May 30 2022
From Elmo R. Oliveira, Oct 23 2024: (Start)
E.g.f.: exp(x)*(4 + 25*x*(2 + x)).
a(n) = A016861(n)*A016897(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A177065 a(n) = (8*n+3)*(8*n+5).

Original entry on oeis.org

15, 143, 399, 783, 1295, 1935, 2703, 3599, 4623, 5775, 7055, 8463, 9999, 11663, 13455, 15375, 17423, 19599, 21903, 24335, 26895, 29583, 32399, 35343, 38415, 41615, 44943, 48399, 51983, 55695, 59535, 63503, 67599, 71823, 76175, 80655, 85263, 89999, 94863, 99855
Offset: 0

Views

Author

Vincenzo Librandi, May 31 2010

Keywords

Comments

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 64*A002061(n+1) - 49. - Bruno Berselli, Aug 24 2010

Links

Crossrefs

Cf. A002061, A004770, A016754, A017101, A125169, A177059.

Programs

Formula

a(n) = 128*n + a(n-1) with n > 0, a(0)=15.
a(n) = A125169(A016754(n) - 1). - Reinhard Zumkeller, Jul 05 2010
a(0)=15, a(1)=143, a(2)=399, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 13 2013
G.f.: (15+98*x+15*x^2)/(1-x)^3. - Vincenzo Librandi, Apr 08 2013
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017101(n)*A004770(n).
Sum_{n>=0} 1/a(n) = (sqrt(2)-1)*Pi/16.
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(tan(3*Pi/16)) + sin(Pi/8) * log(cot(Pi/16)))/4.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/8)*cos(Pi/(4*sqrt(2))).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/8). (End)
E.g.f.: exp(x)*(15 + 64*x*(2 + x)). - Elmo R. Oliveira, Oct 25 2024

Extensions

Edited by N. J. A. Sloane, Jun 22 2010

A177072 a(n) = (9*n+2)*(9*n+7).

Original entry on oeis.org

14, 176, 500, 986, 1634, 2444, 3416, 4550, 5846, 7304, 8924, 10706, 12650, 14756, 17024, 19454, 22046, 24800, 27716, 30794, 34034, 37436, 41000, 44726, 48614, 52664, 56876, 61250, 65786, 70484, 75344, 80366, 85550, 90896, 96404, 102074, 107906, 113900, 120056
Offset: 0

Views

Author

Vincenzo Librandi, May 31 2010

Keywords

Comments

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 67. - Bruno Berselli, Aug 24 2010

Links

Crossrefs

Cf. A002061, A017185, A017245, A177059.

Programs

  • Magma
    I:=[14, 176, 500]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Apr 08 2013
  • Mathematica
    CoefficientList[Series[2(7 + 67 x + 7 x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 08 2013 *)
Table[(9*n + 2)*(9*n + 7), {n, 0, 40}] (* Amiram Eldar, Feb 19 2023 *)
LinearRecurrence[{3,-3,1},{14,176,500},50] (* Harvey P. Dale, Jun 10 2023 *)
  • PARI
    a(n)=(9*n+2)*(9*n+7) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 162*n + a(n-1) with n > 0, a(0)=14.
From Vincenzo Librandi, Apr 08 2013: (Start)
G.f.: 2*(7+67*x+7*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017185(n)*A017245(n).
Sum_{n>=0} 1/a(n) = cot(2*Pi/9)*Pi/45.
Product_{n>=0} (1 - 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(29)*Pi/18).
Product_{n>=0} (1 + 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(21)*Pi/18). (End)
E.g.f.: exp(x)*(14 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

Extensions

Edited by N. J. A. Sloane, Jun 22 2010

A177073 a(n) = (9*n+4)*(9*n+5).

Original entry on oeis.org

20, 182, 506, 992, 1640, 2450, 3422, 4556, 5852, 7310, 8930, 10712, 12656, 14762, 17030, 19460, 22052, 24806, 27722, 30800, 34040, 37442, 41006, 44732, 48620, 52670, 56882, 61256, 65792, 70490, 75350, 80372, 85556, 90902, 96410, 102080, 107912, 113906, 120062
Offset: 0

Views

Author

Vincenzo Librandi, May 31 2010

Keywords

Comments

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 61. - Bruno Berselli, Aug 24 2010

Links

Crossrefs

Cf. A002061, A017209, A017221, A177059.

Programs

  • Magma
    [(9*n+4)*(9*n+5): n in [0..50]]; // Vincenzo Librandi, Apr 08 2013
  • Mathematica
    f[n_] := Module[{c = 9n}, (c+4)(c+5)]; Array[f, 40, 0] (* or *) LinearRecurrence[{3, -3, 1}, {20, 182, 506}, 40] (* Harvey P. Dale, Jun 24 2011 *)
  • PARI
    a(n)=(9*n+4)*(9*n+5) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 162*n + a(n-1) with n > 0, a(0)=20.
From Harvey P. Dale, Jun 24 2011: (Start)
a(0)=20, a(1)=182, a(2)=506, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -2*(x*(10*x+61)+10)/(x-1)^3. (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017209(n)*A017221(n).
Sum_{n>=0} 1/a(n) = tan(Pi/18)*Pi/9.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/18)*cos(sqrt(5)*Pi/18).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/18)*cosh(sqrt(3)*Pi/18). (End)
E.g.f.: exp(x)*(20 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

Extensions

Edited by N. J. A. Sloane, Jun 22 2010

A177060 a(n) = (7*n+2)*(7*n+5) = 49*n^2 + 49*n + 10.

Original entry on oeis.org

10, 108, 304, 598, 990, 1480, 2068, 2754, 3538, 4420, 5400, 6478, 7654, 8928, 10300, 11770, 13338, 15004, 16768, 18630, 20590, 22648, 24804, 27058, 29410, 31860, 34408, 37054, 39798, 42640, 45580, 48618, 51754, 54988, 58320, 61750, 65278, 68904, 72628, 76450
Offset: 0

Views

Author

Vincenzo Librandi, May 31 2010

Keywords

Comments

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 49*A002061(n+1) - 39. - Bruno Berselli, Aug 24 2010

Examples

			For n=1, a(1) = 98 + 10 = 108.
For n=2, a(2) = 98*2 + 108 = 304.
For n=3, a(3) = 98*3 + 304 = 598.

Links

Crossrefs

Cf. A002061, A017005, A017041, A177059.

Programs

Formula

a(n) = 98*n + a(n-1) with a(0) = 10.
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017005(n)*A017041(n).
Sum_{n>=0} 1/a(n) = tan(3*Pi/14)*Pi/21.
Product_{n>=0} (1 - 1/a(n)) = sec(3*Pi/14)*cos(sqrt(13)*Pi/14).
Product_{n>=0} (1 + 1/a(n)) = sec(3*Pi/14)*cos(sqrt(5)*Pi/14). (End)
From Elmo R. Oliveira, Oct 24 2024: (Start)
G.f.: 2*(5 + 39*x + 5*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(10 + 49*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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