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.

A158490 a(n) = 100*n^2 - 10.

Original entry on oeis.org

90, 390, 890, 1590, 2490, 3590, 4890, 6390, 8090, 9990, 12090, 14390, 16890, 19590, 22490, 25590, 28890, 32390, 36090, 39990, 44090, 48390, 52890, 57590, 62490, 67590, 72890, 78390, 84090, 89990, 96090, 102390, 108890, 115590, 122490, 129590, 136890, 144390, 152090
Offset: 1

Views

Author

Vincenzo Librandi, Mar 20 2009

Keywords

Comments

The identity (20*n^2 - 1)^2 - (100*n^2 - 10)*(2*n)^2 = 1 can be written as A158491(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Crossrefs

Cf. A005843, A158447, A158491.

Programs

  • Magma
    I:=[90, 390, 890]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
  • Mathematica
    LinearRecurrence[{3,-3,1},{90,390,890},20]
100*Range[40]^2-10 (* Harvey P. Dale, Apr 03 2019 *)
  • PARI
    a(n)=100*n^2-10 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 10*x*(-9-12*x+x^2)/(x-1)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(10))*Pi/sqrt(10))/20.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(10))*Pi/sqrt(10) - 1)/20. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: 10*(exp(x)*(10*x^2 + 10*x - 1) + 1).
a(n) = 10*A158447(n). (End)

A158598 a(n) = 40*n^2 - 1.

Original entry on oeis.org

39, 159, 359, 639, 999, 1439, 1959, 2559, 3239, 3999, 4839, 5759, 6759, 7839, 8999, 10239, 11559, 12959, 14439, 15999, 17639, 19359, 21159, 23039, 24999, 27039, 29159, 31359, 33639, 35999, 38439, 40959, 43559, 46239, 48999, 51839, 54759, 57759, 60839, 63999, 67239
Offset: 1

Views

Author

Vincenzo Librandi, Mar 22 2009

Keywords

Comments

The identity (40*n^2 - 1)^2 - (400*n^2 - 20)*(2*n)^2 = 1 can be written as a(n)^2 - A158597(n)*A005843(n)^2 = 1.

Links

Crossrefs

Cf. A005843, A158447, A158597.

Programs

  • Magma
    I:=[39, 159, 359]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 16 2012
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {39, 159, 359}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
40*Range[40]^2-1 (* Harvey P. Dale, Jan 31 2022 *)
  • PARI
    for(n=1, 40, print1(40*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 16 2012

Formula

G.f.: x*(-39 - 42*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 16 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)) - 1)/2. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(40*x^2 + 40*x - 1) + 1.
a(n) = A158447(2*n). (End)

Extensions

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

A158446 a(n) = 25*n^2 - 5.

Original entry on oeis.org

20, 95, 220, 395, 620, 895, 1220, 1595, 2020, 2495, 3020, 3595, 4220, 4895, 5620, 6395, 7220, 8095, 9020, 9995, 11020, 12095, 13220, 14395, 15620, 16895, 18220, 19595, 21020, 22495, 24020, 25595, 27220, 28895, 30620, 32395, 34220, 36095, 38020, 39995, 42020, 44095
Offset: 1

Views

Author

Vincenzo Librandi, Mar 19 2009

Keywords

Comments

The identity (10*n^2 - 1)^2 - (25*n^2 - 5)*(2*n)^2 = 1 can be written as A158447(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Crossrefs

Cf. A005843, A134538, A158447.

Programs

  • Magma
    I:=[20, 95, 220]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
  • Mathematica
    Table[25n^2-5,{n,50}]
LinearRecurrence[{3,-3,1},{20,95,220},40] (* Harvey P. Dale, May 05 2019 *)
  • PARI
    a(n) = 25*n^2 - 5.

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 5*x*(4+7*x-x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(5))*Pi/sqrt(5))/10.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(5))*Pi/sqrt(5) - 1)/10. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 5*(exp(x)*(5*x^2 + 5*x - 1) + 1).
a(n) = 5*A134538(n). (End)

A330082 a(n) = 5*A064038(n+1).

Original entry on oeis.org

0, 5, 15, 15, 25, 75, 105, 70, 90, 225, 275, 165, 195, 455, 525, 300, 340, 765, 855, 475, 525, 1155, 1265, 690, 750, 1625, 1755, 945, 1015, 2175, 2325, 1240, 1320, 2805, 2975, 1575, 1665, 3515, 3705, 1950, 2050, 4305, 4515, 2365, 2475, 5175, 5405, 2820, 2940
Offset: 0

Views

Author

Paul Curtz, Dec 01 2019

Keywords

Comments

Main column of a pentagonal spiral for A026741:
(25)
49 (15) 31
24 29 (15) 8 16
47 14 7 ( 5) 3 17 33
23 27 13 2 ( 0) 1 7 9 17
45 13 6 3 1 4 19 35
22 25 11 5 9 10 18
43 12 23 11 21 37
21 41 20 39 19
a(n) = 5 * A064038(n+1) from a pentagonal spiral.
Compare to A319127 = 6 * A002620 in the hexagonal spiral:
22 23 23 22 (24)
20 12 13 13 (12) 25
21 10 5 4 ( 6) 14 25
21 11 5 1 ( 0) 7 15 24
20 11 4 1 ( 0) 2 7 15 26
18 10 2 3 3 6 14 27
19 8 9 9 8 16 27
19 18 16 17 17 26
30 28 29 29 28

Links

Crossrefs

Cf. A026741, A028895, A064038. A033429, A062786, A087348, A147874, A158447, A168668 are in the spiral.

Programs

  • Mathematica
    A330082[n_]:=5Numerator[n(n+1)/4];Array[A330082,100,0] (* Paolo Xausa, Dec 04 2023 *)
  • PARI
    concat(0, Vec(5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3) + O(x^50))) \\ Colin Barker, Dec 08 2019

Formula

a(n) = A026741(A028895(n)).
From Colin Barker, Dec 08 2019: (Start)
G.f.: 5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3).
a(n) = 3*a(n-1) - 6*a(n-2) + 10*a(n-3) - 12*a(n-4) + 12*a(n-5) - 10*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) for n>8.
a(n) = (-5/16 + (5*i)/16)*(((-3-3*i) + (-i)^n + i^(1+n))*n*(1+n)) where i=sqrt(-1).
(End)

Extensions

More terms from Colin Barker, Dec 22 2019
Name corrected by Paolo Xausa, Dec 04 2023
Showing 1-4 of 4 results.

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