A158491 -id:A158491 - cp's OEIS frontend

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

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

Vincenzo Librandi, Mar 20 2009

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.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158447, A158491.

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]];

LinearRecurrence[{3,-3,1},{90,390,890},20]
100*Range[40]^2-10 (* Harvey P. Dale, Apr 03 2019 *)

a(n)=100*n^2-10 \\ Charles R Greathouse IV, Jun 17 2017

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)

A158597 a(n) = 400*n^2 - 20.

Original entry on oeis.org

380, 1580, 3580, 6380, 9980, 14380, 19580, 25580, 32380, 39980, 48380, 57580, 67580, 78380, 89980, 102380, 115580, 129580, 144380, 159980, 176380, 193580, 211580, 230380, 249980, 270380, 291580, 313580, 336380, 359980, 384380, 409580, 435580, 462380, 489980, 518380

Offset: 1

Vincenzo Librandi, Mar 22 2009

The identity (40*n^2 - 1)^2 - (400*n^2 - 20)*(2*n)^2 = 1 can be written as A158598(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158491, A158598.

I:=[380, 1580, 3580]; [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

LinearRecurrence[{3, -3, 1}, {380, 1580, 3580}, 50] (* Vincenzo Librandi, Feb 16 2012 *)
400*Range[40]^2-20 (* Harvey P. Dale, Nov 04 2015 *)

for(n=1, 40, print1(400*n^2- 20", ")); \\ Vincenzo Librandi, Feb 16 2012

G.f.: 20*x*(-19 - 22*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 14 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)))/40.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) - 1)/40. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 20*(exp(x)*(20*x^2 + 20*x - 1) + 1).
a(n) = 20*A158491(n). (End)

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

A158774 a(n) = 80*n^2 - 1.

Original entry on oeis.org

79, 319, 719, 1279, 1999, 2879, 3919, 5119, 6479, 7999, 9679, 11519, 13519, 15679, 17999, 20479, 23119, 25919, 28879, 31999, 35279, 38719, 42319, 46079, 49999, 54079, 58319, 62719, 67279, 71999, 76879, 81919, 87119, 92479, 97999, 103679, 109519, 115519, 121679

Offset: 1

Vincenzo Librandi, Mar 26 2009

The identity (80*n^2 - 1)^2 - (1600*n^2 - 40)*(2*n)^2 = 1 can be written as a(n)^2 - A158773(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158491, A158773.

I:=[79, 319, 719]; [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 20 2012

LinearRecurrence[{3, -3, 1}, {79, 319, 719}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
80*Range[40]^2-1 (* Harvey P. Dale, Apr 21 2018 *)

for(n=1, 40, print1(80*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 20 2012

From R. J. Mathar, Jul 26 2009: (Start)
G.f.: x*(-79 - 82*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 24 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(4*sqrt(5)))*Pi/(4*sqrt(5)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(4*sqrt(5)))*Pi/(4*sqrt(5)) - 1)/2. (End)
From Elmo R. Oliveira, Jan 25 2025: (Start)
E.g.f.: exp(x)*(80*x^2 + 80*x - 1) + 1.
a(n) = A158491(2*n). (End)

Edited by R. J. Mathar, Jul 26 2009

A193448 a(n) = 4(5n^2 - 5*n + 1).

Original entry on oeis.org

4, 44, 124, 244, 404, 604, 844, 1124, 1444, 1804, 2204, 2644, 3124, 3644, 4204, 4804, 5444, 6124, 6844, 7604, 8404, 9244, 10124, 11044, 12004, 13004, 14044, 15124, 16244, 17404, 18604, 19844, 21124, 22444, 23804, 25204, 26644, 28124, 29644, 31204, 32804

Offset: 1

Giovanni Teofilatto, Jul 26 2011

The natural numbers of the form 5*n^2-1, with n odd. See also A158491 for the cases where n is even. - Giovanni Teofilatto, Oct 10 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A062786, A158491.

[4*(5*n^2-5*n+1): n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

A193448:=n->4*(5*n^2 - 5*n + 1): seq(A193448(n), n=1..50); # Wesley Ivan Hurt, Nov 21 2015

Table[4*(5*n^2 - 5*n + 1), {n, 50}] (* Wesley Ivan Hurt, Nov 21 2015 *)

a(n) = 4*(5*n^2 - 5*n + 1) \\ Anders Hellström, Nov 21 2015

a(n) = 4*A062786(n).
G.f.: -4*x*(1+8*x+x^2) / (x-1)^3. - R. J. Mathar, Aug 26 2011
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3. - Wesley Ivan Hurt, Nov 21 2015

cp's OEIS Frontend

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

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A158597 a(n) = 400*n^2 - 20.

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Magma

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A158774 a(n) = 80*n^2 - 1.

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Magma

Mathematica

PARI

Formula

Extensions

A193448 a(n) = 4*(5*n^2 - 5*n + 1).

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Formula

A193448 a(n) = 4(5n^2 - 5*n + 1).