A158560 -id:A158560 - cp's OEIS frontend

A158559 a(n) = 225*n^2 - 15.

210, 885, 2010, 3585, 5610, 8085, 11010, 14385, 18210, 22485, 27210, 32385, 38010, 44085, 50610, 57585, 65010, 72885, 81210, 89985, 99210, 108885, 119010, 129585, 140610, 152085, 164010, 176385, 189210, 202485, 216210, 230385, 245010, 260085, 275610, 291585, 308010

Offset: 1

Vincenzo Librandi, Mar 21 2009

The identity (30*n^2 - 1)^2 - (225*n^2 - 15) * (2*n)^2 = 1 can be written as A158560(n)^2 - a(n) * A005843(n)^2 = 1.

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

Cf. A005843, A158560.

I:=[210, 885, 2010]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 14 2012

15(15Range[40]^2-1) (* or *) LinearRecurrence[{3,-3,1},{210,885,2010},40] (* Harvey P. Dale, Jan 24 2012 *)

for(n=1, 40, print1(225*n^2 - 15", ")); \\ Vincenzo Librandi, Feb 05 2012

G.f.: 15*x*(-14 - 17*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(15))*Pi/sqrt(15))/30.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(15))*Pi/sqrt(15) - 1)/30. (End)

Comment rewritten by R. J. Mathar, Oct 16 2009

A158669 a(n) = 900*n^2 - 30.

Original entry on oeis.org

870, 3570, 8070, 14370, 22470, 32370, 44070, 57570, 72870, 89970, 108870, 129570, 152070, 176370, 202470, 230370, 260070, 291570, 324870, 359970, 396870, 435570, 476070, 518370, 562470, 608370, 656070, 705570, 756870, 809970, 864870, 921570, 980070, 1040370, 1102470

Offset: 1

Vincenzo Librandi, Mar 24 2009

The identity (60*n^2 - 1)^2 - (900*n^2 - 30)*(2*n)^2 = 1 can be written as A158670(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, A158560, A158670.

I:=[870, 3570, 8070]; [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 18 2012

LinearRecurrence[{3, -3, 1}, {870, 3570, 8070}, 50] (* Vincenzo Librandi, Feb 18 2012 *)

for(n=1, 40, print1(900*n^2 - 30", ")); \\ Vincenzo Librandi, Feb 18 2012

G.f.: 30*x*(-29 - 32*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 20 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(30))*Pi/sqrt(30))/60.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(30))*Pi/sqrt(30) - 1)/60. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 30*(exp(x)*(30*x^2 + 30*x - 1) + 1).
a(n) = 30*A158560(n). (End)

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009

A283867 Numbers n such that 30n^2 - 1 and 30n^2 + 1 are (twin) primes.

Original entry on oeis.org

1, 3, 10, 14, 18, 38, 62, 73, 116, 118, 143, 183, 221, 232, 242, 330, 333, 413, 430, 455, 470, 496, 507, 533, 538, 556, 606, 622, 645, 675, 687, 701, 720, 777, 792, 819, 846, 879, 881, 895, 913, 1000, 1019, 1030, 1092, 1155, 1214, 1238, 1253, 1261, 1313, 1337, 1350, 1407, 1418, 1429, 1431

Offset: 1

Juri-Stepan Gerasimov, Mar 17 2017

nonn

			3 is in this sequence because 30*3^2 - 1 = 269 and 30*3^2 + 1 = 271 are twin primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000290, A001097, A077800, A158558, A158560.

[n: n in [1..1500] | IsPrime(30*n^2-1) and IsPrime(30*n^2+1)];

Select[Range@ 1431, PrimeQ[30*#^2 + 1] && PrimeQ[30*#^2 - 1] &] (* Indranil Ghosh, Mar 17 2017 *)

is(n)=isprime(30*n^2-1) && isprime(30*n^2+1) \\ Charles R Greathouse IV, Mar 17 2017

from sympy import isprime
[i for i in range(1, 1501) if isprime(30*i**2 - 1) and isprime(30*i**2 + 1)] # Indranil Ghosh, Mar 17 2017

a(n) >> n log^2 n. - Charles R Greathouse IV, Mar 17 2017

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A158559 a(n) = 225*n^2 - 15.

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A158669 a(n) = 900*n^2 - 30.

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A283867 Numbers n such that 30n^2 - 1 and 30n^2 + 1 are (twin) primes.

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cp's OEIS Frontend

A158559 a(n) = 225*n^2 - 15.

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Magma

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A158669 a(n) = 900*n^2 - 30.

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Magma

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A283867 Numbers n such that 30*n^2 - 1 and 30*n^2 + 1 are (twin) primes.

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A283867 Numbers n such that 30n^2 - 1 and 30n^2 + 1 are (twin) primes.