A247541 -id:A247541 - cp's OEIS frontend

A158556 a(n) = 28*n^2 + 1.

1, 29, 113, 253, 449, 701, 1009, 1373, 1793, 2269, 2801, 3389, 4033, 4733, 5489, 6301, 7169, 8093, 9073, 10109, 11201, 12349, 13553, 14813, 16129, 17501, 18929, 20413, 21953, 23549, 25201, 26909, 28673, 30493, 32369, 34301, 36289, 38333, 40433, 42589, 44801, 47069

Offset: 0

Vincenzo Librandi, Mar 21 2009

The identity (28*n^2 + 1)^2 - (196*n^2 + 14) * (2*n)^2 = 1 can be written as a(n)^2 - A158555(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158555, A247541.

I:=[1, 29, 113]; [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, Mar 02 2012

LinearRecurrence[{3, -3, 1}, {1, 29, 113}, 50] (* Vincenzo Librandi, Mar 02 2012 *)
28*Range[0,40]^2+1 (* Harvey P. Dale, Jun 30 2022 *)

for(n=0, 40, print1(28*n^2+1", ")); \\ Vincenzo Librandi, Mar 02 2012

G.f.: (1 + 26*x + 29*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(28))*Pi/sqrt(28) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(28))*Pi/sqrt(28) + 1)/2. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(1 + 28*x + 28*x^2).
a(n) = A247541(2*n). (End)

Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009

A158481 a(n) = 49*n^2 + 7.

Original entry on oeis.org

56, 203, 448, 791, 1232, 1771, 2408, 3143, 3976, 4907, 5936, 7063, 8288, 9611, 11032, 12551, 14168, 15883, 17696, 19607, 21616, 23723, 25928, 28231, 30632, 33131, 35728, 38423, 41216, 44107, 47096, 50183, 53368, 56651, 60032, 63511, 67088, 70763, 74536, 78407

Offset: 1

Vincenzo Librandi, Mar 20 2009

The identity (14*n^2 + 1)^2 - (49*n^2 + 7)*(2*n)^2 = 1 can be written as A158482(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, A158482, A247541.

I:=[56, 203, 448];[n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

LinearRecurrence[{3,-3,1},{56,203,448},40]

PARI
```
a(n)=49*n^2+7.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 7*x*(8+5*x+x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(7))*Pi/sqrt(7) - 1)/14.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(7))*Pi/sqrt(7))/14. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 7*(exp(x)*(7*x^2 + 7*x + 1) - 1).
a(n) = 7*A247541(n). (End)

cp's OEIS Frontend

A158556 a(n) = 28*n^2 + 1.

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