A157448 -id:A157448 - cp's OEIS frontend

A157446 a(n) = 16*n^2 - n.

15, 62, 141, 252, 395, 570, 777, 1016, 1287, 1590, 1925, 2292, 2691, 3122, 3585, 4080, 4607, 5166, 5757, 6380, 7035, 7722, 8441, 9192, 9975, 10790, 11637, 12516, 13427, 14370, 15345, 16352, 17391, 18462, 19565, 20700, 21867, 23066, 24297, 25560

Offset: 1

Vincenzo Librandi, Mar 01 2009

The identity (2048*n^2 - 128*n + 1)^2 - (16*n^2 - n)*(512*n - 16)^2 = 1 can be written as A157448(n)^2 - a(n)*A157447(n)^2 = 1. - Vincenzo Librandi, Jan 26 2012
This is the case s=4 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - Bruno Berselli, Jan 26 2012
Sequence found by reading the line from 15, in the direction 15, 62, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
The continued fraction expansion of sqrt(a(n)) is [4n-1; {1, 6, 1, 8n-2}]. For n=1, this collapses to [3; {1, 6}]. - Magus K. Chu, Sep 22 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A157447, A157448.

I:=[15, 62, 141]; [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, Jan 26 2012

LinearRecurrence[{3,-3,1},{15,62,141},40] (* Vincenzo Librandi, Jan 26 2012 *)

for(n=1, 22, print1(16*n^2 - n", ")); \\ Vincenzo Librandi, Jan 26 2012

G.f.: x*(15 + 17*x)/(1-x)^3. - Vincenzo Librandi, Jan 26 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 26 2012

A157447 a(n) = 512*n - 16.

Original entry on oeis.org

496, 1008, 1520, 2032, 2544, 3056, 3568, 4080, 4592, 5104, 5616, 6128, 6640, 7152, 7664, 8176, 8688, 9200, 9712, 10224, 10736, 11248, 11760, 12272, 12784, 13296, 13808, 14320, 14832, 15344, 15856, 16368, 16880, 17392, 17904, 18416, 18928

Offset: 1

Vincenzo Librandi, Mar 01 2009

The identity (2048*n^2 - 128*n + 1)^2 - (16*n^2 - n)*(512*n - 16)^2 = 1 can be written as A157448(n)^2 - A157446(n)*a(n)^2 = 1 (see also second comment at A157446). - Vincenzo Librandi, Jan 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A157446, A157448.

I:=[496, 1008]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jan 26 2012

LinearRecurrence[{2,-1},{496,1008},40] (* Vincenzo Librandi, Jan 26 2012 *)
512*Range[50]-16 (* Harvey P. Dale, Apr 09 2019 *)

for(n=1, 22, print1(512*n - 16", ")); \\ Vincenzo Librandi, Jan 26 2012

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jan 26 2012

G.f.: x*(16*x + 496)/(x-1)^2. - Vincenzo Librandi, Jan 26 2012

cp's OEIS Frontend

A157446 a(n) = 16*n^2 - n.

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