A154357 a(n) = 25*n^2 - 14*n + 2.

2, 13, 74, 185, 346, 557, 818, 1129, 1490, 1901, 2362, 2873, 3434, 4045, 4706, 5417, 6178, 6989, 7850, 8761, 9722, 10733, 11794, 12905, 14066, 15277, 16538, 17849, 19210, 20621, 22082, 23593, 25154, 26765, 28426, 30137, 31898, 33709, 35570, 37481, 39442, 41453, 43514

Offset

0,1

Comments

The identity (1250*n^2 - 700*n + 99)^2 - (25*n^2 - 14*n + 2)*(250*n - 70)^2 = 1 can be written as A154359(n)^2 - a(n)*A154361(n)^2 = 1.

Numbers of the form (4*n-1)^2 + (3*n-1)^2. - Bruno Berselli, Dec 11 2011

From Bruno Berselli, Dec 13 2011: (Start)

More generally, considering together this sequence and A154355, A154358-A154361, for

r = (1/4)*(1250*(n-1)*(n-2) + 75*(2*n-3)(-1)^n + 321) with n>=0, i.e. the interleaving of A154358 and A154359 (649, 99, 99, 649, 2049, 3699,...)

s = (5/2)*(50*n+3*(-1)^n-75), the interleaving of A154360 and A154361 (-180, -70, 70, 180, 320, 430,...)

t = (1/8)*(50*(n-1)*(n-2) + 3*(2*n-3)*(-1)^n + 13), the interleaving of A154355 and A154357 (13, 2, 2, 13, 41, 74,...)

we verify that r^2 - t*s^2 = 1.

For n even we obtain (1250*n^2 - 1800*n + 649)^2 - (25*n^2 - 36*n + 13)*(250*n - 180)^2 = 1; for n odd we have the identity shown in the first comment. (End)

sqrt(A154357(n)) for n >= 1 has the continued fraction x; [1 1 1 1 2x] where x = 5n - 2 (the part in [] being repeated). - Robert Israel, May 26 2013

For n >= 1, the continued fraction expansion of sqrt(4*a(n)) is [10n-3; {4, 1, 5n-3, 1, 4, 20n-6}] - Magus K. Chu, Sep 16 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: (2 + 7*x + 41*x^2)/(1-x)^3. - R. J. Mathar, Jan 05 2011

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Feb 08 2012

E.g.f.: (2 + 11*x + 25*x^2)*exp(x). - G. C. Greubel, Sep 14 2016

Mathematica

LinearRecurrence[{3, -3, 1}, {2, 13, 74}, 40] (* Vincenzo Librandi, Feb 08 2012 *)

Prog

(PARI) a(n)=25*n^2-14*n+2 \\ Charles R Greathouse IV, Dec 23 2011
(Magma) [25*n^2-14*n+2: n in [0..40]]; // Bruno Berselli, Sep 15 2016

Crossrefs

Cf. A154355, A154358, A154359, A154360, A154361.

Sequence in context: A004027 A263840 A307288 * A161130 A192700 A007509

Adjacent sequences: A154354 A154355 A154356 * A154358 A154359 A154360

Keyword

nonn,easy

Author

Vincenzo Librandi, Jan 08 2009

Extensions

One entry and offset corrected by R. J. Mathar, Jan 05 2011

First comment rewritten by Bruno Berselli, Dec 11 2011

Status

approved