A158058 a(n) = 16*n^2 - 2*n.

14, 60, 138, 248, 390, 564, 770, 1008, 1278, 1580, 1914, 2280, 2678, 3108, 3570, 4064, 4590, 5148, 5738, 6360, 7014, 7700, 8418, 9168, 9950, 10764, 11610, 12488, 13398, 14340, 15314, 16320, 17358, 18428, 19530, 20664, 21830, 23028, 24258, 25520

Offset: 1

Author

Vincenzo Librandi, Mar 12 2009

Keywords

• nonn
• easy

Comments

The identity (16*(n-1) + 15)^2 - (16*n^2 - 2*n)*4^2 = 1 can be written as A125169(n-1)^2 - a(n)*4^2 = 1. - Vincenzo Librandi, Feb 01 2012

Sequence found by reading the line from 14, in the direction 14, 60, ... 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, 2, 1, 8n-2}]. - Magus K. Chu, Nov 08 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(4^2*t-2)).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A125169.

Programs

Magma

[16*n^2-2*n: n in [1..40]]

Maple

seq(16*n^2-2*n,n=1..40); # Nathaniel Johnston, Jun 26 2011

Mathematica

LinearRecurrence[{3,-3,1},{14,60,138},40]

PARI

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

Formula

G.f.: x*(-14 - 18*x)/(x-1)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).