OFFSET
1,2
COMMENTS
The equation a(t)*(3*a(t)-2) = m*m is equivalent to the Pell equation (3*a(t)-1)*(3*a(t)-1) - 3*m*m = 1. -Paul Weisenhorn,May 12 2009
As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> infinity} a(n)/a(n-1) = (2 + sqrt(3))^2 = 7 + 4 * sqrt(3). -Ant King,Nov 16 2011
Also numbers n such that the octagonal number N(n) is equal to the sum of two consecutive triangular numbers. -Colin Barker,Dec 11 2014
Also nonnegative integers y in the solutions to 2*x^2 - 6*y^2 + 4*x + 4*y + 2 + 2 = 0, the corresponding values of x beingA251963.-Colin Barker,Dec 11 2014
LINKS
Vincenzo Librandi,Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics,Octagonal Square Number.
Index entries for linear recurrences with constant coefficients,signature (15,-15,1).
FORMULA
Nearest integer to (1/6) * (2+sqrt(3))^(2n-1). -Ralf Stephan,Feb 24 2004
a(n) =A001835(n)^2. -Lekraj Beedassy,Jul 21 2006
FromPaul Weisenhorn,May 12 2009: (Start)
With A=(2+sqrt(3))^2=7+4*sqrt(3) the equation x*x-3*m*m=1 has solutions
x(t) + sqrt(3)*m(t) = (2+sqrt(3))*A^t and the recurrences
x(t+2) = 14*x(t+1) - x(t) with <x(t)> = 2, 26, 362, 5042
m(t+2) = 14*m(t+1) - m(t) with <m(t)> = 1, 15, 209, 2911
a(t+2) = 14*a(t+1) - a(t) - 4 with <a(t)> = 1, 9, 121, as above. (End)
FromAnt King,Nov 15 2011: (Start)
a(n) = 14*a(n-1) - a(n-2) - 4.
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3).
a(n) = (1/6)*( (2+sqrt(3))^(2n-1) + (2-sqrt(3))^(2n-1) + 2 ).
a(n) = ceiling( (1/6)*(2 + sqrt(3))^(2n-1) ).
a(n) = (1/6)*( (tan(5*Pi/12))^(2n-1) + (tan(Pi/12))^(2n-1) + 2 ).
a(n) = ceiling ( (1/6)*(tan(5*Pi/12))^(2n-1) ).
G.f.: x*(1-6*x+x^2) / ((1-x)*(1-14*x+x^2)). (End)
a(n) =A006253(2n-2). -Andrey Goder,Oct 17 2021
MATHEMATICA
LinearRecurrence[ {15, -15, 1}, {1, 9, 121}, 17 ] (*Ant King,Nov 16 2011 *)
CoefficientList[Series[x (1-6x+x^2)/((1-x)(1-14x+x^2)), {x, 0, 30}], x] (*Harvey P. Dale,Sep 01 2021 *)
PROG
(Magma) I:=[1, 9, 121]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..20]]; //Vincenzo Librandi,Nov 17 2011
(PARI) Vec(x*(1-6*x+x^2) / ((1-x)*(1-14*x+x^2)) + O(x^100)) \\Colin Barker,Dec 11 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved