%I #9 Aug 01 2015 10:00:57

%S 1,20,667,22646,769285,26133032,887753791,30157495850,1024467105097,

%T 34801724077436,1182234151527715,40161159427864862,

%U 1364297186395877581,46345943178031972880,1574397770866691200327,53483178266289468838226,1816853663282975249299345

%N Indices of decagonal numbers that are also hexagonal.

%C As n increases, this sequence is approximately geometric with common ratio (1+sqrt(2))^4=17+12*sqrt(2).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35, -35, 1).

%F G.f.: x(1-15*x+2*x^2) / ((1-x)*(1-34*x+x^2))

%F a(n) = 35*a(n-1)-35*a(n-2)+a(n-3)

%F a(n) = 34*a(n-1)-a(n-2)-12

%F a(n) = 1/16 *((3-sqrt(2))*(1+sqrt(2))^(4*n-2)+(3+sqrt(2))*(1-sqrt(2))^(4*n-2)+6)

%F a(n) = ceiling(1/16*(3-sqrt(2))*(1+sqrt(2))^(4*n-2))

%e The second decagonal number which is also hexagonal is A001107(20) = 1540. Hence a(2) = 20.

%t LinearRecurrence[{35, -35, 1}, {1, 20, 667}, 17]

%Y Cf. A203134, A203135, A000384, A001107.

%K nonn,easy

%O 1,2

%A _Ant King_, Dec 30 2011