login
Numbers n such that 2*n+1 and 3*n+1 are both triangular numbers.
5

%I #21 Jul 08 2016 11:13:15

%S 45,4455,436590,42781410,4192141635,410787098865,40252943547180,

%T 3944377680524820,386508759747885225,37873914077612227275,

%U 3711257070846250387770,363665319028854925774230,35635490007756936475486815,3491914355441150919671933685

%N Numbers n such that 2*n+1 and 3*n+1 are both triangular numbers.

%C Inspired by A274579.

%C a(n+1) / a(n) goes to 49 + 20*sqrt(6) when n goes to infinity.

%C Intersection of A045943 and A074377. - _Colin Barker_, Jun 30 2016

%H Colin Barker, <a href="/A274603/b274603.txt">Table of n, a(n) for n = 1..500</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).

%F From _Colin Barker_, Jun 30 2016: (Start)

%F a(n) = 99*a(n-1)-99*a(n-2)+a(n-3) for n>3.

%F G.f.: 45*x / ((1-x)*(1-98*x+x^2)).

%F (End)

%e 45 is a term because 2*45 + 1 = 91 and 3*45 + 1 = 136 are both triangular numbers.

%o (PARI) isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(3*n+1, 3);

%o (PARI) Vec(45*x/((1-x)*(1-98*x+x^2)) + O(x^20)) \\ _Colin Barker_, Jun 30 2016

%Y Cf. A000217, A045943, A074377, A274579.

%K nonn,easy

%O 1,1

%A _Altug Alkan_, Jun 30 2016

%E More terms from _Colin Barker_, Jun 30 2016