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%I #14 Dec 05 2016 05:27:42
%S 4455,30537,461938302,3166172226,47894687058501,328275068740587,
%T 4965816943137597372,34036215673995404100,514865832250497683700195,
%U 3528942913182916419190605,53382319214430283898266055610,365887859090594924500524938502
%N Numbers k such that 2*k+1 and 10*k+1 are both triangular numbers (A000217).
%H Colin Barker, <a href="/A279042/b279042.txt">Table of n, a(n) for n = 1..350</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,103682,-103682,-1,1).
%F a(n) = a(n-1) + 103682*a(n-2) - 103682*a(n-3) - a(n-4) + a(n-5) for n>5.
%F G.f.: 81*x*(55 + 322*x + 55*x^2) / ((1 - x)*(1 - 322*x + x^2)*(1 + 322*x + x^2)).
%e 4455 is in the sequence because 2*4455+1 = 8911 and 10*4455+1 = 44551 are both triangular numbers.
%t LinearRecurrence[{1, 103682, -103682, -1, 1}, {4455, 30537, 461938302, 3166172226, 47894687058501}, 20] (* _Vincenzo Librandi_, Dec 05 2016 *)
%o (PARI) Vec(81*x*(55 + 322*x + 55*x^2) / ((1 - x)*(1 - 322*x + x^2)*(1 + 322*x + x^2)) + O(x^15))
%o (PARI) isok(k) = ispolygonal(2*k+1, 3) & ispolygonal(10*k+1, 3)
%Y Cf. A000217, A124174, A274579, A274603, A274680, A274756, A274832.
%K nonn,easy
%O 1,1
%A _Colin Barker_, Dec 04 2016