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A201008 Triangular numbers, T(m), that are five-sixths of another triangular number: T(m) such that 6*T(m)=5*T(k) for some k.

%I A201008 #43 Nov 08 2024 11:37:49
%S A201008 0,55,26565,12804330,6171660550,2974727580825,1433812522297155,
%T A201008 691094661019647940,333106192798948009980,160556493834431921162475,
%U A201008 77387896922003387052303025,37300805759911798127288895630
%N A201008 Triangular numbers, T(m), that are five-sixths of another triangular number: T(m) such that 6*T(m)=5*T(k) for some k.
%H A201008 Vincenzo Librandi, <a href="/A201008/b201008.txt">Table of n, a(n) for n = 0..229</a>
%H A201008 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (483,-483,1).
%F A201008 For n > 1, a(n) = 482*a(n-1) - a(n-2) + 55. See A200993 for generalization.
%F A201008 From _Bruno Berselli_, Dec 21 2011: (Start)
%F A201008 G.f.: 55*x/((1-x)*(1-482*x+x^2)).
%F A201008 a(n) = a(-n-1) = 483*a(n-1)-483*a(n-2)+a(n-3).
%F A201008 a(n) = ((11-2*r)^(2*n+1)+(11+2*r)^(2*n+1)-22)/192, where r=sqrt(30). (End)
%e A201008 6*0 = 5*0;
%e A201008 6*55 = 5*66;
%e A201008 6*26565 = 5*31878;
%e A201008 6*12804330 = 5*15365196.
%t A201008 LinearRecurrence[{483,-483,1},{0,55,26565},30] (* _Vincenzo Librandi_, Dec 22 2011 *)
%o A201008 (Maxima) makelist(expand(((11-2*sqrt(30))^(2*n+1)+(11+2*sqrt(30))^(2*n+1)-22)/192), n, 0, 11); /* _Bruno Berselli_, Dec 21 2011 */
%o A201008 (Magma) I:=[0, 55, 26565]; [n le 3 select I[n] else 483*Self(n-1)-483*Self(n-2)+Self(n-3): n in [1..15]]; // _Vincenzo Librandi_, Dec 22 2011
%o A201008 (PARI) concat(0,Vec(55/(1-x)/(1-482*x+x^2)+O(x^98))) \\ _Charles R Greathouse IV_, Dec 23 2011
%Y A201008 Cf. A001652, A029549, A053141, A075528, A200993-A201008.
%K A201008 nonn,easy
%O A201008 0,2
%A A201008 _Charlie Marion_, Dec 20 2011
%E A201008 a(11) corrected by _Bruno Berselli_, Dec 21 2011
%E A201008 a(6) corrected by _Vincenzo Librandi_, Dec 22 2011