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A133216 Integers that are simultaneously triangular (A000217) and decagonal (A001107).

%I A133216 #24 Apr 04 2019 13:44:23
%S A133216 0,1,10,1540,11935,1777555,13773376,2051297326,15894464365,
%T A133216 2367195337045,18342198104230,2731741367653000,21166880717817451,
%U A133216 3152427171076225351,24426562006163234620,3637898223680596402450,28188231388231654934425,4198131397700237172202345
%N A133216 Integers that are simultaneously triangular (A000217) and decagonal (A001107).
%C A133216 Positive terms are of the form (m^2-9)/16 where m runs over the elements of A077443 that are congruent to 5 modulo 8. Correspondingly, for n>1, sqrt(16*a(n)+9) form a subsequence of A077443, while sqrt(8*a(n)+1) form a subsequence of A077442 with indices congruent to 2,3 modulo 4. [_Max Alekseyev_]
%H A133216 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1154, -1154, -1, 1).
%F A133216 a(n) = A000217(A133218(n)) = A001107(A133217(n)).
%F A133216 For n>5, a(n) = 1154*a(n-2) - a(n-4) + 396.
%F A133216 For n>6, a(n) = a(n-1) + 1154*a(n-2) - 1154*a(n-3) - a(n-4) + a(n-5).
%F A133216 For n>1, a(n) = 1/64 * ( (9 + 4* sqrt(2)*(-1)^n)*(1+sqrt(2))^(4*n-6) + (9 - 4* sqrt(2)*(-1)^n)*(1-sqrt(2))^(4*n-6) - 22).
%F A133216 a(n) = floor ( 1/64 * (9 + 4*sqrt(2)*(-1)^n) * (1+sqrt(2))^(4*n-6) ).
%F A133216 G.f.: (x^5 + 9*x^4 + 376*x^3 + 9*x^2 + x)/((1 - x)*(x^2 - 34*x + 1)*(x^2 + 34*x + 1)). [corrected by _Peter Luschny_, Apr 04 2019]
%F A133216 Lim (n -> Infinity, a(2n+1)/a(2n)) = (1/49)*(3649+2580*sqrt(2)).
%F A133216 Lim (n -> Infinity, a(2n)/a(2n-1)) = (1/49)*(193+132*sqrt(2)).
%e A133216 The initial terms of the sequences of triangular (A000217) and decagonal (A001107) numbers are 0, 1, 3, 6, 10, 15, ... and 0, 1, 10, 27, ... respectively. As the third number which is common to both sequences is 10, we have a(3) = 10.
%t A133216 LinearRecurrence[{1, 1154, -1154, -1, 1} , {0, 1, 10, 1540, 11935, 1777555}, 17] (* first term 0 corrected by _Georg Fischer_, Apr 02 2019 *)
%Y A133216 Cf. A000217, A001107, A133217, A133218, A077443, A077442.
%K A133216 nonn
%O A133216 1,3
%A A133216 _Richard Choulet_, Oct 11 2007; _Ant King_, Nov 04 2011
%E A133216 Entry revised by _N. J. A. Sloane_, Nov 06 2011
%E A133216 Term 0 prepended and entry revised accordingly by _Max Alekseyev_, Nov 06 2011