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a(1)=1; thereafter a(n+1) = T(8*a(n)), where T(i)=i*(i+1)/2 is the i-th triangular number.
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%I #28 Apr 25 2023 06:17:13

%S 1,36,41616,55420693056,98286503002057414584576,

%T 309127573515950117423442905473334441338685531136

%N a(1)=1; thereafter a(n+1) = T(8*a(n)), where T(i)=i*(i+1)/2 is the i-th triangular number.

%C All terms are square and triangular; a subsequence of A001110.

%D C. Alsina and R. B. Nelson, Charming Proofs: A Journey into Elegant Mathematics, MAA, 2010. See p. 4.

%F a(n) = (1/8)*sinh(2^(n-2)*arccosh(17))^2. [_Alexander R. Povolotsky_, Aug 14 2011]

%F a(n+2) = 4*a(n+1)*(a(n+1)/(2*a(n))-1)^2, a(1)=1, a(2)=36. [_Alexander R. Povolotsky_, Aug 15 2011]

%F a(n) = A001110(2^(n-1)). - _Ivan N. Ianakiev_, Mar 11 2014

%p T:=n->n*(n+1)/2; t1:=[1]; for n from 1 to 7 do t1:= [op(t1), T(8*t1[nops(t1)])]; od: t1;

%t NestList[(8#(8#+1))/2&,1,7] (* _Harvey P. Dale_, Jan 20 2015 *)

%Y Cf. A000217, A001110, A185096.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Feb 18 2011