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%I #22 Jan 19 2020 06:26:33
%S 1,3,14,124,2096,69056,4486656,578711552,148724449280,76295068188672,
%T 78202296743231488,160236429879963287552,656488575092059763900416,
%U 5378610735570941915498020864,88128536246001466497105446043648
%N a(n) = (2^n)*a(n-1) + (2^(n-1))*a(n-2), a(0)=1, a(1)=3.
%C This is the sequence of numerators of self-convergents to the number 1.40861... whose self-continued fraction is (1,2,4,8,16,...)=A000079. See A096658 for denominators and A096654 for definitions.
%F a(n) is asymptotic to c*2^(n(n+1)/2) where c = 2.1726687508496636560169136... - _Benoit Cloitre_, Jul 02 2004
%F c = 1 + Sum_{k>=1} (Product_{j=1..k} 1/(2^(j-1)*(2^j-1))) = 2.172668750849663656016913609859312820656436935109608860295... . - _Vaclav Kotesovec_, Nov 27 2015
%F a(n) = Sum_{k=0..n+1} q-binomial(n-k+1,k)*2^(binomial(n-k+1,2)+binomial(k,2)), where q-binomial is triangle A022166, that is, with q=2. - _Vladimir Kruchinin_, Jan 19 2020
%e a(2)=4*3+2*1=14, a(3)=8*14+4*3=124.
%t a[0] = 1; a[1] = 3; a[n_] := (2^n)*a[n-1] + (2^(n-1))*a[n-2]; Table[ a[n], {n, 0, 14}] (* _Robert G. Wilson v_, Jul 03 2004 *)
%t b[n_, k_] := k^2 - k (1 + n) + n (1 + n)/2;
%t a[n_] := Sum[2^b[n, k] QBinomial[n - k + 1, k, 2], {k, 0, n + 1}] ;
%t Table[a[n], {n, 0, 14}] (* After _Vladimir Kruchinin_, _Peter Luschny_, Jan 19 2020 *)
%Y Cf. A000079, A096654, A096658.
%K nonn
%O 0,2
%A _Clark Kimberling_, Jul 01 2004
%E More terms from _Benoit Cloitre_, Jul 02 2004
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