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%I #29 Sep 08 2022 08:46:05
%S 0,1,1,9,25,313,1913,41977,531705,22023929,566489849,45671496441,
%T 2366013917945,376506912762617,39141278944373497,12376519796349807353,
%U 2577539376694811306745,1624792742123856760679161,677311275106408471956040441,852536648457739021814912002809
%N Recurrence a(n) = a(n-1) + 2^n*a(n-2) with a(0)=0, a(1)=1.
%C Generally (if p>0, q>1), recurrence a(n) = b*a(n-1) + (p*q^n+d)*a(n-2), a(n) is asymptotic to c*q^(n^2/4)*(p*q)^(n/2), where c is for fixed parameters b, p, d, q, a(0), a(1) constant, independent on n.
%H Vincenzo Librandi, <a href="/A228465/b228465.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) ~ c * 2^(n^2/4 + n/2), where c = 0.548441579870783378573455400152590154... if n is even and c = 0.800417244834941368929416800341853541... if n is odd.
%F a(n) = Sum_{k=1..floor(n/2+1/2)} qbinomial(n-k,k-1)*2^(k^2-1), where q-binomial is triangle A022166, that is, with q=2. - _Vladimir Kruchinin_, Jan 20 2020
%t RecurrenceTable[{a[n]==a[n-1]+2^n*a[n-2],a[0]==0,a[1]==1},a,{n,0,20}]
%t (* Alternative: *)
%t a[n_] := Sum[2^(k^2-1) QBinomial[n - k , k - 1, 2], {k, 1, n}];
%t Table[a[n], {n, 0, 19}] (* After _Vladimir Kruchinin_. _Peter Luschny_, Jan 20 2020 *)
%o (Magma) [n le 2 select (n-1) else Self(n-1)+Self(n-2)*2^(n-1): n in [1..20]]; // _Vincenzo Librandi_, Aug 23 2013
%o (SageMath)
%o def a(n):
%o return sum(2^(k^2 - 1)*q_binomial(n-k , k-1, 2) for k in (1..n))
%o print([a(n) for n in range(20)]) # _Peter Luschny_, Jan 20 2020
%Y Cf. A006116, A006117, A006118, A006119, A006120, A006121, A006122.
%K nonn
%O 0,4
%A _Vaclav Kotesovec_, Aug 22 2013
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