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%I #15 Jan 19 2019 04:15:43
%S 3,8,21,56,148,393,1041,2761,7318,19403,51436,136366,361513,958413,
%T 2540831,6735996,17857733,47342548,125509476,332737401
%N Sum of terms in level n of TRIP - Stern sequence associated with permutation triple (e,13,132).
%H I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, M. Stoffregen, <a href="https://arxiv.org/abs/1509.05239">Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences</a>, arXiv:1509.05239 [math.CO], 17 Sep 2015.
%F Conjecture: G.f.: ( -3-5*x-x^2 ) / ( -1+x+4*x^2+x^3 ). - _R. J. Mathar_, Dec 02 2016
%p A278616T := proc(n)
%p option remember;
%p local an, nrecur ;
%p if n = 1 then
%p [1, 1, 1] ;
%p else
%p an := procname(floor(n/2)) ;
%p if type(n, 'even') then
%p # apply F0
%p [op(1, an)+ op(3, an),op(3, an), op(2, an)] ;
%p else
%p # apply F1
%p [op(2, an), op(1, an)+ op(3, an),op(1, an)] ;
%p end if;
%p end if;
%p end proc;
%p A278616 := proc(n)
%p local a, l;
%p a := 0 ;
%p for l from 2^n to 2^(n+1)-1 do
%p L := A278616T(l) ;
%p # a := a+ L[1]+L[2]+L[3] ;
%p a := a+ L[2];
%p end do:
%p a ;
%p end proc: # _R. J. Mathar_, Dec 02 2016
%t AT[n_] := AT[n] = Module[{an}, If[n == 1, {1, 1, 1}, an = AT[Floor[n/2]]; If[EvenQ[n], {an[[1]] + an[[3]], an[[3]], an[[2]]}, {an[[2]], an[[1]] + an[[3]], an[[1]] } ]]];
%t a[n_] := a[n] = Module[{a = 0, l, L}, For[l = 2^n, l <= 2^(n + 1) - 1, l++, L = AT[l]; a = a + L[[1]] + L[[2]] + L[[3]]]; a];
%t Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 19}] (* _Jean-François Alcover_, Nov 22 2017, after _R. J. Mathar_ *)
%Y Cf. A278612, A278613, A278614, A278615.
%K nonn,more
%O 0,1
%A _Ilya Amburg_, Nov 23 2016
%E More terms from _R. J. Mathar_, Dec 02 2016
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