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A278616 Sum of terms in level n of TRIP - Stern sequence associated with permutation triple (e,13,132).

%I A278616 #15 Jan 19 2019 04:15:43
%S A278616 3,8,21,56,148,393,1041,2761,7318,19403,51436,136366,361513,958413,
%T A278616 2540831,6735996,17857733,47342548,125509476,332737401
%N A278616 Sum of terms in level n of TRIP -  Stern sequence associated with permutation triple (e,13,132).
%H A278616 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 A278616 Conjecture: G.f.: ( -3-5*x-x^2 ) / ( -1+x+4*x^2+x^3 ). - _R. J. Mathar_, Dec 02 2016
%p A278616 A278616T := proc(n)
%p A278616     option remember;
%p A278616     local an, nrecur ;
%p A278616     if n = 1 then
%p A278616         [1, 1, 1] ;
%p A278616     else
%p A278616         an := procname(floor(n/2)) ;
%p A278616         if type(n, 'even') then
%p A278616             # apply F0
%p A278616             [op(1, an)+ op(3, an),op(3, an), op(2, an)] ;
%p A278616         else
%p A278616             # apply F1
%p A278616             [op(2, an), op(1, an)+ op(3, an),op(1, an)] ;
%p A278616         end if;
%p A278616     end if;
%p A278616 end proc;
%p A278616 A278616 := proc(n)
%p A278616     local a, l;
%p A278616     a := 0 ;
%p A278616     for l from 2^n to 2^(n+1)-1 do
%p A278616         L := A278616T(l) ;
%p A278616         # a := a+ L[1]+L[2]+L[3] ;
%p A278616         a := a+ L[2];
%p A278616     end do:
%p A278616     a ;
%p A278616 end proc: # _R. J. Mathar_, Dec 02 2016
%t A278616 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 A278616 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 A278616 Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 19}] (* _Jean-François Alcover_, Nov 22 2017, after _R. J. Mathar_ *)
%Y A278616 Cf. A278612, A278613, A278614, A278615.
%K A278616 nonn,more
%O A278616 0,1
%A A278616 _Ilya Amburg_, Nov 23 2016
%E A278616 More terms from _R. J. Mathar_, Dec 02 2016