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A278614 Sum of terms in level n of TRIP - Stern sequence associated with permutation triple (e,12,12).

%I A278614 #22 Jan 19 2019 04:12:56
%S A278614 3,8,22,62,176,502,1434,4100,11726,33542,95952,274494,785266,2246484,
%T A278614 6426742,18385646,52597744,150471910,430470890,1231493604
%N A278614 Sum of terms in level n of TRIP -  Stern sequence associated with permutation triple (e,12,12).
%H A278614 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 A278614 Conjecture: G.f.: ( 3-x-5*x^2 ) / ( 1-3*x-x^2+4*x^3 ). - _R. J. Mathar_, Dec 02 2016
%p A278614 A278614T := proc(n)
%p A278614     option remember;
%p A278614     local an, nrecur ;
%p A278614     if n = 1 then
%p A278614         [1, 1, 1] ;
%p A278614     else
%p A278614         an := procname(floor(n/2)) ;
%p A278614         if type(n, 'even') then
%p A278614             # apply F0
%p A278614             [op(3, an), op(2, an),op(1, an)+ op(3, an)] ;
%p A278614         else
%p A278614             # apply F1
%p A278614             [op(2, an), op(1, an), op(1, an)+op(3, an)] ;
%p A278614         end if;
%p A278614     end if;
%p A278614 end proc;
%p A278614 A278614 := proc(n)
%p A278614     local a, l;
%p A278614     a := 0 ;
%p A278614     for l from 2^n to 2^(n+1)-1 do
%p A278614         L := A278614T(l) ;
%p A278614         a := a+ L[1]+L[2]+L[3] ;
%p A278614     end do:
%p A278614     a ;
%p A278614 end proc: # _R. J. Mathar_, Dec 02 2016
%t A278614 A278614T[n_] := A278614T[n] = Module[{an}, If[n == 1, {1, 1, 1}, an = A271487T[Floor[n/2]]; If[EvenQ[n], {an[[3]], an[[2]], an[[1]] + an[[3]]}, {an[[2]], an[[1]], an[[1]] + an[[3]]}]]];
%t A278614 a[n_] := a[n] = Module[{a = 0, l, L}, For[l = 2^n, l <= 2^(n + 1) - 1, l++, L = A278614T[l]; a = a + L[[1]] + L[[2]] + L[[3]]]; a];
%t A278614 Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 19}] (* _Jean-François Alcover_, Nov 20 2017, after _R. J. Mathar_ *)
%Y A278614 Cf. A278612, A278613, A278615, A278616.
%K A278614 nonn,more
%O A278614 0,1
%A A278614 _Ilya Amburg_, Nov 23 2016
%E A278614 More terms from _R. J. Mathar_, Dec 02 2016