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A301877 Group the natural numbers into groups (1),(2),(3),(4),(5,6),(7,8,9),... so that the n-th group contains N(n) terms, where N(n) is the Narayana's cows sequence (A000930). Sequence contains the sum of the terms in the n-th group.

%I A301877 #8 Jun 03 2018 11:33:06
%S A301877 1,2,3,4,11,24,46,99,216,455,969,2086,4469,9570,20548,44118,94689,
%T A301877 203318,436653,937720,2013884,4325391,9290080,19953405,42857019,
%U A301877 92051300,197714721,424668244,912140480,1959179226,4208109535,9038581200,19413940167,41699153408,89565528714,192377651011,413207678264
%N A301877 Group the natural numbers into groups (1),(2),(3),(4),(5,6),(7,8,9),... so that the n-th group contains N(n) terms, where N(n) is the Narayana's cows sequence (A000930). Sequence contains the sum of the terms in the n-th group.
%C A301877 a(n) is the sum of all nodes at height n-1 within a binary tree structure recursively built from the Hofstadter H-sequence (see comments for A005374).
%H A301877 J. Ramírez, V. Sirvent, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_45_from91to105.pdf">A note on the k-Narayana sequence</a>, Annales Mathematicae et Informaticae 45 (2015) pp. 91-105.
%H A301877 Wikipedia, <a href="https://www.wikipedia.org/wiki/Hofstadter_sequence">Hofstadter sequence</a>.
%F A301877 a(1) = 1 and for n > 1, a(n) = (2N(n)+N(n-2)+1)*N(n-2)/2, where N(n) is the Narayana's cows sequence (A000930).
%F A301877 Conjectures from _Colin Barker_, Mar 28 2018: (Start)
%F A301877 G.f.: x*(1 - x)*(1 + x - 5*x^3 - 5*x^4 - 3*x^5 + x^6 + 2*x^7 + x^8) / ((1 + x^2 - x^3)*(1 - x - x^3)*(1 - x - 2*x^2 - x^3)).
%F A301877 a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 6*a(n-4) - 6*a(n-5) + 5*a(n-6) - a(n-7) for n>10.
%F A301877 (End)
%e A301877 a(8) = 14 + 15 + 16 + 17 + 18 + 19 = (2N(8)+N(6)+1)*N(6)/2 = 99.
%t A301877 Narayana[n_] := Which[n==0, 1, n==1, 1, n==2, 1, True, Narayana[n]=Narayana[n-3] + Narayana[n-1]]; a[n_] := If[n==1, 1, (2 Narayana[n]+Narayana[n-2]+1)Narayana[n-2]/2]; Array[a, 50]
%Y A301877 Cf. A000930, A005374.
%K A301877 nonn
%O A301877 1,2
%A A301877 _Frank M Jackson_, Mar 28 2018