OFFSET
1,2
COMMENTS
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).
LINKS
J. RamÃrez, V. Sirvent, A note on the k-Narayana sequence, Annales Mathematicae et Informaticae 45 (2015) pp. 91-105.
Wikipedia, Hofstadter sequence.
FORMULA
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).
Conjectures from Colin Barker, Mar 28 2018: (Start)
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)).
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.
(End)
EXAMPLE
a(8) = 14 + 15 + 16 + 17 + 18 + 19 = (2N(8)+N(6)+1)*N(6)/2 = 99.
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Mar 28 2018
STATUS
approved