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.

1, 2, 3, 4, 11, 24, 46, 99, 216, 455, 969, 2086, 4469, 9570, 20548, 44118, 94689, 203318, 436653, 937720, 2013884, 4325391, 9290080, 19953405, 42857019, 92051300, 197714721, 424668244, 912140480, 1959179226, 4208109535, 9038581200, 19413940167, 41699153408, 89565528714, 192377651011, 413207678264

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

Table of n, a(n) for n=1..37.

J. Ramírez, 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

Cf. A000930, A005374.

Sequence in context: A037396 A037432 A357360 * A369438 A221172 A116054

Adjacent sequences: A301874 A301875 A301876 * A301878 A301879 A301880

Keyword

nonn

Author

Frank M Jackson, Mar 28 2018

Status

approved