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A301809 Group the natural numbers such that the first group is (1) then (2),(3),(4,5),(6,7,8),... with the n-th group containing F(n) sequential terms where F(n) is the n-th Fibonacci number (A000045(n)). Sequence gives the sum of terms in the n-th group.

%I A301809 #38 Jul 07 2025 11:50:57
%S A301809 1,2,3,9,21,55,140,364,945,2465,6435,16821,43992,115102,301223,788425,
%T A301809 2063817,5402651,14143524,37026936,96935685,253777537,664392743,
%U A301809 1739393929,4553778096,11921922650,31211961195,81713914569,213929707485,560075086495,1466295355580,3838810662436,10050136117497
%N A301809 Group the natural numbers such that the first group is (1) then (2),(3),(4,5),(6,7,8),... with the n-th group containing F(n) sequential terms where F(n) is the n-th Fibonacci number (A000045(n)). Sequence gives the sum of terms in the n-th group.
%C A301809 a(n) is the sum of all nodes at height n-1 within a binary tree structure recursively built from the Hofstadter G-sequence (see comments for A005206).
%H A301809 Colin Barker, <a href="/A301809/b301809.txt">Table of n, a(n) for n = 1..1000</a>
%H A301809 Wikipedia, <a href="https://www.wikipedia.org/wiki/Hofstadter_sequence">Hofstadter sequence</a>.
%H A301809 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-5,-1, 1).
%F A301809 a(1) = 1 and for n > 1, a(n) = (F(n+2)+1)*F(n-1)/2, where F(n) is the n-th Fibonacci number (A000045).
%F A301809 From _Colin Barker_, Mar 27 2018: (Start)
%F A301809 G.f.: x*(1 - x)*(1 - 4*x^2 - x^3 + x^4) / ((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)).
%F A301809 a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5) for n>6. (End)
%F A301809 a(n) = A033192(n+1) - A033192(n) for n > 1. - _J.S. Seneschal_, Jul 07 2025
%e A301809 a(7) = 14 + 15 + 16 + ... + 21 = (F(9)+1)*F(6)/2 = 140.
%t A301809 a[n_] := If[n==1, 1, (Fibonacci[n+2]+1)Fibonacci[n-1]/2]; Array[a, 50]
%t A301809 Join[{1}, LinearRecurrence[{3, 1, -5, -1, 1}, {2, 3, 9, 21, 55}, 40]] (* _Vincenzo Librandi_, Apr 18 2018 *)
%o A301809 (Magma) [1] cat [(Fibonacci(n+2)+1)*Fibonacci(n-1) div 2 : n in [2..35] ]; // _Vincenzo Librandi_, Apr 18 2018
%o A301809 (PARI) a(n) = if (n==1, 1, (fibonacci(n+2)+1)*fibonacci(n-1)/2); \\ _Michel Marcus_, Apr 21 2018
%o A301809 (PARI) Vec(x*(1 - x)*(1 - 4*x^2 - x^3 + x^4) / ((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)) + O(x^60)) \\ _Colin Barker_, May 11 2018
%Y A301809 Cf. A000045, A005206, A122931.
%Y A301809 Cf. A000217, A033192.
%K A301809 nonn,easy
%O A301809 1,2
%A A301809 _Frank M Jackson_, Mar 27 2018