|
%I #9 Jul 06 2023 15:17:08
%S 1,1,1,3,5,9,17,29,51,85,145,239,401,657,1087,1773,2911,4735,7731,
%T 12551,20427,33123,53789,87151,141341,228893,370891,600441,972419,
%U 1573947,2548139,4123859,6674909,10801679,17481323,28287737,45776791,74072259,119861601
%N a(n) = a(n-1) + a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.
%C a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). Guide to related sequences:
%C ****
%C sequence recurrence a(0),a(1),a(2)
%C A298338 a(n) = a(n-1)+a(n-2)+a([n/2]) 1,1,1
%C A298339 a(n) = a(n-1)+a(n-2)+a([n/2]) 1,2,3
%C A298400 a(n) = a(n-1)+a(n-2)-a([n/2]) 1,1,1
%C A298401 a(n) = a(n-1)+a(n-2)-a([n/2]) 1,2,3
%C A298340 a(n) = a(n-1)+a(n-2)+a([n/3]) 1,1,1
%C A298341 a(n) = a(n-1)+a(n-2)+a([n/3]) 1,2,3
%C A298342 a(n) = a(n-1)+a(n-2)+a([2n/3]) 1,1,1
%C A298343 a(n) = a(n-1)+a(n-2)+a([2n/3]) 1,2,3
%C A298344 a(n) = a(n-1)+a(n-2)+a([n/3]) + a([2n/3]) 1,1,1
%C A298345 a(n) = a(n-1)+a(n-2)+a([n/3]) + a([2n/3]) 1,2,3
%C A298346 a(n) = a(n-1)+a(n-2)+2 a([n/2]) 1,1,1
%C A298347 a(n) = a(n-1)+a(n-2)+2 a([n/2]) 1,2,3
%C A298348 a(n) = a(n-1)+a(n-2)+2 a([(n+1)/2]) 1,1,1
%C A298349 a(n) = a(n-1)+a(n-2)+2 a([(n+1)/2]) 1,2,3
%C A298350 a(n) = a(n-1)+a(n-2)+2 a(ceiling(n/2)) 1,1,1
%C A298351 a(n) = a(n-1)+a(n-2)+2 a(ceiling(n/2)) 1,2,3
%C A298352 a(n) = a(n-1)+a(n-2)+a([(n-1)/2]) 1,1,1
%C A298353 a(n) = a(n-1)+a(n-2)+a([(n-1)/2]) 1,2,3
%C A298354 a(n) = a(n-1)+a(n-2)+2 a([(n-1)/2]) 1,1,1
%C A298355 a(n) = a(n-1)+a(n-2)+2 a([(n-1)/2]) 1,2,3
%C A298356 a(n) = a(n-1)+a(n-2)+a([n/2])+a([n/3]+...+a([n/n}) 1,1,1
%C A298357 a(n) = a(n-1)+a(n-2)+a([n/2])+a([n/3]+...+a([n/n}) 1,2,3
%C A298369 a(n) = a(n-1)+a(n-2)+2a([n/2])+3a([n/3]+...+4 a([n/n}) 1,1,1
%C A298370 a(n) = a(n-1)+a(n-2)+2a([n/2])+3a([n/3]+...+4a([n/n]) 1,2,3
%C A298402 a(n) = 2*a(n-1)-a(n-3)+a([n/2]) 1,1,1
%C A298403 a(n) = 2*a(n-1)-a(n-3)+a([n/2]) 1,2,3
%C A298404 a(n) = 2*a(n-1)-a(n-3)+a(ceiling(n/2)) 1,1,1
%C A298405 a(n) = 2*a(n-1)-a(n-3)+a(ceiling(n/2)) 1,2,3
%C A298406 a(n) = 2*a(n-1)-a(n-3)+a([n/2])+a([n/3]+...+ a([n/n]) 1,1,1
%C A298407 a(n) = 2*a(n-1)-a(n-3)+a([n/2])+a([n/3]+...+ a([n/n]) 1,2,3
%C A298408 a(n) = 2*a(n-1)-a(n-3)+2a([n/2])+3a([n/3]+...+ 4a([n/n]) 1,1,1
%C A298409 a(n) = 2*a(n-1)-a(n-3)+2a([n/2])+3a([n/3]+...+ 4a([n/n]) 1,2,3
%H Clark Kimberling, <a href="/A298338/b298338.txt">Table of n, a(n) for n = 0..1000</a>
%H Evangelos G. Filothodoros, <a href="https://arxiv.org/abs/2306.14652">Strongly coupled fermions in odd dimensions and the running cut-off Lambda_d</a>, arXiv:2306.14652 [hep-th], 2023.
%t a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[n/2]];
%t Table[a[n], {n, 0, 30}] (* A298338 *)
%Y Cf. A001622, A000045, A298339.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, Feb 09 2018
|
