A298338 a(n) = a(n-1) + a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1.

1, 1, 1, 3, 5, 9, 17, 29, 51, 85, 145, 239, 401, 657, 1087, 1773, 2911, 4735, 7731, 12551, 20427, 33123, 53789, 87151, 141341, 228893, 370891, 600441, 972419, 1573947, 2548139, 4123859, 6674909, 10801679, 17481323, 28287737, 45776791, 74072259, 119861601

Offset: 0

Author

Clark Kimberling, Feb 09 2018

Keywords

• nonn
• easy

Comments

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:

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sequence recurrence a(0),a(1),a(2)

A298338 a(n) = a(n-1)+a(n-2)+a([n/2]) 1,1,1

A298339 a(n) = a(n-1)+a(n-2)+a([n/2]) 1,2,3

A298400 a(n) = a(n-1)+a(n-2)-a([n/2]) 1,1,1

A298401 a(n) = a(n-1)+a(n-2)-a([n/2]) 1,2,3

A298340 a(n) = a(n-1)+a(n-2)+a([n/3]) 1,1,1

A298341 a(n) = a(n-1)+a(n-2)+a([n/3]) 1,2,3

A298342 a(n) = a(n-1)+a(n-2)+a([2*n/3]) 1,1,1

A298343 a(n) = a(n-1)+a(n-2)+a([2*n/3]) 1,2,3

A298344 a(n) = a(n-1)+a(n-2)+a([n/3])+a([2*n/3]) 1,1,1

A298345 a(n) = a(n-1)+a(n-2)+a([n/3])+a([2*n/3]) 1,2,3

A298346 a(n) = a(n-1)+a(n-2)+2*a([n/2]) 1,1,1

A298347 a(n) = a(n-1)+a(n-2)+2*a([n/2]) 1,2,3

A298348 a(n) = a(n-1)+a(n-2)+2*a([(n+1)/2]) 1,1,1

A298349 a(n) = a(n-1)+a(n-2)+2*a([(n+1)/2]) 1,2,3

A298350 a(n) = a(n-1)+a(n-2)+2*a(ceiling(n/2)) 1,1,1

A298351 a(n) = a(n-1)+a(n-2)+2*a(ceiling(n/2)) 1,2,3

A298352 a(n) = a(n-1)+a(n-2)+a([(n-1)/2]) 1,1,1

A298353 a(n) = a(n-1)+a(n-2)+a([(n-1)/2]) 1,2,3

A298354 a(n) = a(n-1)+a(n-2)+2*a([(n-1)/2]) 1,1,1

A298355 a(n) = a(n-1)+a(n-2)+2*a([(n-1)/2]) 1,2,3

A298356 a(n) = a(n-1)+a(n-2)+a([n/2])+a([n/3])+...+a([n/n]) 1,1,1

A298357 a(n) = a(n-1)+a(n-2)+a([n/2])+a([n/3])+...+a([n/n]) 1,2,3

A298369 a(n) = a(n-1)+a(n-2)+2*a([n/2])+3*a([n/3])+...+n*a([n/n]) 1,1,1

A298370 a(n) = a(n-1)+a(n-2)+2*a([n/2])+3*a([n/3])+...+n*a([n/n]) 1,2,3

A298402 a(n) = 2*a(n-1)-a(n-3)+a([n/2]) 1,1,1

A298403 a(n) = 2*a(n-1)-a(n-3)+a([n/2]) 1,2,3

A298404 a(n) = 2*a(n-1)-a(n-3)+a(ceiling(n/2)) 1,1,1

A298405 a(n) = 2*a(n-1)-a(n-3)+a(ceiling(n/2)) 1,2,3

A298406 a(n) = 2*a(n-1)-a(n-3)+a([n/2])+a([n/3])+...+a([n/n]) 1,1,1

A298407 a(n) = 2*a(n-1)-a(n-3)+a([n/2])+a([n/3])+...+a([n/n]) 1,2,3

A298408 a(n) = 2*a(n-1)-a(n-3)+2*a([n/2])+3*a([n/3])+...+n*a([n/n]) 1,1,1

A298409 a(n) = 2*a(n-1)-a(n-3)+2*a([n/2])+3*a([n/3])+...+n*a([n/n]) 1,2,3

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Evangelos G. Filothodoros, Strongly coupled fermions in odd dimensions and the running cut-off Lambda_d, arXiv:2306.14652 [hep-th], 2023.

Crossrefs

Cf. A001622, A000045, A298339.

Programs

Mathematica

a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[n/2]];
Table[a[n], {n, 0, 30}]  (* A298338 *)