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A298338 a(n) = a(n-1) + a(n-2) + a([n/2]), where a(0) = 1, a(1) = 1, a(2) = 1. 42
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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:

****

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([2n/3])                           1,1,1

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

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

A298345  a(n) = a(n-1)+a(n-2)+a([n/3]) +  a([2n/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)+2a([n/2])+3a([n/3]+...+4 a([n/n})   1,1,1

A298370  a(n) = a(n-1)+a(n-2)+2a([n/2])+3a([n/3]+...+4a([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)+2a([n/2])+3a([n/3]+...+ 4a([n/n]) 1,1,1

A298409  a(n) = 2*a(n-1)-a(n-3)+2a([n/2])+3a([n/3]+...+ 4a([n/n]) 1,2,3

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

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 *)

CROSSREFS

Cf. A001622, A000045, A298339.

Sequence in context: A032679 A154607 A287207 * A018162 A077879 A078140

Adjacent sequences:  A298335 A298336 A298337 * A298339 A298340 A298341

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 09 2018

STATUS

approved

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Last modified January 25 17:16 EST 2020. Contains 331245 sequences. (Running on oeis4.)