login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A131295 a(n)=ds_4(a(n-1))+ds_4(a(n-2)), a(0)=0, a(1)=1; where ds_4=digital sum base 4. 14
0, 1, 1, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3, 5, 5, 4, 3, 4, 4, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The digital sum analog (in base 4) of the Fibonacci recurrence.

When starting from index n=3, periodic with Pisano period A001175(3)=8.

Also a(n)==A004090(n) modulo 3 (A004090(n)=digital sum of Fib(n)).

For general bases p>2, the inequality 2<=a(n)<=2p-3 holds for n>2. Actually, a(n)<=5=A131319(4) for the base p=4.

a(n) and Fib(n)=A000045(n) are congruent modulo 3 which implies that (a(n) mod 3) is equal to (Fib(n) mod 3)=A082115(n-1) (for n>0). Thus (a(n) mod 3) is periodic with the Pisano period = A001175(3)=8 too. - Hieronymus Fischer

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for Colombian or self numbers and related sequences

FORMULA

a(n)=a(n-1)+a(n-2)-3*(floor(a(n-1)/4)+floor(a(n-2)/4)).

a(n)=floor(a(n-1)/4)+floor(a(n-2)/4)+(a(n-1)mod 4)+(a(n-2)mod 4).

a(n)=A002265(a(n-1))+A002265(a(n-2))+A010873(a(n-1))+A010873(a(n-2)).

a(n)=Fib(n)-3*sum{1<k<n, Fib(n-k+1)*floor(a(k)/4)}, where Fib(n)=A000045(n).

EXAMPLE

a(8)=3, since a(6)=5=11(base 4), ds_4(5)=2,

a(7)=4=10(base 4), ds_4(4)=1 and so a(8)=2+1.

MATHEMATICA

nxt[{a_, b_}]:={b, Total[IntegerDigits[a, 4]]+Total[IntegerDigits[b, 4]]}; NestList[ nxt, {0, 1}, 110][[All, 1]] (* Harvey P. Dale, Jul 30 2018 *)

CROSSREFS

Cf. A000045, A010073, A010074, A010075, A010076, A010077, A131294, A131296, A131297, A131318, A131319, A131320.

Sequence in context: A133293 A262565 A096289 * A102642 A183228 A133304

Adjacent sequences:  A131292 A131293 A131294 * A131296 A131297 A131298

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Jun 27 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 18 05:32 EDT 2019. Contains 327165 sequences. (Running on oeis4.)