

A271079


Residues (mod 32) of partial sums of Fibonacci numbers starting with F(2).


0



1, 3, 6, 11, 19, 0, 21, 23, 14, 7, 23, 0, 25, 27, 22, 19, 11, 0, 13, 15, 30, 15, 15, 0, 17, 19, 6, 27, 3, 0, 5, 7, 14, 23, 7, 0, 9, 11, 22, 3, 27, 0, 29, 31, 30, 31, 31, 0
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OFFSET

0,2


COMMENTS

If one Pisano period (of 48 members) is partitioned sequentially as 8 groups of 6, then we observe that in each group, the 6th member is 0.
There are 8 instances of this residue 0.
The set of residues {3,7,11,15,19,23,27,31} is seen in the 2nd members of each group of 6, and similarly seen in the 4th, and in the 5th members. This set appears in a total of 3 instances.
The set of residues {6,14,22,30} has 2 instances, seen when considering the 3rd members of each group of 6.
The set of residues {1,5,9,13,17,21,25,29} appears as 1 instance when considering the 1st members of each group of 6.
For the 8 instances case, each residue is congruent with 8 (mod 4).
For the 3 instances case, each residue is congruent with 3 (mod 4).
For the 2 instances case, each residue is congruent with 2 (mod 4).
For the 1 instances case, each residue is congruent with 1 (mod 4).
8,3,2,1 are Fibonacci numbers.


REFERENCES

C. N. Menhinick, The Fibonacci Resonance and other new Golden Ratio discoveries, Onperson, (2015), pages 419420.


LINKS

Table of n, a(n) for n=0..47.
E. T. Jacobson, Distribution of the Fibonacci numbers mod 2^k, Fibonacci Quarterly, 30:3, (1992), pages 211215.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,1,0,1, 0,1,0,1,0,1,0,1,0,1, 0,1,0,1,0,1, 0,1,0,1,0,1,0,1,0,1,0,1, 0,1,0,1,0,1,0,1,0,1).


FORMULA

a(n) = (F(n+4)2) mod 32. (Based on the F(n) partial sums formula: F(n+2)1, while here omitting F(1)=1 and F(0)=0.)


MATHEMATICA

Table[Mod[Fibonacci[n + 4]  2, 32], {n, 0, 64}] (* Michael De Vlieger, Mar 31 2016 *)
Mod[Accumulate[Fibonacci[Range[2, 50]]], 32] (* Harvey P. Dale, Jul 19 2018 *)


PROG

(PARI) a(n)=(fibonacci(n%48+4)2)%32 \\ Charles R Greathouse IV, Mar 31 2016


CROSSREFS

Cf. A000045.
Sequence in context: A264923 A321381 A238903 * A091094 A116100 A295066
Adjacent sequences: A271076 A271077 A271078 * A271080 A271081 A271082


KEYWORD

nonn,easy


AUTHOR

Clive N. Menhinick, Mar 30 2016


STATUS

approved



