A271079 Residues (mod 32) of partial sums of Fibonacci numbers starting with F(2).

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

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 419-420.

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 211-215.

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 A332446 A116100

Adjacent sequences: A271076 A271077 A271078 * A271080 A271081 A271082

Keyword

nonn,easy,changed

Author

Clive N. Menhinick, Mar 30 2016

Status

approved