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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 (list; graph; refs; listen; history; text; internal format)
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 A116100 A295066

Adjacent sequences:  A271076 A271077 A271078 * A271080 A271081 A271082

KEYWORD

nonn,easy

AUTHOR

Clive N. Menhinick, Mar 30 2016

STATUS

approved

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Last modified June 16 06:46 EDT 2019. Contains 324145 sequences. (Running on oeis4.)