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%I #16 Jul 19 2018 20:07:38
%S 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,
%T 19,6,27,3,0,5,7,14,23,7,0,9,11,22,3,27,0,29,31,30,31,31,0
%N Residues (mod 32) of partial sums of Fibonacci numbers starting with F(2).
%C 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.
%C There are 8 instances of this residue 0.
%C 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.
%C The set of residues {6,14,22,30} has 2 instances, seen when considering the 3rd members of each group of 6.
%C 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.
%C For the 8 instances case, each residue is congruent with 8 (mod 4).
%C For the 3 instances case, each residue is congruent with 3 (mod 4).
%C For the 2 instances case, each residue is congruent with 2 (mod 4).
%C For the 1 instances case, each residue is congruent with 1 (mod 4).
%C 8,3,2,1 are Fibonacci numbers.
%D C. N. Menhinick, The Fibonacci Resonance and other new Golden Ratio discoveries, Onperson, (2015), pages 419-420.
%H E. T. Jacobson, <a href="http://www.fq.math.ca/Scanned/30-3/jacobson.pdf">Distribution of the Fibonacci numbers mod 2^k</a>, Fibonacci Quarterly, 30:3, (1992), pages 211-215.
%H <a href="/index/Rec#order_46">Index entries for linear recurrences with constant coefficients</a>, 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).
%F 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.)
%t Table[Mod[Fibonacci[n + 4] - 2, 32], {n, 0, 64}] (* _Michael De Vlieger_, Mar 31 2016 *)
%t Mod[Accumulate[Fibonacci[Range[2,50]]],32] (* _Harvey P. Dale_, Jul 19 2018 *)
%o (PARI) a(n)=(fibonacci(n%48+4)-2)%32 \\ _Charles R Greathouse IV_, Mar 31 2016
%Y Cf. A000045.
%K nonn,easy
%O 0,2
%A _Clive N. Menhinick_, Mar 30 2016
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