

A319053


a(n) is the exponent of the largest power of 2 that appears in the factorization of the entries in the matrix {{3,1},{1,1}}^n.


2



0, 1, 5, 3, 4, 8, 6, 7, 12, 9, 10, 15, 12, 13, 18, 15, 16, 20, 18, 19, 25, 21, 22, 28, 24, 25, 31, 27, 28, 32, 30, 31, 36, 33, 34, 39, 36, 37, 42, 39, 40, 44, 42, 43, 50, 45, 46, 53, 48, 49, 56, 51, 52, 56, 54, 55, 60, 57, 58, 63, 60, 61, 66, 63, 64, 68, 66, 67, 73, 69
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OFFSET

1,3


COMMENTS

a(n) appears to equal n1 for n not a multiple of 3.
The matrix entries of M^n, with n >= 0, are M^n(1, 1) = 2^(n1)*F(n+3) = A063782(n), M^n(2, 2) = 2^(n1)*F(n3) = A319196(n), M^n(1, 2) = M^n(2, 1) = 2^(n1)*F(n) = A085449(n), where i = sqrt(1), F = A000045, and F(1) = 1, F(2) = 1, F(3) = 2. Proof by CayleyHamilton, with S(n, i) = (i)^n*F(n+1), where S(n, x) is given in A049310.  Wolfdieter Lang, Oct 08 2018
The above conjecture is true. From the preceding formulas for the elements of M^n this claims that the Fibonacci numbers F(n3), F(n) and F(n+3) are always odd for n == 1 or 2 (mod 3). This is true because F(n) is even iff n == 0 (mod 3) (see e.g. Vajda, p.73), and each of the three indices is == 1 or 2 (mod 3) for n == 1 or 2 (mod 3), respectively.  Wolfdieter Lang, Oct 09 2018


REFERENCES

S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989, p. 73.


LINKS

Table of n, a(n) for n=1..70.


EXAMPLE

For n = 3, the matrix {{3,1},{1,1}}^3 = {{32,8},{8,0}} and the largest power of 2 appearing in the factorization of any entry is 2^5 = 32. Hence, a(3) = 5.


MATHEMATICA

Join[{0, 1, 5}, Table[Max[ IntegerExponent[Flatten[MatrixPower[{{3, 1}, {1, 1}}, n]], 2]], {n, 4, 40}]]


PROG

(PARI) a(n) = vecmax(apply(x>if (x, valuation(x, 2), 0), [3, 1; 1, 1]^n)); \\ Michel Marcus, Sep 09 2018


CROSSREFS

Cf. A000045, A049310, A063782, A085449, A130481, A319196.
Sequence in context: A004494 A193090 A004162 * A109681 A196406 A070367
Adjacent sequences: A319050 A319051 A319052 * A319054 A319055 A319056


KEYWORD

nonn


AUTHOR

Greg Dresden, Sep 09 2018


STATUS

approved



