A266295 2-free tetranacci sequence beginning 1,3,5,7.

1, 3, 5, 7, 1, 1, 7, 1, 5, 7, 5, 9, 13, 17, 11, 25, 33, 43, 7, 27, 55, 33, 61, 11, 5, 55, 33, 13, 53, 77, 11, 77, 109, 137, 167, 245, 329, 439, 295, 327, 695, 439, 439, 475, 1, 677, 199, 169, 523, 49, 235, 61, 217, 281, 397, 239, 567, 371, 787

Offset

1,2

Comments

For n>4, a(n) = (a(n-1) + a(n-2) + a(n-3) + a(n-4)) / 2^d, where 2^d is the largest power of 2 dividing a(n-1) + a(n-2) + a(n-3) + a(n-4). In other words, sum the previous four terms, then divide by two until the result is odd.

References

Alm, Herald, Miller, and Sexton, 2-Free Tetranacci Sequences, unpublished.

Links

Jeremy F. Alm, Table of n, a(n) for n = 1..10000

Formula

a(n) = (a(n-1) + a(n-2) + a(n-3) + a(n-4)) / 2^d, where 2^d is the largest power of 2 dividing a(n-1) + a(n-2) + a(n-3) + a(n-4).

a(n) = A000265(a(n-1) + a(n-2) + a(n-3) + a(n-4)). - Michel Marcus, Dec 29 2015

Mathematica

nxt[{a_, b_, c_, d_}] := {b, c, d, (a + b + c + d)/2^IntegerExponent[ a + b + c + d, 2]}; NestList[nxt, {1, 3, 5, 7}, 60][[All, 1]] (* Harvey P. Dale, Nov 09 2020 *)

Prog

(Python)
### CREATES A b-FILE ###
def main():
    name = "b266295.txt"
    file = open(name, 'w')
    file.write('1' + ' ' + '1\n')
    file.write('2' + ' ' + '3\n')
    file.write('3' + ' ' + '5\n')
    file.write('4' + ' ' + '7\n')
    a, b, c, d = 1, 3, 5, 7
    for i in range(5, 10001):
        x=a+b+c+d
        while x%2==0:
            x /= 2
        a, b, c, d = b, c, d, x
        file.write(str(i) + ' ' + str(int(d)) + '\n')
    file.close()
main()
(PARI) lista(nn) = {print1(x = 1, ", "); print1(y = 3, ", "); print1(z = 5, ", "); print1(t = 7, ", "); for (n=5, nn, tt = (x+y+z+t); tt /= 2^valuation(tt, 2); print1(tt, ", "); x=y; y=z; z=t; t=tt; ); } \\ Michel Marcus, Dec 29 2015

Crossrefs

Cf. A000265, A232666, A233248, A233526.

Sequence in context: A031057 A133069 A192327 * A172049 A021032 A212120

Adjacent sequences: A266292 A266293 A266294 * A266296 A266297 A266298

Keyword

nonn

Author

Jeremy F. Alm, Dec 28 2015

Status

approved