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

Author

Jeremy F. Alm, Dec 28 2015

Keywords

• nonn

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• Alm, Herald, Miller, and Sexton, 2-Free Tetranacci Sequences, unpublished.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

Crossrefs

Cf. A000265, A232666, A233248, A233526.

Programs

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 *)

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

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()

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