A181767 a(n) = a(n-7) XOR a(n-5) XOR a(n-4) XOR a(n-3) XOR a(n-2) XOR a(n-1) with initial seven terms 95, 111, 119, 123, 125, 62, and 31.

95, 111, 119, 123, 125, 62, 31, 15, 71, 99, 113, 56, 92, 46, 87, 43, 85, 42, 21, 74, 37, 82, 41, 84, 106, 117, 122, 61, 94, 47, 23, 75, 101, 50, 25, 76, 102, 115, 57, 28, 78, 103, 51, 89, 44, 86, 107, 53, 26, 13, 70, 35, 17, 8, 68, 34, 81, 104, 116, 58, 93, 110, 55, 27, 77, 38, 19, 9, 4, 66, 97, 112, 120, 60, 30, 79, 39, 83, 105, 52, 90, 109, 118, 59, 29, 14, 7, 3, 65, 32, 80, 40, 20, 10, 69, 98, 49, 88, 108, 54, 91

Offset

0,1

Comments

The restriction to {1,...,127} is a permutation of that set. The same is true for the sequence obtained from any other starting value a(0) in this set, and for companion matrices associated to any polynomial whose coefficients mod 2 are the binary representation of 3,9,15,17,29,39,43,57,63,65,75,83,85,101,111,113,119 or 125 = 127-2 (the present case). - M. F. Hasler, Nov 13 2010

The terms represent vectors v[n] = binary(a(n)) in (Z/2Z)^7, encoded as 7-bit integers (lsb first), i.e. binary(a(n+1)) = M.binary(a(n)) (mod 2), with the matrix M given below. (Thus the sign of the entries of M has no incidence on the terms of the sequence.) The sequence is periodic with period 127, i.e. a(n+127)=a(n) for all n. - M. F. Hasler, Nov 13 2010

Links

Table of n, a(n) for n=0..100.

Wikipedia, Companion matrix.

Index entries for linear recurrences with constant coefficients, order 127.

Formula

a(n) = [1,2,4,8,16,32,64].(M^n.v % 2), where % denotes the binary remainder operation, v=(1, 1, 1, 1, 1, 0, 1)^T (column vector) and M is described below. [M. F. Hasler, Nov 13 2010]

Mathematica

F = {{0, 1, 0, 0, 0, 0, 0},
      {0, 0, 1, 0, 0, 0, 0},
      {0, 0, 0, 1, 0, 0, 0},
      {0, 0, 0, 0, 1, 0, 0},
      {0, 0, 0, 0, 0, 1, 0},
      {0, 0, 0, 0, 0, 0, 1},
      {-1, 0, -1, -1, -1, -1, -1}};
f[x_] = CharacteristicPolynomial[F, x];
v[0] = IntegerDigits[-f[2] - 128, 2];
a = Table[Sum[Mod[v[n][[m]], 2]*2^(m - 1), {m, 1, 7}], {n, 0, 100}]

Prog

(PARI) a(n)={ vector(7, i, 1<<i)\2* lift( Mod( matrix( 7, 7, i, j, (j==i+1)-(i==7 & j!=2)), 2)^(n%127)*vector(7, i, i!=6)~)} \\ M. F. Hasler, Nov 13 2010

Crossrefs

Sequence in context: A338582 A338581 A306303 * A331663 A046005 A339522

Adjacent sequences: A181764 A181765 A181766 * A181768 A181769 A181770

Keyword

nonn,easy,less

Author

Roger L. Bagula

Extensions

Meaningful name by Dominic McCarty, Jul 03 2025

Status

approved