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URL: https://oeis.org/A181767

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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.
0
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
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
KEYWORD
nonn,easy,less
EXTENSIONS
Meaningful name by Dominic McCarty, Jul 03 2025
STATUS
approved