A186187 Period 8 sequence [ 2, 2, 1, 2, 4, 2, 1, 2, ...] except a(0) = 1.

1, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2

Offset

0,2

Comments

Also continued fraction expansion of sqrt(2717)/38. - Bruno Berselli, Mar 07 2011

Links

Table of n, a(n) for n=0..104.

Michael Somos, Rational Function Multiplicative Coefficients, 2014.

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).

Formula

Euler transform of length 8 sequence [ 2, -2, 2, 0, 0, -2, 0, 1].

Moebius transform is length 8 sequence [ 2, -1, 0, 3, 0, 0, 0, -2].

a(n) = 2 * b(n) where b() is multiplicative with b(2) = 1/2, b(4) = 2, b(2^e) = 1 if e > 2, b(p^e) = 1 if p > 2.

G.f.: (1 + x)^4 * (1 - x + x^2)^2 / (1 - x^8) = (1-x+x^2)^2*(1+x)^3 / ((1-x) *(1+x^2) *(1+x^4)).

a(-n) = a(n). a(2*n + 1) = 2, a(4*n + 2) = 1, a(8*n + 4) = 4, a(8*n) = 2 except a(0) = 1.

a(n) = A056594(n) - A014017(n) + 2 for n > 0. - Bruno Berselli, Feb 15 2011

E.g.f.: 2*exp(x) - cos(x/sqrt(2))*cosh(x/sqrt(2)) + cos(x) - 1. - Amiram Eldar, Dec 02 2025

Example

1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 2*x^5 + x^6 + 2*x^7 + 2*x^8 + 2*x^9 + ...

Mathematica

PadRight[{1}, 108, {2, 2, 1, 2, 4, 2, 1, 2}] (* Harvey P. Dale, Mar 22 2012 *)

Prog

(PARI) {a(n) = - (n==0) + [ 2, 2, 1, 2, 4, 2, 1, 2] [n%8 + 1]}
(PARI) {a(n) = polcoeff( (1 + x)^4 * (1 - x + x^2)^2 / (1 - x^8) + x * O(x^abs(n)), abs(n))}
(Magma) [1] cat &cat[ [2, 1, 2, 4, 2, 1, 2, 2]: n in [1..13]];  // Bruno Berselli, Mar 07 2011

Crossrefs

Cf. A014017, A056594.

Sequence in context: A290091 A059149 A273917 * A013943 A164281 A082693

Adjacent sequences: A186184 A186185 A186186 * A186188 A186189 A186190

Keyword

nonn,easy

Author

Michael Somos, Feb 14 2011

Status

approved