A176040 Periodic sequence: Repeat 3, 1.

3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3

Offset

0,1

Comments

Interleaving of A010701 and A000012.

Also continued fraction expansion of (3+sqrt(21))/2.

Also decimal expansion of 31/99.

Essentially first differences of A014601.

Inverse binomial transform of 3 followed by A020707.

Second inverse binomial transform of A052919 without initial term 2.

Third inverse binomial transform of A007582 without initial term 1.

Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + ... is the o.g.f. for A008619. - Peter Bala, Mar 13 2015

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,1).

Formula

a(n) = 2+(-1)^n.

a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 1.

a(n) = -a(n-1)+4 for n > 0; a(0) = 3.

a(n) = 3*((n+1) mod 2)+(n mod 2).

a(n) = A010684(n+1).

G.f.: (3+x)/((1-x)*(1+x)).

From Amiram Eldar, Jan 01 2023: (Start)

Multiplicative with a(2^e) = 3, and a(p^e) = 1 for p >= 3.

Dirichlet g.f.: zeta(s)*(1+2^(1-s)). (End)

Mathematica

PadRight[{}, 120, {3, 1}] (* or *) LinearRecurrence[{0, 1}, {3, 1}, 120] (* Harvey P. Dale, Mar 11 2015 *)

Prog

(Magma) &cat[ [3, 1]: n in [0..52] ];
[ 2+(-1)^n: n in [0..104] ];

Crossrefs

Cf. A010701 (all 3's sequence), A000012 (all 1's sequence), A090458 (decimal expansion of (3+sqrt(21))/2), A010684 (repeat 1, 3), A014601 (congruent to 0 or 3 mod 4), A020707 (2^(n+2)), A052919, A007582 (2^(n-1)*(1+2^n)), A008619.

Sequence in context: A153284 A112030 A010684 * A125768 A377301 A377300

Adjacent sequences: A176037 A176038 A176039 * A176041 A176042 A176043

Keyword

cofr,cons,easy,nonn,mult

Author

Klaus Brockhaus, Apr 07 2010

Status

approved