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

Author

Klaus Brockhaus, Apr 07 2010

Keywords

• cofr
• cons
• easy
• nonn
• mult

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

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

Crossrefs

Cf. A153284, 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.

Programs

Magma

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

Mathematica

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

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)