A176260 Periodic sequence: Repeat 5, 1.

5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5

Offset: 0

Author

Klaus Brockhaus, Apr 13 2010

Keywords

• cofr
• cons
• easy
• nonn
• mult

Comments

Interleaving of A010716 and A000012.

Also continued fraction expansion of (5+3*sqrt(5))/2.

Also decimal expansion of 17/33.

Essentially first differences of A047264.

Binomial transform of 5 followed by -A122803 without initial terms 1, -2.

Inverse binomial transform of 5 followed by A007283 without initial term 3.

Second inverse binomial transform of A168607 without initial term 3.

Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 3*x^3 + 6*x^4 + 6*x^5 + ... is the o.g.f. for A008805. - Peter Bala, Mar 13 2015

Links

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

Crossrefs

Cf. A010716 (all 5's sequence), A000012 (all 1's sequence), A090550 (decimal expansion of (5+3*sqrt(5))/2), A010686 (repeat 1, 5), A047264 (congruent to 0 or 5 mod 6), A122803 (powers of -2), A007283 (3*2^n), A168607 (3^n+2), A008805.

Programs

Magma

&cat[ [5, 1]: n in [0..52] ];
[ 3+2*(-1)^n: n in [0..104] ];

Formula

a(n) = 3+2*(-1)^n.

a(n) = a(n-2) for n > 1; a(0) = 5, a(1) = 1.

a(n) = -a(n-1)+6 for n > 0; a(0) = 5.

a(n) = 5*((n+1) mod 2)+(n mod 2).

a(n) = A010686(n+1).

G.f.: (5+x)/(1-x^2).

From Amiram Eldar, Jan 01 2023: (Start)

Multiplicative with a(2^e) = 5, and a(p^e) = 1 for p >= 3.

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

E.g.f.: 5*cosh(x) + sinh(x). - Stefano Spezia, Feb 09 2025