A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

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

Offset: 0

Author

Paul Barry, May 15 2003

Keywords

• easy
• nonn

Comments

Partial sums of A084099. Inverse binomial transform of A000749 (without leading zeros).

From Klaus Brockhaus, May 31 2010: (Start)

Periodic sequence: Repeat 1, 3, 3, 1.

Interleaving of A010684 and A176040.

Continued fraction expansion of (7 + 5*sqrt(29))/26.

Decimal expansion of 121/909.

a(n) = A143432(n+3) + 1 = 2*A021913(n+1) + 1 = 2*A133872(n+3) + 1.

a(n) = A165207(n+1) - 1.

First differences of A047538.

Binomial transform of A084102. (End)

From Wolfdieter Lang, Feb 09 2012: (Start)

a(n) = A045572(n+1) (Modd 5) := A203571(A045572(n+1)), n >= 0.

For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the five residue classes Modd 5, called [m] for m=0,1,...,4, are shown in the array A090298 if there the last row is taken as class [0] after inclusion of 0.

(End)

Examples

			From _Wolfdieter Lang_, Feb 09 2012: (Start)
Modd 5 of nonnegative odd numbers restricted mod 5:
A045572: 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, ...
Modd 5:  1, 3, 3, 1,  1,  3,  3,  1,  1,  3, ...
(End)
		

Links

• Guenther Schrack, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1, -1, 1).

Crossrefs

Cf. A084102.

Cf. A010684 (repeat 1, 3), A176040 (repeat 3, 1), A178593 (decimal expansion of (7+5*sqrt(29))/26), A143432 (expansion of (1+x^4)/((1-x)*(1+x^2))), A021913 (repeat 0, 0, 1, 1), A133872 (repeat 1, 1, 0, 0), A165207 (repeat 2, 2, 4, 4), A047538 (congruent to 0, 1, 4 or 7 mod 8), A084099 (expansion of (1+x)^2/(1+x^2)), A000749 (expansion of x^3/((1-x)^4-x^4)). - Klaus Brockhaus, May 31 2010

Programs

Magma

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x)^2/((1-x)*(1+x^2)) )); // G. C. Greubel, Feb 28 2019

Mathematica

CoefficientList[Series[(1+x)^2/((1-x)(1+x^2)),{x,0,110}],x] (* or *) PadRight[{},110,{1,3,3,1}] (* Harvey P. Dale, Nov 21 2012 *)

PARI

x='x+O('x^100); Vec((1+x)^2/((1-x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

Sage

((1+x)^2/((1-x)*(1+x^2))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Feb 28 2019

Formula

a(n) = binomial(3, n mod 4). - Paul Barry, May 25 2003

From Klaus Brockhaus, May 31 2010: (Start)

a(n) = a(n-4) for n > 3; a(0) = a(3) = 1, a(1) = a(2) = 3.

a(n) = (4 - (1+i)*i^n - (1-i)*(-i)^n)/2 where i = sqrt(-1). (End)

E.g.f.: 2*exp(x) + sin(x) - cos(x). - Arkadiusz Wesolowski, Nov 04 2017

a(n) = 2 - (-1)^(n*(n+1)/2). - Guenther Schrack, Feb 26 2019