A115291 Expansion of (1+x)^3/(1-x).

1, 4, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Author

Paul Barry, Jan 19 2006

Keywords

• nonn
• easy

Comments

Partial sums are A086570. Partial sums of squares are A115295. Correlation triangle is A115292.

Let m=4. We observe that a(n) = Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A113311 and A040000: it is the same formula with respectively m=3 and m=2. We can generalize this result with the sequence whose G.f is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009

Also continued fraction expansion of (132-sqrt(17))/103. - Bruno Berselli, Sep 23 2011

Also decimal expansion of 1331/9000. - Vincenzo Librandi, Sep 23 2011

Links

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

Crossrefs

Cf. A040000, A113311, A171418, A171440, A171441, A171442, A171443.

Programs

Mathematica

CoefficientList[Series[(1+x)^3/(1-x),{x,0,100}],x] (* or *) PadRight[ {1,4,7},120,{8}] (* Harvey P. Dale, May 23 2016 *)

Formula

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

a(n) = Sum_{k=0..n} C(3, k).

a(n) = A004070(n, 3).

From Elmo R. Oliveira, Aug 09 2024: (Start)

E.g.f.: 8*exp(x) - 7 - 4*x - x^2/2.

a(n) = 8, n > 2. (End)