A113311 Expansion of (1+x)^2/(1-x).

1, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Author

Paul Barry, Oct 25 2005

Keywords

• nonn
• easy

Comments

Row sums of A113310.

Let m=3. We observe that a(n)=Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A040000 and A115291: it is the same formula with respectively m=2 and m=4. 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 (3+sqrt(5))/4. - Bruno Berselli, Sep 23 2011

Also decimal expansion of 121/900. - Vincenzo Librandi, Sep 24 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A040000, A115291, A171418, A171440-A171443.

Programs

Mathematica

CoefficientList[Series[(1+x)^2/(1-x),{x,0,110}],x] (* Harvey P. Dale, Aug 19 2011 *)

PARI

a(n)=if(n>1,4,2*n+1) \\ Charles R Greathouse IV, Jun 12 2015

Formula

a(n) = Sum_{k=0..n} Sum_{i=0..n-k} (-1)^i*C(i+k-2, i).

E.g.f.: 4*exp(x) - x - 3. - Elmo R. Oliveira, Aug 08 2024

Extensions

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010