OFFSET
0,2
COMMENTS
As the definition indicates, each term is the sum of numbers from Pascal's Triangle in an (n+1) X (n+1) square arrangement.
Example for a(2):
1
1 1
|1 2 1 |
|1 3 3 |1
|1 4 6 |4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
...
and
(1+2+1) + (1+3+3) + (1+4+6) = 22 = a(2). Similarly, a(1) = (1+1)+(1+2) = 5. - John Molokach, Sep 17 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = 4^n - 2^n + C(2*n+2, n+1)/2. - Vaclav Kotesovec, Oct 28 2012
(n+1)*a(n) +2*(-6*n-1)*a(n-1) +4*(13*n-9)*a(n-2) +8*(-12*n+19)*a(n-3) +32*(2*n-5)*a(n-4)=0. - R. J. Mathar, Oct 01 2013
EXAMPLE
a(1)=binomial(1,0)+binomial(2,0)+binomial(1,1)+binomial(2,1)=1+1+1+2=5
MAPLE
a := n->add(add(binomial(i, j), i=n..2*n), j=0..n); seq(a(n), n=0..25);
MATHEMATICA
Table[4^n-2^n+Binomial[2*n+2, n+1]/2, {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2012 *)
PROG
(PARI) a(n)=4^n-2^n+binomial(2*n+2, n+1)/2; \\ Joerg Arndt, May 10 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 25 2003
STATUS
approved