OFFSET
0,2
COMMENTS
Substituting x*(1-x)/(1-2x) into x^2/(1-x^2) yields x^2*(g.f. of sequence).
The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 2, s(n+1) = 3. - Herbert Kociemba, Jun 02 2004
Diagonal sums of triangle in A125171. - Philippe Deléham, Jan 14 2014
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).
FORMULA
G.f.: (1-x)^2/((1-x-x^2)*(1-3*x+x^2)). - Floor van Lamoen and N. J. A. Sloane, Jan 21 2001
a(n) = Sum_{k=0..n} T(n, k)*T(n, n+k), T given by A027926.
a(n) = 2*a(n-1) + Sum_{m < n-1} a(m) + F(n-1) = A059512(n+2) - F(n) where F(n) is the n-th Fibonacci number (A000045). - Floor van Lamoen, Jan 21 2001
a(n) = (2/5)*Sum_{k=1..4} sin(2*Pi*k/5)*sin(3*Pi*k/5)*(1+2*cos(Pi*k/5))^(n+1)). - Herbert Kociemba, Jun 02 2004
MATHEMATICA
Table[(Fibonacci[2n+3]-Fibonacci[n])/2, {n, 0, 30}] (* or *) LinearRecurrence[{4, -3, -2, 1}, {1, 2, 6, 16}, 30] (* Harvey P. Dale, Apr 28 2022 *)
PROG
(PARI) a(n)=(fibonacci(2*n+3)-fibonacci(n))/2
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved