OFFSET
0,4
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (5th line of Table 1 is a(n-2)).
A. Shriki and O. Liba, Polygons with Fibonacci Number Coordinates: Problem B-1167, Fib. Quart. 54,2 May 2016, p. 180-181.
Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).
FORMULA
a(n) = +5*a(n-1) -8*a(n-2) +5*a(n-3) -1*a(n-4).
G.f.: x^2/((1-x)^2*(1-3*x+x^2)).
G.f.: x^2*Fibe(x)/(1-x)^2, with Fibe(x) := 1/(1-3*x+x^2) = g.f. A001906(n+1) (Fibonacci numbers F(2(n+1))).
Fourth diagonal of array defined by T(i, 1) = T(1, j) = 1, T(i, j) = Max(T(i-1, j) + T(i-1, j-1); T(i-1, j-1) + T(i, j-1)). - Benoit Cloitre, Aug 05 2003
a(n) = Sum_{k=0..n-2} binomial(2*n-k-1, k). - Johannes W. Meijer, Aug 12 2013
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} binomial(i+j, i-j). - Wesley Ivan Hurt, Mar 25 2015
a(n) = Sum_{k=0..n} (binomial(n+1,k+2)*Fibonacci(k)). - Vladimir Kruchinin, Oct 21 2016
a(n) = (-((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n)/sqrt(5) - n. - Colin Barker, Jan 28 2017
MAPLE
a[0]:=0: a[1]:=1: for n from 2 to 50 do a[n] := 3*a[n-1]-a[n-2] od: seq(a[n]-n, n=0..27); # Zerinvary Lajos, Mar 20 2008
with(combinat): seq(fibonacci(2*n)-n, n=0..27); # Zerinvary Lajos, Jun 19 2008
g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)-n), n=0..27); # Zerinvary Lajos, Mar 22 2009
MATHEMATICA
CoefficientList[Series[x^2 / ((1 - x)^2 (1 - 3 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 26 2015 *)
PROG
(Sage) [(lucas_number1(n, 3, 1)-lucas_number1(n, 2, 1)) for n in range(1, 28)]# Zerinvary Lajos, Mar 13 2009
(Magma) I:=[0, 0, 1, 5]; [n le 4 select I[n] else 5*Self(n-1)-8*Self(n-2)+5*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 26 2015
(Maxima)
makelist(sum(fib(k)*binomial(n+1, k+2), k, 0, n), n, 0, 20); /* Vladimir Kruchinin, Oct 21 2016 */
(PARI) concat(vector(2), Vec(x^2/((1-x)^2*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Jan 28 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang, Apr 27 2000
EXTENSIONS
More terms from James A. Sellers, Apr 28 2000
a(0) added by Arkadiusz Wesolowski, Jun 07 2011
STATUS
approved