OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Feng-Zhen Zhao, The log-behavior of some sequences related to the generalized Leonardo numbers, Integers (2024) Vol. 24, Art. No. A82.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
FORMULA
G.f.: (1+3*x-x^2)/((1-x)*(1-x-x^2)), so the first differences are (essentially) A022087. - R. J. Mathar, May 04 2014
a(n) = 4*Fibonacci(n+2)-3. - Gerry Martens, Jul 04 2015
MATHEMATICA
(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, n_]:= 1; p[n_, x_]:= x*p[n-1, x] +3n +2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192746 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192747 *) (* Clark Kimberling, Jul 09 2011 *)
(* Additional programs *)
a[0]=1; a[1]=5; a[n_]:=a[n]=a[n-1]+a[n-2]+3; Table[a[n], {n, 0, 36}] (* Gerry Martens, Jul 04 2015 *)
4*Fibonacci[Range[0, 40]+2]-3 (* G. C. Greubel, Jul 24 2019 *)
PROG
(PARI) vector(30, n, n--; 4*fibonacci(n+2)-3) \\ G. C. Greubel, Jul 24 2019
(Magma) [4*Fibonacci(n+2)-3: n in [0..30]]; // G. C. Greubel, Jul 24 2019
(Sage) [4*fibonacci(n+2)-3 for n in (0..30)] # G. C. Greubel, Jul 24 2019
(GAP) List([0..30], n-> 4*Fibonacci(n+2)-3); # G. C. Greubel, Jul 24 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 09 2011
STATUS
approved