OFFSET
0,3
COMMENTS
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
FORMULA
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1 - 2*x + 3*x^2 - x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n^2 + 5*n + 10)/2. - G. C. Greubel, Jul 11 2019
MATHEMATICA
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n(n+1)/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}] (* A030119 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192966 *)
LinearRecurrence[{4, -5, 1, 2, -1}, {0, 1, 2, 6, 14}, 40] (* Vincenzo Librandi, Nov 16 2011 *)
Table[Fibonacci[n+4] +2*Fibonacci[n+2] -(n^2+5*n+10)/2, {n, 0, 40}] (* G. C. Greubel, Jul 11 2019 *)
PROG
(Magma) I:=[0, 1, 2, 6, 14]; [n le 5 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011
(Magma) F:=Fibonacci; [F(n+4) +2*F(n+2) -(n^2+5*n+10)/2: n in [0..40]]; // G. C. Greubel, Jul 11 2019
(PARI) vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(n^2+5*n+10)/2) \\ G. C. Greubel, Jul 11 2019
(Sage) f=fibonacci; [f(n+4) +2*f(n+2) -(n^2+5*n+10)/2 for n in (0..40)] # G. C. Greubel, Jul 11 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+4) +2*F(n+2) -(n^2+5*n+10)/2); # G. C. Greubel, Jul 11 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved