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A192920 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence). 2
0, 5, 8, 26, 63, 170, 440, 1157, 3024, 7922, 20735, 54290, 142128, 372101, 974168, 2550410, 6677055, 17480762, 45765224, 119814917, 313679520, 821223650, 2149991423, 5628750626, 14736260448, 38580030725, 101003831720, 264431464442 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
See A192872.
LINKS
FORMULA
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = A192883(n+1).
G.f.: x*(5-2*x)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, Aug 01 2011
a(n) = (A005248(n+2) - 7*(-1)^n)/5. - R. J. Mathar, Aug 01 2011
a(n) = Fibonacci(n+2)^2 - (-1)^n. - G. C. Greubel, Feb 06 2019
Sum_{n>=1} 1/a(n) = 7/18. - Amiram Eldar, Oct 05 2020
MATHEMATICA
(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 3; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^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}] (* A192919 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192920 *)
(* Second program *)
LinearRecurrence[{2, 2, -1}, {0, 5, 8}, 30] (* G. C. Greubel, Feb 06 2019 *)
PROG
(PARI) vector(30, n, n--; fibonacci(n+2)^2 -(-1)^n) \\ G. C. Greubel, Feb 06 2019, modified Jul 28 2019
(Magma) [Fibonacci(n+2)^2 -(-1)^n: n in [0..30]]; // G. C. Greubel, Feb 06 2019, modified Jul 28 2019
(Sage) [fibonacci(n+2)^2 -(-1)^n for n in (0..30)] # G. C. Greubel, Feb 06 2019, modified Jul 28 2019
(GAP) a:=[0, 5, 8];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Feb 06 2019
CROSSREFS
Sequence in context: A067973 A101584 A112577 * A162562 A182543 A026539
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 12 2011
STATUS
approved

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)