A192969 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

1, 2, 6, 12, 23, 41, 71, 120, 200, 330, 541, 883, 1437, 2334, 3786, 6136, 9939, 16093, 26051, 42164, 68236, 110422, 178681, 289127, 467833, 756986, 1224846, 1981860, 3206735, 5188625, 8395391, 13584048, 21979472, 35563554, 57543061, 93106651

Offset

0,2

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n(n+3)/2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Links

Daniel Suteu, Table of n, a(n) for n = 0..999

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).

G.f.: (1 - x + 2*x^2 - x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014

a(0) = 1; a(1) = 2; a(n) = 1 + n + a(n-1) + a(n-2). - Daniel Suteu, Jan 12 2016

a(n) = 2*Fibonacci(n+2) + 3*Fibonacci(n+1) - n - 4. - G. C. Greubel, Jul 11 2019

Maple

F:= gfun:-rectoproc({a(0) = 1, a(1) = 2, a(n) = 1 + n + a(n-1) + a(n-2)}, a(n), remember):
map(F, [$0..100]); # Robert Israel, Jan 18 2016

Mathematica

(* First progream *)
q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n (n + 3)/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}] (* A192969 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192970 *)
(* Second program *)
Table[2*Fibonacci[n+2]+3*Fibonacci[n+1]-n-4, {n, 0, 40}] (* G. C. Greubel, Jul 11 2019 *)

Prog

(Sidef)
func a((0)) { 1 }
func a((1)) { 2 }
func a(n) is cached { 1 + n + a(n-1) + a(n-2) }
100.times { |i| say a(i-1) }
# Daniel Suteu, Jan 12 2016
(PARI) vector(40, n, n--; f=fibonacci; 2*f(n+2)+3*f(n+1)-n-4) \\ G. C. Greubel, Jul 11 2019
(Magma) F:=Fibonacci; [2*F(n+2)+3*F(n+1)-n-4: n in [0..40]]; // G. C. Greubel, Jul 11 2019
(Sage) f=fibonacci; [2*f(n+2)+3*f(n+1)-n-4 for n in (0..40)] # G. C. Greubel, Jul 11 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+2)+3*F(n+1)-n-4); # G. C. Greubel, Jul 11 2019

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192970.

Sequence in context: A069956 A062476 A192703 * A294532 A323950 A291014

Adjacent sequences: A192966 A192967 A192968 * A192970 A192971 A192972

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved