A192969 - OEIS

%I #32 Sep 08 2022 08:45:58

%S 1,2,6,12,23,41,71,120,200,330,541,883,1437,2334,3786,6136,9939,16093,

%T 26051,42164,68236,110422,178681,289127,467833,756986,1224846,1981860,

%U 3206735,5188625,8395391,13584048,21979472,35563554,57543061,93106651

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

%C 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.

%H Daniel Suteu, <a href="/A192969/b192969.txt">Table of n, a(n) for n = 0..999</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).

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

%F G.f.: (1 - x + 2*x^2 - x^3)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, May 11 2014

%F a(0) = 1; a(1) = 2; a(n) = 1 + n + a(n-1) + a(n-2). - _Daniel Suteu_, Jan 12 2016

%F a(n) = 2*Fibonacci(n+2) + 3*Fibonacci(n+1) - n - 4. - _G. C. Greubel_, Jul 11 2019

%p F:= gfun:-rectoproc({a(0) = 1, a(1) = 2, a(n) = 1 + n + a(n-1) + a(n-2)},a(n),remember):

%p map(F, [$0..100]); # _Robert Israel_, Jan 18 2016

%t (* First progream *)

%t q = x^2; s = x + 1; z = 40;

%t p[0, x] := 1;

%t p[n_, x_] := x*p[n - 1, x] + n (n + 3)/2;

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192969 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192970 *)

%t (* Second program *)

%t Table[2*Fibonacci[n+2]+3*Fibonacci[n+1]-n-4, {n,0,40}] (* _G. C. Greubel_, Jul 11 2019 *)

%o (Sidef)

%o func a((0)) { 1 }

%o func a((1)) { 2 }

%o func a(n) is cached { 1 + n + a(n-1) + a(n-2) }

%o 100.times { |i| say a(i-1) }

%o # _Daniel Suteu_, Jan 12 2016

%o (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+2)+3*f(n+1)-n-4) \\ _G. C. Greubel_, Jul 11 2019

%o (Magma) F:=Fibonacci; [2*F(n+2)+3*F(n+1)-n-4: n in [0..40]]; // _G. C. Greubel_, Jul 11 2019

%o (Sage) f=fibonacci; [2*f(n+2)+3*f(n+1)-n-4 for n in (0..40)] # _G. C. Greubel_, Jul 11 2019

%o (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+2)+3*F(n+1)-n-4); # _G. C. Greubel_, Jul 11 2019

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

%K nonn

%O 0,2

%A _Clark Kimberling_, Jul 13 2011