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A192814 Constant term in the reduction of the polynomial (2*x+1)^n by x^3 -> x^2 + x + 1. See Comments.

%I A192814 #15 Sep 08 2022 08:45:58
%S A192814 1,1,1,9,49,225,1041,4873,22817,106753,499425,2336585,10931921,
%T A192814 51145825,239289457,1119533257,5237818689,24505519873,114650876097,
%U A192814 536402551689,2509598769265,11741342323937,54932733173713,257006830281609
%N A192814 Constant term in the reduction of the polynomial (2*x+1)^n by x^3 -> x^2 + x + 1.  See Comments.
%C A192814 For discussions of polynomial reduction, see A192232 and A192744.
%H A192814 G. C. Greubel, <a href="/A192814/b192814.txt">Table of n, a(n) for n = 0..1000</a>
%H A192814 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,7).
%F A192814 a(n) = 5*a(n-1) - 3*a(n-2) + 7*a(n-3).
%F A192814 G.f.: (1 -4*x -x^2) / (1 -5*x +3*x^2 -7*x^3). - _R. J. Mathar_, May 06 2014
%p A192814 seq(coeff(series((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3),x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Jan 03 2019
%t A192814 q = x^3; s = x^2 + x + 1; z = 40;
%t A192814 p[n_, x_] := (2 x + 1)^n;
%t A192814 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192814 reduce[{p1_, q_, s_, x_}] :=
%t A192814 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192814        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192814 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192814 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192814 *)
%t A192814 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192815 *)
%t A192814 u2 = u2/2  (* A192816 *)
%t A192814 LinearRecurrence[{5,-3,7}, {1,1,1}, 30] (* _G. C. Greubel_, Jan 03 2019 *)
%o A192814 (PARI) my(x='x+O('x^30)); Vec((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3)) \\ _G. C. Greubel_, Jan 03 2019
%o A192814 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-4*x-x^2)/(1-5*x+3*x^2-7*x^3) )); // _G. C. Greubel_, Jan 03 2019
%o A192814 (Sage) ((1-4*x-x^2)/(1-5*x+3*x^2-7*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 03 2019
%o A192814 (GAP) a:=[1,1,1];; for n in [4..25] do a[n]:=5*a[n-1]-3*a[n-2]+7*a[n-3]; od; Print(a); # _Muniru A Asiru_, Jan 03 2019
%Y A192814 Cf. A192744, A192232, A192815.
%K A192814 nonn
%O A192814 0,4
%A A192814 _Clark Kimberling_, Jul 10 2011