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A192806 a(n) = 7*a(n-1) - 5*a(n-2) + a(n-3), with initial values a(0) = a(1) = 1, a(2)=4.

%I A192806 #38 Sep 08 2022 08:45:58
%S A192806 1,1,4,24,149,927,5768,35890,223317,1389537,8646064,53798080,
%T A192806 334745777,2082876103,12960201916,80641778674,501774317241,
%U A192806 3122171529233,19426970897100,120879712950776,752145307699165,4680045560037375,29120472094716576
%N A192806 a(n) = 7*a(n-1) - 5*a(n-2) + a(n-3), with initial values a(0) = a(1) = 1, a(2)=4.
%C A192806 Old definition was "Constant term in the reduction of (x^2+x+1)^n by x^3 -> x^2+x+1." For discussions of polynomial reduction, see A192232 and A192744.
%C A192806 From _Bob Selcoe_, Jun 10 2014: (Start)
%C A192806 a(n) is the trinomial transform of tribonacci numbers. (i.e., A027907(n) transform of A000073(n+2)).
%C A192806 Let the m-nacci numbers be denoted by M"(n). Examples: Fibonacci numbers are 2"(n); tribonacci numbers are 3"(n); 137-nacci numbers are 137"(n). Then the m-nomial transform of M" is M"(m*n), where M"(0)=1 and M"(n)=0 when n<0. Therefore a(n) = 3"(3n). (End)
%H A192806 G. C. Greubel, <a href="/A192806/b192806.txt">Table of n, a(n) for n = 0..1000</a>
%H A192806 László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%H A192806 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-5,1).
%F A192806 G.f.: (1 - 6*x + 2*x^2)/(1 - 7*x + 5*x^2 - x^3). - _R. J. Mathar_, May 06 2014
%F A192806 a(n) = A000073(3n+2), n>0. - _Bob Selcoe_, Jun 10 2014
%e A192806 G.f. = 1 + x + 4*x^2 + 24*x^3 + 149*x^4 + 927*x^5 + 5768*x^6 + ...
%t A192806 q = x^3; s = x^2 + x + 1; z = 40;
%t A192806 p[n_, x_] := (x^2 + x + 1)^n;
%t A192806 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192806 reduce[{p1_, q_, s_, x_}] :=
%t A192806 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192806        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192806 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192806 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]   (* A192806 *)
%t A192806 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]   (* A192807 *)
%t A192806 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]   (* A099464 *)
%t A192806 LinearRecurrence[{7,-5,1}, {1,1,4}, 50] (* _G. C. Greubel_, Jan 02 2019 *)
%o A192806 (PARI) {a(n) = polcoeff( lift( (1 + x + x^2)^n * Mod(1, x^3 - x^2 - x - 1)), 0)}; /* _Michael Somos_, Jun 17 2014 */
%o A192806 (PARI) my(x='x+O('x^30)); Vec((1-6*x+2*x^2)/(1-7*x+5*x^2-x^3)) \\ _G. C. Greubel_, Jan 02 2019
%o A192806 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-6*x+2*x^2)/(1-7*x+5*x^2-x^3) )); // _G. C. Greubel_, Jan 02 2019
%o A192806 (Sage) ((1-6*x+2*x^2)/(1-7*x+5*x^2-x^3)).series(x,20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 02 2019
%o A192806 (GAP) a:=[1,1,4];; for n in [4..25] do a[n]:=7*a[n-1]-5*a[n-2]+a[n-3]; od; Print(a); # _Muniru A Asiru_, Jan 02 2019
%Y A192806 Cf. A192744, A192232, A192807, A000073, A027907.
%K A192806 nonn,easy
%O A192806 0,3
%A A192806 _Clark Kimberling_, Jul 10 2011
%E A192806 Edited by _N. J. A. Sloane_, Jun 03 2018