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A337283 a(n) = Sum_{i=0..n} i*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

%I A337283 #36 Sep 08 2022 08:46:25
%S A337283 0,0,2,5,21,101,395,1578,6186,23610,89220,333431,1234343,4536551,
%T A337283 16567157,60172532,217524468,783111476,2809027334,10043413545,
%U A337283 35805255545,127314522569,451629771519,1598650868766,5647706073630,19916305738030,70117445671624,246478579433947,865201260035147
%N A337283 a(n) = Sum_{i=0..n} i*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.
%D A337283 Raphael Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202.
%H A337283 Colin Barker, <a href="/A337283/b337283.txt">Table of n, a(n) for n = 0..1000</a>
%H A337283 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2,-2,-35,3,0,48,-11,7,-14,2,-1,1).
%F A337283 From _Colin Barker_, Sep 13 2020: (Start)
%F A337283 G.f.: x^2*(1 + x)*(2 - 7*x + 7*x^2 + 3*x^3 + 9*x^4 + 7*x^5 + x^6 + x^7 + x^8) / ((1 - x)*(1 + x + x^2 - x^3)^2*(1 - 3*x - x^2 - x^3)^2).
%F A337283 a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3) - 35*a(n-4) + 3*a(n-5) + 48*a(n-7) - 11*a(n-8) + 7*a(n-9) - 14*a(n-10) + 2*a(n-11) - a(n-12) + a(n-13) for n>12.
%F A337283 (End)
%F A337283 a(n) = Sum_{j=0..n} j*A085697(j). - _G. C. Greubel_, Nov 20 2021
%t A337283 T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
%t A337283 a[n_]:= a[n]= Sum[j*T[j]^2, {j,0,n}];
%t A337283 Table[a[n], {n,0,30}] (* _G. C. Greubel_, Nov 20 2021 *)
%o A337283 (PARI) concat([0,0], Vec(x^2*(1+x)*(2 -7*x +7*x^2 +3*x^3 +9*x^4 +7*x^5 +x^6 + x^7 +x^8)/((1-x)*(1 +x +x^2 -x^3)^2*(1 -3*x -x^2 -x^3)^2) + O(x^30))) \\ _Colin Barker_, Sep 19 2020
%o A337283 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2*(1-x^2)*(2-7*x+7*x^2+3*x^3+9*x^4+7*x^5+x^6+x^7+x^8)/((1-x)*(1+x+x^2-x^3)*(1-3*x-x^2-x^3))^2 )); // _G. C. Greubel_, Nov 20 2021
%o A337283 (Sage)
%o A337283 @CachedFunction
%o A337283 def T(n): # A000073
%o A337283     if (n<2): return 0
%o A337283     elif (n==2): return 1
%o A337283     else: return T(n-1) +T(n-2) +T(n-3)
%o A337283 def A337283(n): return sum( j*T(j)^2 for j in (0..n) )
%o A337283 [A337283(n) for n in (0..40)] # _G. C. Greubel_, Nov 20 2021
%Y A337283 Cf. A000073, A085697, A337282, A337284.
%K A337283 nonn,easy
%O A337283 0,3
%A A337283 _N. J. A. Sloane_, Sep 12 2020