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A337285 a(n) = Sum_{i=1..n} (i-1)^2*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.

%I A337285 #18 Apr 25 2024 09:13:58
%S A337285 0,1,5,41,297,1522,7606,35830,159734,691175,2911275,11995471,48573775,
%T A337285 193800376,763577276,2976338876,11493413820,44020618429,167385941185,
%U A337285 632387189285,2375420846885,8876467428110,33013780952786,122261706093330,451010242361106,1657768413841731,6073328651742855
%N A337285 a(n) = Sum_{i=1..n} (i-1)^2*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.
%D A337285 R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202. (Note that this paper uses an offset for the tribonacci numbers that is different from that used in A000073.)
%H A337285 G. C. Greubel, <a href="/A337285/b337285.txt">Table of n, a(n) for n = 1..1000</a>
%H A337285 <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (7,-9,-7,-56,96,108,252,-162,-114,-318,126,-16,136,-36,12,-21,3,-1,1).
%F A337285 From _G. C. Greubel_, Nov 22 2021: (Start)
%F A337285 a(n) = A337286(n) - 2*A337283(n) + A107239(n).
%F A337285 a(n) = Sum_{j=0..n-1} j^2*A000073(j+1)^2.
%F A337285 G.f.: x^2*(1 -2*x +15*x^2 +62*x^3 -97*x^4 +96*x^5 +73*x^6 -64*x^7 -57*x^8 -194*x^9 -127*x^10 -138*x^11 -55*x^12 -12*x^13 -9*x^14 -4*x^15)/((1-x)*(1 +x +x^2 -x^3)^3*(1 -3*x -x^2 -x^3)^3). (End)
%t A337285 T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
%t A337285 A337285[n_]:= Sum[j^2*T[j+1]^2, {j,0,n-1}];
%t A337285 Table[A337285[n], {n, 40}] (* _G. C. Greubel_, Nov 22 2021 *)
%o A337285 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1 -2*x+15*x^2+62*x^3-97*x^4+96*x^5+73*x^6-64*x^7-57*x^8-194*x^9-127*x^10-138*x^11 -55*x^12-12*x^13-9*x^14-4*x^15)/((1-x)*(1+x+x^2-x^3)^3*(1-3*x-x^2 -x^3)^3) )); // _G. C. Greubel_, Nov 22 2021
%o A337285 (Sage)
%o A337285 @CachedFunction
%o A337285 def T(n): # A000073
%o A337285     if (n<2): return 0
%o A337285     elif (n==2): return 1
%o A337285     else: return T(n-1) +T(n-2) +T(n-3)
%o A337285 def A337285(n): return sum( j^2*T(j+1)^2 for j in (0..n-1) )
%o A337285 [A337285(n) for n in (1..40)] # _G. C. Greubel_, Nov 22 2021
%Y A337285 Cf. A000073, A085697, A107239, A337282, A337283, A337284, A337286.
%K A337285 nonn
%O A337285 1,3
%A A337285 _N. J. A. Sloane_, Sep 12 2020