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%I #21 Apr 25 2024 09:14:27
%S 0,0,4,13,77,477,2241,10522,47386,204202,860302,3546623,14357567,
%T 57286271,225714755,879795380,3397426356,13012405492,49478890936,
%U 186932228945,702169068945,2623863676449,9758799153349,36140284390030,133317609306766,490032600916766,1795262239190210,6557012850772931
%N a(n) = Sum_{i=0..n} i^2*T(i)^2, where T(i) = A000073(i) is the i-th tribonacci number.
%D 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 G. C. Greubel, <a href="/A337286/b337286.txt">Table of n, a(n) for n = 0..1000</a>
%H <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 G.f.: x^2*(4 - 15*x + 22*x^2 + 83*x^3 - 90*x^4 + 11*x^5 - 128*x^6 - 207*x^7 - 224*x^8 - 233*x^9 - 162*x^10 - 147*x^11 - 58*x^12 - 3*x^13 - 4*x^14 - 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
%t T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] +T[n-2] +T[n-3]]]; (* A000073 *)
%t A337286[n_]:= Sum[j^2*T[j]^2, {j,0,n}];
%t Table[A337286[n], {n, 0, 50}] (* _G. C. Greubel_, Nov 22 2021 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(4-15*x+22*x^2+83*x^3-90*x^4+11*x^5-128*x^6-207*x^7-224*x^8-233*x^9-162*x^10- 147*x^11-58*x^12-3*x^13-4*x^14-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 (Sage)
%o @CachedFunction
%o def T(n): # A000073
%o if (n<2): return 0
%o elif (n==2): return 1
%o else: return T(n-1) +T(n-2) +T(n-3)
%o def A337286(n): return sum( j^2*T(j)^2 for j in (0..n) )
%o [A337286(n) for n in (0..40)] # _G. C. Greubel_, Nov 22 2021
%Y Cf. A000073, A085697, A337282, A337283, A337284, A337285.
%K nonn,changed
%O 0,3
%A _N. J. A. Sloane_, Sep 12 2020
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