A337284 - OEIS

%I #22 Sep 08 2022 08:46:25

%S 0,1,3,15,79,324,1338,5370,20858,79907,301917,1127753,4175945,

%T 15347222,56045572,203563012,735880196,2649245173,9502874215,

%U 33976624115,121128306995,430701953720,1527852568478,5408197139806,19106052817630,67376379676855,237205619596129,833831061604429,2926954896983117

%N a(n) = Sum_{i=1..n} (i-1)*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="/A337284/b337284.txt">Table of n, a(n) for n = 1..1000</a>

%F Schumacher (on page 194) gives two explicit formulas for a(n) in terms of tribonacci numbers.

%F From _Colin Barker_, Sep 14 2020: (Start)

%F G.f.: x^2*(1 - 2*x + 2*x^2 + 12*x^3 + 8*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 2*x^9) / ((1 - x)*(1 + x + x^2 - x^3)^2*(1 - 3*x - x^2 - x^3)^2)

%F 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>13.

%F (End)

%F a(n) = A337283(n) - A107239(n). - _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]]];

%t a[n_]:= a[n]= Sum[(j-1)*T[j]^2, {j,0,n}];

%t Table[a[n], {n,40}] (* _G. C. Greubel_, Nov 22 2021 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x^2*(1-2*x+2*x^2+12*x^3+8*x^5+2*x^6+4*x^7+3*x^8+2*x^9)/((1-x)*(1-2*x-3*x^2-6*x^3+x^4+x^6)^2) )); // _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 A337284(n): return sum( (j-1)*T(j)^2 for j in (0..n) )

%o [A337284(n) for n in (1..40)] # _G. C. Greubel_, Nov 22 2021

%Y Cf. A000073, A085697, A107239, A337282, A337283, A337285.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Sep 12 2020