|
%I #48 Jul 23 2023 18:22:26
%S 0,0,1,1,4,16,49,169,576,1936,6561,22201,75076,254016,859329,2907025,
%T 9834496,33269824,112550881,380757169,1288092100,4357584144,
%U 14741602225,49870482489,168710633536,570743986576,1930813074369,6531893843049
%N a(n) = T(n)^2, where T(n) = A000073(n) is the n-th tribonacci number.
%C In general, squaring the terms of a third-order linear recurrence with signature (x,y,z) will result in a sixth-order recurrence with signature (x^2 + y, x^2*y + z*x + y^2, x^3*z + 4*x*y*z - y^3 + 2*z^2, x^2*z^2 - x*y^2*z - z^2*y, z^2*y^2 - z^3*x, -z^4). - _Gary Detlefs_, Jan 10 2023
%D R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202.
%H G. C. Greubel, <a href="/A085697/b085697.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,6,-1,0,-1).
%F G.f.: x^2*( 1-x-x^2-x^3 )/( (1-3*x-x^2-x^3)*(1+x+x^2-x^3) ).
%F a(n+6) = 2*a(n+5) + 3*a(n+4) + 6*a(n+3) - a(n+2) - a(n).
%F a(n) = (-A057597(n-2) + 3*A057597(n-1) + 6*A057597(n) + 5*A113300(n-1) - A099463(n-2))/11. - _R. J. Mathar_, Aug 19 2008
%t LinearRecurrence[{2,3,6,-1,0,-1},{0,0,1,1,4,16},30] (* _Harvey P. Dale_, Oct 26 2020 *)
%o (Maxima)
%o t[0]:0$ t[1]:0$ t[2]:1$
%o t[n]:=t[n-1]+t[n-2]+t[n-3]$
%o makelist(t[n]^2,n,0,40); /* _Emanuele Munarini_, Mar 01 2011 */
%o (Magma)
%o R<x>:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(1-x-x^2-x^3)/((1-3*x-x^2-x^3)*(1+x+x^2-x^3)) )); // _G. C. Greubel_, Nov 20 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 A085697(n): return T(n)^2
%o [A085697(n) for n in (0..40)] # _G. C. Greubel_, Nov 20 2021
%Y Cf. A000073, A057597, A099463, A107239, A113300.
%K easy,nonn
%O 0,5
%A _Emanuele Munarini_, Jul 18 2003
%E Offset corrected to match A000073 by _N. J. A. Sloane_, Sep 12 2020
%E Name corrected to match corrected offset by _Michael A. Allen_, Jun 10 2021
|
