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 A085697 a(n) = T(n)^2, where T(n) = A000073(n) is the n-th tribonacci number. 17
 0, 0, 1, 1, 4, 16, 49, 169, 576, 1936, 6561, 22201, 75076, 254016, 859329, 2907025, 9834496, 33269824, 112550881, 380757169, 1288092100, 4357584144, 14741602225, 49870482489, 168710633536, 570743986576, 1930813074369, 6531893843049 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 REFERENCES R. Schumacher, Explicit formulas for sums involving the squares of the first n Tribonacci numbers, Fib. Q., 58:3 (2020), 194-202. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,3,6,-1,0,-1). FORMULA G.f.: x^2*( 1-x-x^2-x^3 )/( (1-3*x-x^2-x^3)*(1+x+x^2-x^3) ). a(n+6) = 2*a(n+5) + 3*a(n+4) + 6*a(n+3) - a(n+2) - a(n). 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 MATHEMATICA LinearRecurrence[{2, 3, 6, -1, 0, -1}, {0, 0, 1, 1, 4, 16}, 30] (* Harvey P. Dale, Oct 26 2020 *) PROG (Maxima) t[0]:0\$ t[1]:0\$ t[2]:1\$ t[n]:=t[n-1]+t[n-2]+t[n-3]\$ makelist(t[n]^2, n, 0, 40); /* Emanuele Munarini, Mar 01 2011 */ (Magma) R:=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 (Sage) @CachedFunction def T(n): # A000073 if (n<2): return 0 elif (n==2): return 1 else: return T(n-1) +T(n-2) +T(n-3) def A085697(n): return T(n)^2 [A085697(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021 CROSSREFS Cf. A000073, A057597, A099463, A107239, A113300. Sequence in context: A188501 A283692 A173712 * A262268 A231156 A271483 Adjacent sequences: A085694 A085695 A085696 * A085698 A085699 A085700 KEYWORD easy,nonn AUTHOR Emanuele Munarini, Jul 18 2003 EXTENSIONS Offset corrected to match A000073 by N. J. A. Sloane, Sep 12 2020 Name corrected to match corrected offset by Michael A. Allen, Jun 10 2021 STATUS approved

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Last modified July 21 00:33 EDT 2024. Contains 374461 sequences. (Running on oeis4.)