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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
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<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
(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
Sequence in context: A188501 A283692 A173712 * A262268 A231156 A271483
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.)