OFFSET
0,4
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
M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(3) (1963), 71-74.
Z. Jakubczyk, Advanced Problems and Solutions, Fib. Quart. 51 (3) (2013) 185, H-715.
Eric Weisstein's World of Mathematics, Tribonacci Number
Index entries for linear recurrences with constant coefficients, signature (3,1,3,-7,1,-1,1).
FORMULA
a(n) = T(0)^2 + T(1)^2 + ... + T(n)^2 where T(n) = A000073(n).
From R. J. Mathar, Aug 19 2008: (Start)
a(n) = Sum_{i=0..n} A085697(i).
G.f.: x^2*(1-x-x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)*(1-x)). (End)
a(n) = 1/4 - (1/11)*Sum_{_R = RootOf(_Z^3-_Z^2-_Z-1)} ((3 + 7*_R + 5*_R^2)/(3*_R^2 - 2*_R - 1)*_R^(-n) - (1/44)*Sum_{_R = RootOf(_Z^3+_Z^2+3*_Z-1)} ((-1 - 2*_R - 9*_R^2)/(3*_R^2 + 2*_R + 3)*_R^(-n). - Robert Israel, Mar 26 2010
a(n+1) = A000073(n)*A000073(n+1) + ( (A000073(n+1) - A000073(n-1))^2 - 1 )/4 for n>0 [Jakubczyk]. - R. J. Mathar, Dec 19 2013
EXAMPLE
a(7) = 71 = 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2 + 7^2
MAPLE
b:= proc(n) option remember; `if`(n<3, [n*(n-1)/2$2],
(t-> [t, t^2+b(n-1)[2]])(add(b(n-j)[1], j=1..3)))
end:
a:= n-> b(n)[2]:
seq(a(n), n=0..30); # Alois P. Heinz, Nov 22 2021
MATHEMATICA
Accumulate[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 30]^2] (* Harvey P. Dale, Sep 11 2011 *)
LinearRecurrence[{3, 1, 3, -7, 1, -1, 1}, {0, 0, 1, 2, 6, 22, 71}, 30] (* Ray Chandler, Aug 02 2015 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0, 0] cat Coefficients(R!( x^2*(1-x-x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)*(1-x)) )); // 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 A107231(n): return sum(T(j)^2 for j in (0..n))
[A107239(n) for n in (0..40)] # G. C. Greubel, Nov 20 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, May 17 2005
STATUS
approved