A107239 - OEIS

%I #49 Sep 08 2022 08:45:18

%S 0,0,1,2,6,22,71,240,816,2752,9313,31514,106590,360606,1219935,

%T 4126960,13961456,47231280,159782161,540539330,1828631430,6186215574,

%U 20927817799,70798300288,239508933824,810252920400,2741065994769,9272959837818,31370198430718

%N Sum of squares of tribonacci numbers (A000073).

%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="/A107239/b107239.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Feinberg, <a href="http://www.fq.math.ca/Scanned/1-3/feinberg.pdf">Fibonacci-Tribonacci</a>, Fib. Quart. 1(3) (1963), 71-74.

%H Z. Jakubczyk, <a href="http://www.fq.math.ca/Problems/August2013advanced.pdf">Advanced Problems and Solutions</a>, Fib. Quart. 51 (3) (2013) 185, H-715.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,3,-7,1,-1,1).

%F a(n) = T(0)^2 + T(1)^2 + ... + T(n)^2 where T(n) = A000073(n).

%F From _R. J. Mathar_, Aug 19 2008: (Start)

%F a(n) = Sum_{i=0..n} A085697(i).

%F 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)

%F 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

%F 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

%e a(7) = 71 = 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2 + 7^2

%p b:= proc(n) option remember; `if`(n<3, [n*(n-1)/2$2],

%p      (t-> [t, t^2+b(n-1)[2]])(add(b(n-j)[1], j=1..3)))

%p     end:

%p a:= n-> b(n)[2]:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 22 2021

%t Accumulate[LinearRecurrence[{1,1,1},{0,0,1},30]^2] (* _Harvey P. Dale_, Sep 11 2011 *)

%t LinearRecurrence[{3,1,3,-7,1,-1,1}, {0,0,1,2,6,22,71}, 30] (* _Ray Chandler_, Aug 02 2015 *)

%o (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

%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 A107231(n): return sum(T(j)^2 for j in (0..n))

%o [A107239(n) for n in (0..40)] # _G. C. Greubel_, Nov 20 2021

%Y Cf. A000073, A000213, A085697.

%Y Cf. A107240, A107241, A107242, A107243, A107244, A107245, A107246, A107247.

%K easy,nonn

%O 0,4

%A _Jonathan Vos Post_, May 17 2005