A107239 Sum of squares of tribonacci numbers (A000073).

0, 0, 1, 2, 6, 22, 71, 240, 816, 2752, 9313, 31514, 106590, 360606, 1219935, 4126960, 13961456, 47231280, 159782161, 540539330, 1828631430, 6186215574, 20927817799, 70798300288, 239508933824, 810252920400, 2741065994769, 9272959837818, 31370198430718

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

Cf. A000073, A000213, A085697.

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

Sequence in context: A002839 A109194 A014334 * A262068 A148496 A217528

Adjacent sequences: A107236 A107237 A107238 * A107240 A107241 A107242

Keyword

easy,nonn

Author

Jonathan Vos Post, May 17 2005

Status

approved