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 A107239 Sum of squares of tribonacci numbers (A000073). 15
 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 (list; graph; refs; listen; history; text; internal format)
 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:=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

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Last modified August 18 18:56 EDT 2022. Contains 356215 sequences. (Running on oeis4.)