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A107239 Sum of squares of tribonacci numbers (A000073). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Not to be confused with A107240 which is based on alternate tribonacci sequence A000213(n), which starts 1,1,1,3. Prime values include: a(4) = 2, a(7) = 71. Semiprime values include: a(5) = 6 = 2 * 3, a(6) = 22 = 2 * 11, a(11) = 9313 = 67 * 139, a(35) = 3 * 15674342521439179.

LINKS

Table of n, a(n) for n=0..26.

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

a(n) = sum(i=0..n-2) A085697(i). G.f.: x^2*(1-x-x^2-x^3)/((x^3-x^2-x-1)(x^3+x^2+3*x-1)(1-x)). - R. J. Mathar, Aug 19 2008

a(n) = 1/4 -1/11*sum((3+7*_R+5*_R^2)/(3*_R^2-2*_R-1)*_R^(-n), _R = RootOf(_Z^3-_Z^2-_Z-1)) -1/44*sum((-1-2*_R-9*_R^2)/(3*_R^2+2*_R+3)*_R^(-n), _R = RootOf(_Z^3+_Z^2+3*_Z-1)). - 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(1) = 0 = 0^2

a(2) = 0 = 0^2 + 0^2

a(3) = 1 = 0^2 + 0^2 + 1^2

a(4) = 2 = 0^2 + 0^2 + 1^2 + 1^2

a(5) = 6 = 0^2 + 0^2 + 1^2 + 1^2 + 2^2

a(6) = 22 = 0^2 + 0^2 + 1^2 + 1^1 + 2^2 + 4^2

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

a(8) = 240 = 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2 + 7^2 + 13^2

a(9) = 816 = 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2 + 7^2 + 13^2 + 24^2

a(10) = 2752 = 44^2 + 816

a(11) = 9313 = 81^2 + 2752

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}, 27] (* Ray Chandler, Aug 02 2015 *)

CROSSREFS

Cf. A000073, A000213, A107240, A107241-A107248.

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 July 15 14:12 EDT 2020. Contains 335772 sequences. (Running on oeis4.)