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A107239
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Sum of squares of tribonacci numbers (A000073).
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9
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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; internal format)
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OFFSET
| 0,4
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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.
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REFERENCES
| M. Feinberg, "Fibonacci-Tribonacci." Fib. Quart. 1, 71-74, 1963.
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LINKS
| Eric Weisstein's World of Mathematics, Tribonacci Number..
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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)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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)) [From Robert Israel (israel(AT)math.ubc.ca), Mar 26 2010]
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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
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MATHEMATICA
| Accumulate[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 30]^2] (* From Harvey P. Dale, Sep 11 2011 *)
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CROSSREFS
| Cf. A000073, A000213, A107240, A107241-A107248.
Sequence in context: A002839 A109194 A014334 * A148496 A181367 A106434
Adjacent sequences: A107236 A107237 A107238 * A107240 A107241 A107242
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), May 17 2005
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