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A005969
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Sum of fourth powers of Fibonacci numbers.
(Formerly M2106)
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11
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1, 2, 18, 99, 724, 4820, 33381, 227862, 1564198, 10714823, 73457064, 503438760, 3450734281, 23651386922, 162109796922, 1111115037483, 7615701104764, 52198777931900, 357775783071021, 2452231602371646, 16807845698458702
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listen;
history;
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OFFSET
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1,2
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REFERENCES
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A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 19.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: x*(1+x)*(x^2-5*x+1)/ ( (x^2+3*x+1)*(x^2-7*x+1)*(x-1)^2 ). - Ralf Stephan, Apr 23 2004
a(n) = (1/25)*(F(4n+2)-(-1)^n*4*F(2n+1)+6n+3) where F(n)=A000045(n). - Benoit Cloitre, Sep 13 2004. [Corrected by David Lambert (dave.lambert(AT)comcast.net), Mar 28 2008]
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MAPLE
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with(combinat): l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+fibonacci(i)^4; printf(`%d, `, l[i]) od: # James A. Sellers, May 29 2000
A005969:=(z+1)*(z**2-5*z+1)/(z**2-7*z+1)/(z**2+3*z+1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation, offset zero
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MATHEMATICA
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CoefficientList[Series[(1+x)*(x^2-5*x+1)/((x^2+3*x+1)*(x^2-7*x+1)*(x- 1)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 02 2017 *)
LinearRecurrence[{6, 10, -30, 10, 6, -1}, {1, 2, 18, 99, 724, 4820}, 30] (* G. C. Greubel, Jan 17 2018 *)
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PROG
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(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; -1, 6, 10, -30, 10, 6]^n*[0; 1; 2; 18; 99; 724])[1, 1] \\ Charles R Greathouse IV, Sep 28 2015
(Magma) [(1/25)*(Fibonacci(4*n+2)-(-1)^n*4*Fibonacci(2*n+1)+6*n+3): n in [1..25]]; // Vincenzo Librandi, Jun 02 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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