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A120718
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Expansion of 3*x/(1 - 2*x^2 - 2*x + x^3).
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1
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0, 3, 6, 18, 45, 120, 312, 819, 2142, 5610, 14685, 38448, 100656, 263523, 689910, 1806210, 4728717, 12379944, 32411112, 84853395, 222149070, 581593818, 1522632381, 3986303328, 10436277600, 27322529475, 71531310822, 187271402994, 490282898157, 1283577291480
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) for n>2.
a(n) = (3/2^(n+1))*( (1-sqrt(5))*(3-sqrt(5))^n + (1+sqrt(5))*(3+sqrt(5))^n + (-2)^(n+1) )/5. (End)
a(n) = (3/5)*(Lucas(2*n+1) - (-1)^n). - G. C. Greubel, Jul 21 2023
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MATHEMATICA
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PROG
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(PARI) a(n) = 3*(fibonacci(2*n+2) + fibonacci(2*n) - (-1)^n)/5 \\ Colin Barker, Oct 01 2016
(PARI) concat(0, Vec(3*x/(1-2*x^2-2*x+x^3) + O(x^40))) \\ Colin Barker, Oct 01 2016
(Magma) [(3/5)*(Lucas(2*n+1) -(-1)^n): n in [0..40]]; // G. C. Greubel, Jul 21 2023
(SageMath) [(3/5)*(lucas_number2(2*n+1, 1, -1) -(-1)^n) for n in range(41)] # G. C. Greubel, Jul 21 2023
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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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