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A097132
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a(n) = Sum_{k=0..n} Fibonacci(k) + (-1)^k*Fibonacci(k-1).
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1
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1, 2, 4, 5, 10, 12, 25, 30, 64, 77, 166, 200, 433, 522, 1132, 1365, 2962, 3572, 7753, 9350, 20296, 24477, 53134, 64080, 139105, 167762, 364180, 439205, 953434, 1149852, 2496121, 3010350, 6534928, 7881197, 17108662, 20633240, 44791057
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1 + x - x^2 - 2*x^3)/((1 - 3*x^2 + x^4)*(1-x));
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - a(n-4) + a(n-5);
a(n) = 1 + (1/2 - sqrt(5)/2)^n*(1/2 - 3*sqrt(5)/10) - (sqrt(5)/2 - 1/2)^n*(3*sqrt(5)/10 + 1/2) + (-sqrt(5)/2 - 1/2)^n*(3*sqrt(5)/10 - 1/2) + (sqrt(5)/2 + 1/2)^n*(3*sqrt(5)/10 + 1/2);
a(2n) = 1 + 3*Fibonacci(2n) = A097136(n);
a(2n+1) = 1 + Fibonacci(2n) + Fibonacci(2n+2) = 1 + Lucas(2n).
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MATHEMATICA
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LinearRecurrence[{1, 3, -3, -1, 1}, {1, 2, 4, 5, 10}, 40] (* Harvey P. Dale, Nov 12 2022 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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