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A131780
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Row sums of triangle A131779.
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3
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1, 4, 5, 10, 15, 26, 41, 68, 109, 178, 287, 466, 753, 1220, 1973, 3194, 5167, 8362, 13529, 21892, 35421, 57314, 92735, 150050, 242785, 392836, 635621, 1028458, 1664079, 2692538, 4356617, 7049156, 11405773, 18454930, 29860703, 48315634, 78176337, 126491972
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OFFSET
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1,2
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COMMENTS
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a(n)/a(n-1) tends to phi; e.g., a(10)/a(9) = 178/109 = 1.633...
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1)
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FORMULA
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From Andrew Howroyd, Sep 01 2018: (Start)
a(n) = 2*Fibonacci(n+1) - (1 - (-1)^n)/2.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n > 4.
G.f.: x*(1 + 3*x - x^2 - 2*x^3)/((1 - x)*(1 + x)*(1 - x - x^2)).
(End)
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EXAMPLE
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a(4) = 10 = sum of row 4 terms of triangle A131779: (3 + 1 + 5 + 1).
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MATHEMATICA
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LinearRecurrence[{1, 2, -1, -1}, {1, 4, 5, 10}, 40] (* or *) CoefficientList[ Series[(1+3*x-x^2-2*x^3)/(1-x-2*x^2+x^3+x^4), {x, 0, 40}], x] (* Harvey P. Dale, Aug 27 2021 *)
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PROG
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(PARI) a(n) = 2*fibonacci(n+1) - (1 - (-1)^n)/2; \\ Andrew Howroyd, Sep 01 2018
(PARI) Vec((1 + 3*x - x^2 - 2*x^3)/((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^40)) \\ Andrew Howroyd, Sep 01 2018
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CROSSREFS
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Row sums of A131779.
Sequence in context: A119040 A287578 A135104 * A354968 A102006 A118735
Adjacent sequences: A131777 A131778 A131779 * A131781 A131782 A131783
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KEYWORD
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nonn,easy
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AUTHOR
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Gary W. Adamson, Jul 14 2007
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EXTENSIONS
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Terms a(11) and beyond from Andrew Howroyd, Sep 01 2018
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STATUS
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approved
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