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A322708
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a(0)=0, a(1)=6 and a(n) = 26*a(n-1) - a(n-2) + 12 for n > 1.
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2
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0, 6, 168, 4374, 113568, 2948406, 76545000, 1987221606, 51591216768, 1339384414374, 34772403556968, 902743108066806, 23436548406180000, 608447515452613206, 15796198853361763368, 410092722671953234374, 10646614590617422330368, 276401886633381027355206
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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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sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(7) + sqrt(6))^n.
sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(7) - sqrt(6))^n.
a(n) = 27*a(n-1) - 27*a(n-2) + a(n-3) for n > 2.
G.f.: 6*x*(1 + x) / ((1 - x)*(1 - 26*x + x^2)).
a(n) = ((13+2*sqrt(42))^(-n) * (-1+(13+2*sqrt(42))^n)^2) / 4.
(End)
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EXAMPLE
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(sqrt(7) + sqrt(6))^2 = 13 + 2*sqrt(42) = sqrt(169) + sqrt(168). So a(2) = 168.
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MATHEMATICA
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LinearRecurrence[{27, -27, 1}, {0, 6, 168}, 20] (* Harvey P. Dale, Apr 30 2022 *)
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PROG
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(PARI) concat(0, Vec(6*x*(1 + x) / ((1 - x)*(1 - 26*x + x^2)) + O(x^20))) \\ Colin Barker, Dec 24 2018
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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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STATUS
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approved
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