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A203629
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Indices of 10-gonal (decagonal) numbers which are also 9-gonal (nonagonal).
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2
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1, 551, 494461, 444025091, 398734036921, 358062721129631, 321539924840371381, 288742494443932370171, 259290438470726428041841, 232842525004217888449202711, 209092328163349193100955992301, 187764677848162571186770031883251
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OFFSET
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1,2
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COMMENTS
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As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity, a(n)/a(n-1)) = (2*sqrt(2)+sqrt(7))^4 = 449+120*sqrt(14).
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LINKS
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FORMULA
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G.f.: x*(1-348*x+11*x^2) / ((1-x)*(1-898*x+x^2)).
a(n) = 898*a(n-1)-a(n-2)-336.
a(n) = 899*a(n-1)-899*a(n-2)+a(n-3).
a(n) = 1/112*((sqrt(7)+7*sqrt(2))*(2*sqrt(2)+sqrt(7))^(4*n-3)-(sqrt(7)-7*sqrt(2))*(2*sqrt(2)-sqrt(7))^(4*n-3)+42).
a(n) = ceiling(1/112*(sqrt(7)+7*sqrt(2))*(2*sqrt(2)+sqrt(7))^(4*n-3)).
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EXAMPLE
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The second number that is both 9-gonal (nonagonal) and 10-gonal (decagonal) is A001107(551) = 1212751. Hence a(2) = 551.
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MATHEMATICA
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LinearRecurrence[{899, -899, 1}, {1, 551, 494461}, 12]
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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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