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A203408
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Numbers which are both heptagonal and decagonal.
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2
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1, 540, 2887450, 1123674201, 6004054625647, 2336525434757970, 12484603034492528512, 4858482201068079159687, 25960009135002449017962445, 10102543266574986692211140472, 53980256514964477791853933850326, 21006844571867038996088473395797925
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OFFSET
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1,2
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COMMENTS
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As n increases, the ratios of consecutive terms settle into an approximate 2-cycle with a(n)/a(n-1) bounded above and below by 1/81*(216401+68432*sqrt(10)) and 1/81*(15761+4984*sqrt(10)) respectively.
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LINKS
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FORMULA
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G.f.: x(1+539*x+807548*x^2+10633*x^3+27*x^4) / ((1-x)*(1-1442*x+x^2)*(1+1442*x+x^2)).
a(n) = 2079362*a(n-2)-a(n-4)+818748.
a(n) = a(n-1)+2079362*a(n-2)-2079362*a(n-3)-a(n-4)+a(n-5).
a(n) = 1/320*((11-2*sqrt(10)*(-1)^n)*(1+sqrt(10))* (3+sqrt(10))^(4*n-3)+(11+2*sqrt(10)*(-1)^n)*(1-sqrt(10))*(3-sqrt(10))^(4*n-3)-126).
a(n) = floor(1/320*(11-2*sqrt(10)*(-1)^n)*(1+sqrt(10))* (3+sqrt(10))^(4*n-3)).
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EXAMPLE
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The second number that is both heptagonal and decagonal is 540. Hence a(2)=540.
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MATHEMATICA
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LinearRecurrence[{1, 2079362, -2079362, -1, 1}, {1, 540, 2887450, 1123674201, 6004054625647}, 15]
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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