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%I #15 Mar 28 2024 09:01:54
%S 1,15,1075,21201,1549717,30571395,2234690407,44083929957,
%T 3222422016745,63568996426167,4646730313455451,91666448762602425,
%U 6700581889580743165,132182955546676270251,9662234438045118188047,190607730231858419099085,13932935359079170846420177
%N Indices of heptagonal numbers that are also decagonal.
%C 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 1/9*(329+104*sqrt(10)) and 1/9*(89+28*sqrt(10)) respectively.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1442,-1442,-1,1).
%F G.f.: x*(1+14*x-382*x^2-62*x^3-3*x^4) / ((1-x)*(1-38*x+x^2)*(1+38*x+x^2)).
%F a(n) = 1442*a(n-2)-a(n-4)-432.
%F a(n) = a(n-1)+1442*a(n-2)-1442*a(n-3)-a(n-4)+a(n-5).
%F a(n) = 1/40*(((-1)^n-sqrt(10))*(2-sqrt(10))*(3+sqrt(10))^(2*n-1)+((-1)^n+sqrt(10))*(2+sqrt(10))*(3-sqrt(10))^(2*n-1)+12).
%F a(n) = ceiling(1/40*((-1)^n-sqrt(10))*(2-sqrt(10))*(3+sqrt(10))^(2*n-1)).
%e The second heptagonal number that is also decagonal is A000566(15)=540. Hence a(2)=15.
%t LinearRecurrence[{1, 1442, -1442, -1, 1}, {1, 15, 1075, 21201, 1549717}, 17]
%Y Cf. A203408, A203410, A001107, A000566.
%K nonn,easy
%O 1,2
%A _Ant King_, Jan 02 2012
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