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%I #26 Dec 07 2019 12:18:28
%S 0,6,726,73926,7405926,740725926,74073925926,7407405925926,
%T 740740725925926,74074073925925926,7407407405925925926,
%U 740740740725925925926,74074074073925925925926,7407407407405925925925926,740740740740725925925925926,74074074074073925925925925926
%N a(n) = 2*(10^n - 1)^2/27.
%C All terms are multiple of 6.
%C Converges in a 10-adic sense to ...925925925926.
%C A transformation of the Wonderful Demlo numbers (A002477).
%C More generally, the ordinary generating function for the transformation of the Wonderful Demlo numbers, is k*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).
%H Ilya Gutkovskiy, <a href="/A271528/a271528.pdf">Transformation of the Wonderful Demlo numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DemloNumber.html">Demlo Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F O.g.f.: 6*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).
%F E.g.f.: 2 (exp(x) - 2*exp(10*x) + exp(100*x))/27.
%F a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
%F a(n) = 6*A002477(n) = 6*A002275(n)^2 = A002276(n)*A002277(n) = sqrt(A075411(n)*A075412(n)).
%F Sum_{n>=1} 1/a(n) = 0.1680577405662077350849154881928636039793563...
%F Lim_{n -> infinity} a(n + 1)/a(n) = 100.
%e n=1: 6 = 2 * 3;
%e n=2: 726 = 22 * 33;
%e n=3: 73926 = 222 * 333;
%e n=4: 7405926 = 2222 * 3333;
%e n=5: 740725926 = 22222 * 33333;
%e n=6: 74073925926 = 222222 * 333333;
%e n=7: 7407405925926 = 2222222 * 3333333;
%e n=8: 740740725925926 = 22222222 * 33333333;
%e n=9: 74074073925925926 = 222222222 * 333333333, etc.
%t Table[2 ((10^n - 1)^2/27), {n, 0, 15}]
%t LinearRecurrence[{111, -1110, 1000}, {0, 6, 726}, 16]
%o (PARI) x='x+O('x^99); concat(0, Vec(6*x*(1+10*x)/(1-111*x+1110*x^2-1000*x^3))) \\ _Altug Alkan_, Apr 09 2016
%o (Python)
%o for n in range(0,10**1):print((int)((2*(10**n-1)**2)/27))
%o # _Soumil Mandal_, Apr 10 2016
%Y Cf. A002275, A002276, A002277, A002477.
%Y Cf. similar sequences of the form k*((10^n - 1)/9)^2: A075411 (k=4), this sequence (k=6), A075412 (k=9), A075413 (k=16), A178630 (k=18), A075414 (k=25), A178631 (k=27), A075415 (k=36), A178632 (k=45), A075416 (k=49), A178633 (k=54), A178634 (k=63), A075417 (k=64), A178635 (k=72), A059988 (k=81).
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Apr 09 2016
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