cp's OEIS Frontend

A271528 a(n) = 2*(10^n - 1)^2/27.

%I A271528 #27 Feb 16 2025 08:33:33
%S A271528 0,6,726,73926,7405926,740725926,74073925926,7407405925926,
%T A271528 740740725925926,74074073925925926,7407407405925925926,
%U A271528 740740740725925925926,74074074073925925925926,7407407407405925925925926,740740740740725925925925926,74074074074073925925925925926
%N A271528 a(n) = 2*(10^n - 1)^2/27.
%C A271528 All terms are multiple of 6.
%C A271528 Converges in a 10-adic sense to ...925925925926.
%C A271528 A transformation of the Wonderful Demlo numbers (A002477).
%C A271528 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 A271528 Ilya Gutkovskiy, <a href="/A271528/a271528.pdf">Transformation of the Wonderful Demlo numbers</a>
%H A271528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DemloNumber.html">Demlo Number</a>
%H A271528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H A271528 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F A271528 O.g.f.: 6*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).
%F A271528 E.g.f.: 2 (exp(x) - 2*exp(10*x) + exp(100*x))/27.
%F A271528 a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
%F A271528 a(n) = 6*A002477(n) = 6*A002275(n)^2 = A002276(n)*A002277(n) = sqrt(A075411(n)*A075412(n)).
%F A271528 Sum_{n>=1} 1/a(n) = 0.1680577405662077350849154881928636039793563...
%F A271528 Lim_{n -> infinity} a(n + 1)/a(n) = 100.
%e A271528 n=1:                  6 = 2 * 3;
%e A271528 n=2:                726 = 22 * 33;
%e A271528 n=3:              73926 = 222 * 333;
%e A271528 n=4:            7405926 = 2222 * 3333;
%e A271528 n=5:          740725926 = 22222 * 33333;
%e A271528 n=6:        74073925926 = 222222 * 333333;
%e A271528 n=7:      7407405925926 = 2222222 * 3333333;
%e A271528 n=8:    740740725925926 = 22222222 * 33333333;
%e A271528 n=9:  74074073925925926 = 222222222 * 333333333, etc.
%t A271528 Table[2 ((10^n - 1)^2/27), {n, 0, 15}]
%t A271528 LinearRecurrence[{111, -1110, 1000}, {0, 6, 726}, 16]
%o A271528 (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 A271528 (Python)
%o A271528 for n in range(0,10**1):print((int)((2*(10**n-1)**2)/27))
%o A271528 # _Soumil Mandal_, Apr 10 2016
%Y A271528 Cf. A002275, A002276, A002277, A002477.
%Y A271528 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 A271528 nonn,easy
%O A271528 0,2
%A A271528 _Ilya Gutkovskiy_, Apr 09 2016