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A178634 a(n) = 63*((10^n - 1)/9)^2.

%I A178634 #41 Aug 02 2025 10:03:09
%S A178634 63,7623,776223,77762223,7777622223,777776222223,77777762222223,
%T A178634 7777777622222223,777777776222222223,77777777762222222223,
%U A178634 7777777777622222222223,777777777776222222222223,77777777777762222222222223,7777777777777622222222222223,777777777777776222222222222223
%N A178634 a(n) = 63*((10^n - 1)/9)^2.
%D A178634 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 33 at p. 62.
%D A178634 Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.
%H A178634 G. C. Greubel, <a href="/A178634/b178634.txt">Table of n, a(n) for n = 1..200</a>
%H A178634 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F A178634 a(n) = 63*A002477(n) = A002283(n)*A002281(n).
%F A178634 a(n) = ((A002281(n-1)*10 + 6)*10^(n-1) + A002276(n-1))*10 + 3.
%F A178634 G.f.: 63*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - _Ilya Gutkovskiy_, Feb 24 2017
%F A178634 E.g.f.: 7*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9. - _Stefano Spezia_, Jul 31 2024
%F A178634 a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. - _Elmo R. Oliveira_, Aug 01 2025
%e A178634 n=1: ..................... 63 = 9 * 7;
%e A178634 n=2: ................... 7623 = 99 * 77;
%e A178634 n=3: ................. 776223 = 999 * 777;
%e A178634 n=4: ............... 77762223 = 9999 * 7777;
%e A178634 n=5: ............. 7777622223 = 99999 * 77777;
%e A178634 n=6: ........... 777776222223 = 999999 * 777777;
%e A178634 n=7: ......... 77777762222223 = 9999999 * 7777777;
%e A178634 n=8: ....... 7777777622222223 = 99999999 * 77777777;
%e A178634 n=9: ..... 777777776222222223 = 999999999 * 777777777.
%t A178634 63((10^Range[15]-1)/9)^2 (* or *) Table[FromDigits[Join[PadRight[{},n,7],{6},PadRight[{},n,2],{3}]],{n,0,15}] (* _Harvey P. Dale_, Apr 23 2012 *)
%o A178634 (Magma) [63*((10^n - 1)/9)^2: n in [1..20]]; // _Vincenzo Librandi_, Dec 28 2010
%o A178634 (PARI) a(n)=63*(10^n\9)^2 \\ _Charles R Greathouse IV_, Jul 02 2013
%o A178634 (Sage) [63*((10^n - 1)/9)^2 for n in (1..20)] # _G. C. Greubel_, Jan 28 2019
%o A178634 (GAP) List([1..20], n -> 63*((10^n - 1)/9)^2); # _G. C. Greubel_, Jan 28 2019
%Y A178634 Cf. A059988, A075412, A075415, A178630, A178631, A178632, A178633, A178635.
%Y A178634 Cf. A002276, A002281, A002283, A002477.
%K A178634 nonn,easy
%O A178634 1,1
%A A178634 _Reinhard Zumkeller_, May 31 2010