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A002477 Wonderful Demlo numbers: a(n) = ((10^n - 1)/9)^2.

%I A002477 M5386 N2339 #126 Aug 03 2025 16:05:49
%S A002477 1,121,12321,1234321,123454321,12345654321,1234567654321,
%T A002477 123456787654321,12345678987654321,1234567900987654321,
%U A002477 123456790120987654321,12345679012320987654321,1234567901234320987654321
%N A002477 Wonderful Demlo numbers: a(n) = ((10^n - 1)/9)^2.
%C A002477 Only the first nine terms of this sequence are palindromes. - _Bui Quang Tuan_, Mar 30 2015
%C A002477 Not all of the terms are Demlo numbers as defined by Kaprekar, i.e., concat(L,M,R) with M and L+R repdigits using the same digit. For example, a(10), a(19), a(28) are not, but a(k) for k = 11, 12, ..., 18 are. - _M. F. Hasler_, Nov 18 2017
%D A002477 D. R. Kaprekar, On Wonderful Demlo numbers, Math. Stud., 6 (1938), 68.
%D A002477 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 29.
%D A002477 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002477 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002477 Vincenzo Librandi, <a href="/A002477/b002477.txt">Table of n, a(n) for n = 1..300</a>
%H A002477 Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, <a href="https://arxiv.org/abs/2008.06864">Antipalindromic numbers</a>, arXiv:2008.06864 [math.CO], 2020. [Mentions this sequence.]
%H A002477 K. R. Gunjikar and D. R. Kaprekar, <a href="/A249605/a249605.pdf">Theory of Demlo numbers</a>, J. Univ. Bombay, Vol. VIII, Part 3, Nov. 1939, pp. 3-9. [Annotated scanned copy]
%H A002477 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A002477 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A002477 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DemloNumber.html">Demlo Number</a>
%H A002477 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H A002477 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F A002477 G.f.: x*(1+10*x) / ((1-x)*(1-10*x)*(1-100*x)). - _Simon Plouffe_ in his 1992 dissertation
%F A002477 a(n+1) = 100*a(n) + 20*A000042(n) + 1; a(1) = 1. - _Reinhard Zumkeller_, May 31 2010
%F A002477 a(n) = A000042(n)^2.
%F A002477 a(n) = A075412(n)/9 = A178630(n)/18 = A178631(n)/27 = A075415(n)/36 = A178632(n)/45 = A178633(n)/54 = A178634(n)/63 = A178635(n)/72 = A059988(n)/81. - _Reinhard Zumkeller_, May 31 2010
%F A002477 a(n+2) = -1000*a(n)+110*a(n+1)+11. - _Alexander R. Povolotsky_, Jun 06 2014
%F A002477 E.g.f.: exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81. - _Stefano Spezia_, May 23 2025
%e A002477 From _José de Jesús Camacho Medina_, Apr 01 2016: (Start)
%e A002477 n=1: ....................... 1 = 9 / 9;
%e A002477 n=2: ..................... 121 = 1089 / 9;
%e A002477 n=3: ................... 12321 = 110889 / 9;
%e A002477 n=4: ................. 1234321 = 11108889 / 9;
%e A002477 n=5: ............... 123454321 = 1111088889 / 9;
%e A002477 n=6: ............. 12345654321 = 111110888889 / 9;
%e A002477 n=7: ........... 1234567654321 = 11111108888889 / 9;
%e A002477 n=8: ......... 123456787654321 = 1111111088888889 / 9;
%e A002477 n=9: ....... 12345678987654321 = 111111110888888889 / 9.        (End)
%e A002477 a(11) = concat(L = 1234567901, R = 20987654321), with L + R = 22222222222 = 2*(10^11-1)/9, of same length as R. - _M. F. Hasler_, Nov 23 2017
%p A002477 A002477 := proc(n)
%p A002477     (10^n-1)^2/81 ;
%p A002477 end proc:
%p A002477 seq(A002477(n),n=1..12) ; # _R. J. Mathar_, Aug 06 2019
%t A002477 Table[FromDigits[PadRight[{},n,1]]^2,{n,15}] (* _Harvey P. Dale_, Oct 16 2012 *)
%t A002477 (10^Range[20] - 1)^2/81 (* _Paolo Xausa_, Aug 03 2025 *)
%o A002477 (PARI) a(n) = (10^n\9)^2 \\ _Charles R Greathouse IV_, Jul 25 2011
%o A002477 (Magma) [((10^n - 1)/9)^2: n in [1..20]]; // _Vincenzo Librandi_, Jul 26 2011
%o A002477 (Maxima) A002477(n):=((10^n - 1)/9)^2$
%o A002477 makelist(A002477(n),n,1,10); /* _Martin Ettl_, Nov 12 2012 */
%Y A002477 Cf. A002275, A080151.
%K A002477 nonn,easy
%O A002477 1,2
%A A002477 _N. J. A. Sloane_
%E A002477 Minor edits from _N. J. A. Sloane_, Aug 18 2009
%E A002477 Further edits from _Reinhard Zumkeller_, May 12 2010