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

1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 1234567900987654321, 123456790120987654321, 12345679012320987654321, 1234567901234320987654321

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Only the first nine terms of this sequence are palindromes. - Bui Quang Tuan, Mar 30 2015

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

Examples

			From _José de Jesús Camacho Medina_, Apr 01 2016: (Start)
n=1: ....................... 1 = 9 / 9;
n=2: ..................... 121 = 1089 / 9;
n=3: ................... 12321 = 110889 / 9;
n=4: ................. 1234321 = 11108889 / 9;
n=5: ............... 123454321 = 1111088889 / 9;
n=6: ............. 12345654321 = 111110888889 / 9;
n=7: ........... 1234567654321 = 11111108888889 / 9;
n=8: ......... 123456787654321 = 1111111088888889 / 9;
n=9: ....... 12345678987654321 = 111111110888888889 / 9.        (End)
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
		

References

• D. R. Kaprekar, On Wonderful Demlo numbers, Math. Stud., 6 (1938), 68.
• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 29.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..300
• Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. [Mentions this sequence.]
• K. R. Gunjikar and D. R. Kaprekar, Theory of Demlo numbers, J. Univ. Bombay, Vol. VIII, Part 3, Nov. 1939, pp. 3-9. [Annotated scanned copy]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Eric Weisstein's World of Mathematics, Demlo Number
• Eric Weisstein's World of Mathematics, Repunit
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A002275, A080151.

Programs

Magma

[((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Jul 26 2011

Maple

A002477 := proc(n)
    (10^n-1)^2/81 ;
end proc:
seq(A002477(n),n=1..12) ; # R. J. Mathar, Aug 06 2019

Mathematica

Table[FromDigits[PadRight[{},n,1]]^2,{n,15}] (* Harvey P. Dale, Oct 16 2012 *)
(10^Range[20] - 1)^2/81 (* Paolo Xausa, Aug 03 2025 *)

Maxima

A002477(n):=((10^n - 1)/9)^2$
makelist(A002477(n),n,1,10); /* Martin Ettl, Nov 12 2012 */

PARI

a(n) = (10^n\9)^2 \\ Charles R Greathouse IV, Jul 25 2011

Formula

G.f.: x*(1+10*x) / ((1-x)*(1-10*x)*(1-100*x)). - Simon Plouffe in his 1992 dissertation

a(n+1) = 100*a(n) + 20*A000042(n) + 1; a(1) = 1. - Reinhard Zumkeller, May 31 2010

a(n) = A000042(n)^2.

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

a(n+2) = -1000*a(n)+110*a(n+1)+11. - Alexander R. Povolotsky, Jun 06 2014

E.g.f.: exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81. - Stefano Spezia, May 23 2025

Extensions

Minor edits from N. J. A. Sloane, Aug 18 2009

Further edits from Reinhard Zumkeller, May 12 2010