A002477 Wonderful Demlo numbers: a(n) = ((10^n - 1)/9)^2. (Formerly M5386 N2339)

1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 1234567900987654321, 123456790120987654321, 12345679012320987654321, 1234567901234320987654321, 123456790123454320987654321, 12345679012345654320987654321, 1234567901234567654320987654321

Offset

1,2

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

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 280.

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).

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

Example

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

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 *)

Prog

(PARI) a(n) = (10^n\9)^2 \\ Charles R Greathouse IV, Jul 25 2011
(Magma) [((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Jul 26 2011
(Maxima) A002477(n):=((10^n - 1)/9)^2$
makelist(A002477(n), n, 1, 10); /* Martin Ettl, Nov 12 2012 */

Crossrefs

Cf. A002275, A080151.

Sequence in context: A062689 A057139 A321687 * A173426 A261570 A068117

Adjacent sequences: A002474 A002475 A002476 * A002478 A002479 A002480

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Further edits from Reinhard Zumkeller, May 12 2010

Status

approved