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A002477 Wonderful Demlo numbers: a(n) = ((10^n - 1)/9)^2.
(Formerly M5386 N2339) 		36
1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 1234567900987654321, 123456790120987654321, 12345679012320987654321, 1234567901234320987654321 (list; graph; refs; listen; history; text; internal format)
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
D. R. Kaprekar, On Wonderful Demlo numbers, Math. Stud., 6 (1938), 68.
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
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 *)
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

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Last modified August 28 02:57 EDT 2024. Contains 375477 sequences. (Running on oeis4.)