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%I #139 Nov 10 2023 09:35:02
%S 0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,111,222,333,444,555,
%T 666,777,888,999,1111,2222,3333,4444,5555,6666,7777,8888,9999,11111,
%U 22222,33333,44444,55555,66666,77777,88888,99999,111111,222222,333333,444444,555555,666666
%N Repdigit numbers, or numbers whose digits are all equal.
%C Complement of A139819. - _David Wasserman_, May 21 2008
%C Subsequence of A134336 and of A178403. - _Reinhard Zumkeller_, May 27 2010
%C Subsequence of A193460. - _Reinhard Zumkeller_, Jul 26 2011
%C Intersection of A009994 and A009996. - _David F. Marrs_, Sep 29 2018
%C Beiler (1964) called these numbers "monodigit numbers". The term "repdigit numbers" was used by Trigg (1974). - _Amiram Eldar_, Jan 21 2022
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, p. 83.
%H Reinhard Zumkeller, <a href="/A010785/b010785.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric F. Bravo, Carlos A. Gómez and Florian Luca, <a href="https://www.emis.de/journals/JIS/VOL22/Gomez/gomez3.html">Product of Consecutive Tribonacci Numbers With Only One Distinct Digit</a>, J. Int. Seq., Vol. 22 (2019), Article 19.6.3.
%H Eric Fernando Bravo, <a href="https://www.mathos.unios.hr/mc/index.php/mc/article/view/4733">On concatenations of Padovan and Perrin numbers</a>, Math. Commun. (2023) Vol 28, 105-119.
%H Mahadi Ddamulira, <a href="https://hal.archives-ouvertes.fr/hal-02405969">Repdigits as sums of three balancing numbers</a>, Mathematica Slovaca, (2019), hal-02405969.
%H Mahadi Ddamulira, <a href="https://arxiv.org/abs/2003.10705">Padovan numbers that are concatenations of two distinct repdigits</a>, arXiv:2003.10705 [math.NT], 2020.
%H Mahadi Ddamulira, <a href="https://hal.archives-ouvertes.fr/hal-02547159">Tribonacci numbers that are concatenations of two repdigits</a>, hal-02547159, Mathematics [math] / Number Theory [math.NT], 2020.
%H Mahadi Ddamulira, <a href="https://doi.org/10.33774/coe-2020-smm9j-v2">Padovan numbers that are concatenations of two distinct repdigits</a>, Mathematica Slovaca, Vol. 71, No. 2 (2021), pp. 275-284.
%H Bart Goddard and Jeremy Rouse, <a href="http://arxiv.org/abs/1607.06681">Sum of two repdigits a square</a>, arXiv:1607.06681 [math.NT], 2016. Mentions this sequence.
%H Bir Kafle, Florian Luca and Alain Togbé, <a href="https://www.fq.math.ca/Abstracts/56-4/kafle.pdf">Triangular Repblocks</a>, Fibonacci Quart., Vol. 56, No. 4 (2018), pp. 325-328.
%H Bir Kafle, Florian Luca and Alain Togbé, <a href="https://doi.org/10.33039/ami.2020.09.002">Pentagonal and heptagonal repdigits</a>, Annales Mathematicae et Informaticae, Vol. 52 (2020), pp. 137-145.
%H Benedict Vasco Normenyo, Bir Kafle, and Alain Togbé, <a href="http://math.colgate.edu/~integers/t55/t55.pdf">Repdigits as Sums of Two Fibonacci Numbers and Two Lucas Numbers</a>, Integers, Vol. 19 (2019), Article A55.
%H Salah Eddine Rihane and Alain Togbé, <a href="https://doi.org/10.1007/s40065-021-00317-1">Repdigits as products of consecutive Padovan or Perrin numbers</a>, Arab. J. Math., Vol. 10 (2021), pp. 469-480.
%H Charles W. Trigg, <a href="https://www.mathstat.dal.ca/FQ/Scanned/12-2/trigg.pdf">Infinite sequences of palindromic triangular numbers</a>, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repdigit.html">Repdigit</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Repdigit">Repdigit</a>.
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10).
%F A037904(a(n)) = 0. - _Reinhard Zumkeller_, Dec 14 2007
%F A178401(a(n)) > 0. - _Reinhard Zumkeller_, May 27 2010
%F From _Reinhard Zumkeller_, Jul 26 2011: (Start)
%F For n > 0: A193459(a(n)) = A000005(a(n)).
%F for n > 10: a(n) mod 10 = floor(a(n)/10) mod 10.
%F A010879(n) = A010879(A059995(n)). (End)
%F A202022(a(n)) = 1. - _Reinhard Zumkeller_, Dec 09 2011
%F a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=4, a(5)=5, a(6)=6, a(7)=7, a(8)=8, a(9)=9, a(10)=11, a(11)=22, a(12)=33, a(13)=44, a(14)=55, a(15)=66, a(16)=77, a(17)=88, a(n) = 11*a(n-9) - 10*a(n-18). - _Harvey P. Dale_, Dec 28 2011
%F A151949(a(n)) = 0; A180410(a(n)) = A227362(a(n)). - _Reinhard Zumkeller_, Jul 09 2013
%F a(n) = (n - 9*floor((n-1)/9))*(10^floor((n+8)/9) - 1)/9. - _José de Jesús Camacho Medina_, Nov 06 2014
%F G.f.: x*(1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6+8*x^7+9*x^8)/((1-x^9)*(1-10*x^9)). - _Robert Israel_, Nov 09 2014
%F A047842(a(n)) = A244112(a(n)). - _Reinhard Zumkeller_, Nov 11 2014
%F Sum_{n>=1} 1/a(n) = (7129/2520) * A065444 = 3.11446261209177581335... - _Amiram Eldar_, Jan 21 2022
%p A010785 := proc(n)
%p (n-9*floor(((n-1)/9)))*((10^(floor(((n+8)/9)))-1)/9) ;
%p end proc:
%p seq(A010785(n), n = 0 .. 100); # _Robert Israel_, Nov 09 2014
%t fQ[n_]:=Module[{id=IntegerDigits[n]}, Length[Union[id]]==1]; Select[Range[0,10000], fQ] (* _Vladimir Joseph Stephan Orlovsky_, Dec 29 2010 *)
%t Union[FromDigits/@Flatten[Table[PadRight[{},i,n],{n,0,9},{i,6}],1]] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10}, {0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88},40] (* _Harvey P. Dale_, Dec 28 2011 *)
%t Union@ Flatten@ Table[k (10^n - 1)/9, {k, 0, 9}, {n, 6}] (* _Robert G. Wilson v_, Oct 09 2014 *)
%t Table[(n - 9 Floor[(n-1)/9]) (10^Floor[(n+8)/9] - 1)/9, {n, 0, 50}] (* _José de Jesús Camacho Medina_, Nov 06 2014 *)
%o (PARI) a(n)=10^((n+8)\9)\9*((n-1)%9+1) \\ _Charles R Greathouse IV_, Jun 15 2011
%o (PARI) nxt(n,t=n%10)=if(t<9,n*(t+1),n*10+9)\t \\ Yields the term a(k+1) following a given term a(k)=n. _M. F. Hasler_, Jun 24 2016
%o (PARI) is(n)={1==#Set(digits(n))}
%o inv(n) = 9*#Str(n) + n%10 - 9 \\ _David A. Corneth_, Jun 24 2016
%o (Haskell)
%o a010785 n = a010785_list !! n
%o a010785_list = 0 : r [1..9] where
%o r (x:xs) = x : r (xs ++ [10*x + x `mod` 10])
%o -- _Reinhard Zumkeller_, Jul 26 2011
%o (Magma) [(n-9*Floor((n-1)/9))*(10^Floor((n+8)/9)-1)/9: n in [0..50]]; // _Vincenzo Librandi_, Nov 10 2014
%o (Python)
%o def a(n): return 0 if n == 0 else int(str((n-1)%9+1)*((n-1)//9+1))
%o print([a(n) for n in range(55)]) # _Michael S. Branicky_, Dec 29 2021
%o (Python)
%o print([0]+[int(d*r) for r in range(1, 7) for d in "123456789"]) # _Michael S. Branicky_, Dec 29 2021
%o (Python) # without string operations
%o def a(n): return 0 if n == 0 else (10**((n-1)//9+1)-1)//9*((n-1)%9+1)
%o print([a(n) for n in range(55)]) # _Michael S. Branicky_, Nov 03 2023
%Y Cf. A000005, A009994, A009996, A010879, A037904, A047842, A059995, A065444, A134336, A139819, A151949, A178401, A178403, A180410, A193459, A193460, A202022, A227362, A244112.
%K nonn,base,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Name clarified by _Jon E. Schoenfield_, Nov 10 2023
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