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A010785
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Repdigit numbers, or numbers whose digits are all equal.
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148
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666
(list;
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OFFSET
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0,3
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COMMENTS
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Beiler (1964) called these numbers "monodigit numbers". The term "repdigit numbers" was used by Trigg (1974). - Amiram Eldar, Jan 21 2022
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REFERENCES
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Albert H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, p. 83.
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LINKS
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Bir Kafle, Florian Luca and Alain Togbé, Triangular Repblocks, Fibonacci Quart., Vol. 56, No. 4 (2018), pp. 325-328.
Eric Weisstein's World of Mathematics, Repdigit.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10).
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FORMULA
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for n > 10: a(n) mod 10 = floor(a(n)/10) mod 10.
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
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
Sum_{n>=1} 1/a(n) = (7129/2520) * A065444 = 3.11446261209177581335... - Amiram Eldar, Jan 21 2022
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MAPLE
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(n-9*floor(((n-1)/9)))*((10^(floor(((n+8)/9)))-1)/9) ;
end proc:
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MATHEMATICA
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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 *)
Union@ Flatten@ Table[k (10^n - 1)/9, {k, 0, 9}, {n, 6}] (* Robert G. Wilson v, Oct 09 2014 *)
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PROG
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(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
(PARI) is(n)={1==#Set(digits(n))}
(Haskell)
a010785 n = a010785_list !! n
a010785_list = 0 : r [1..9] where
r (x:xs) = x : r (xs ++ [10*x + x `mod` 10])
(Magma) [(n-9*Floor((n-1)/9))*(10^Floor((n+8)/9)-1)/9: n in [0..50]]; // Vincenzo Librandi, Nov 10 2014
(Python)
def a(n): return 0 if n == 0 else int(str((n-1)%9+1)*((n-1)//9+1))
(Python)
print([0]+[int(d*r) for r in range(1, 7) for d in "123456789"]) # Michael S. Branicky, Dec 29 2021
(Python) # without string operations
def a(n): return 0 if n == 0 else (10**((n-1)//9+1)-1)//9*((n-1)%9+1)
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CROSSREFS
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Cf. A000005, A009994, A009996, A010879, A037904, A047842, A059995, A065444, A134336, A139819, A151949, A178401, A178403, A180410, A193459, A193460, A202022, A227362, A244112.
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KEYWORD
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nonn,base,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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