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A173712 Numbers n such that (n+n+1) divides the decimal concatenation [n, n+1]. 1
1, 4, 16, 49, 166, 499, 1666, 4999, 16666, 49999, 166666, 499999, 1666666, 4999999, 16666666, 49999999, 166666666, 499999999, 1666666666, 4999999999, 16666666666, 49999999999, 166666666666, 499999999999, 1666666666666, 4999999999999 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

If n+1 has k digits, the concatenation is 10^k n + n + 1, which is

congruent to n (10^k - 1) mod 2n+1. Since n and 2n+1 are coprime,

the condition is just that 2n+1 divides 10^k - 1. Now n+1 having k

digits means 10^(k-1) <= n+1 <= 10^k - 1, so d = (10^k-1)/(2n+1) can only

be 1, 2, 3, 4 or 5. But it can't be 2, 4 or 5 because 10^k - 1 is

not divisible by 2 or 5. The case d=1 corresponds to n = 5*10^(k-1) - 1

(thus 4, 49, 499, ...) and the case d=3 corresponds to n = (10^k-4)/6

(thus 1, 16, 166, 1666, ...). [Robert Israel]

REFERENCES

Eric Angelini, Posting to the Sequence Fans Mailing List, Sep 21 2010

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,11,0,-10).

FORMULA

From Wesley Ivan Hurt, Apr 30 2016: (Start)

G.f.: x*(1 + 4 x + 5 x^2 + 5 x^3)/(1 - 11 x^2 + 10 x^4).

a(n) = 11*a(n-2) - 10*a(n-4) for n>4.

a(n) = (10^((n+1)/2)-4)*(1-(-1)^n)/12 + (10^(n/2)-2)*(1+(-1)^n)/4. (End)

EXAMPLE

1+2 divides 12 -> 12/3 = 4;

4+5 divides 45 -> 45/9 = 5;

16+17 divides 1617 -> 1617/33 = 49;

49+50 divides 4950 -> 4950/99 = 50;

166+167 divides 166167 -> 166167/333 = 499.

MAPLE

A173712:=n->(10^((n+1)/2)-4)*(1-(-1)^n)/12+(10^(n/2)-2)*(1+(-1)^n)/4: seq(A173712(n), n=1..40); # Wesley Ivan Hurt, Apr 30 2016

MATHEMATICA

CoefficientList[Series[(1 + 4 x + 5 x^2 + 5 x^3)/(1 - 11 x^2 + 10 x^4), {x, 0, 30}], x] (* Wesley Ivan Hurt, Apr 30 2016 *)

LinearRecurrence[{0, 11, 0, -10}, {1, 4, 16, 49}, 50] (* G. C. Greubel, Nov 24 2016 *)

PROG

(PARI) last=0; for(n=1, 99999, eval(Str(n, n+1))%(1+2*n)  | print1(n, ", ")) \\ M. F. Hasler

(MAGMA) [(10^((n+1) div 2)-4)*(1-(-1)^n)/12+(10^(n div 2)-2)*(1+(-1)^n)/4 : n in [1..30]]; // Wesley Ivan Hurt, Apr 30 2016

CROSSREFS

Sequence in context: A188516 A188501 A283692 * A085697 A262268 A231156

Adjacent sequences:  A173709 A173710 A173711 * A173713 A173714 A173715

KEYWORD

nonn,easy,base

AUTHOR

N. J. A. Sloane, Nov 25 2010

STATUS

approved

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Last modified May 11 13:41 EDT 2021. Contains 343791 sequences. (Running on oeis4.)