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A099150
Positive integers k such that f(k)+f(k)=concatenation of k and k, where f(k)=k(k+3)/2 (A000096).
6
8, 98, 998, 9998, 99998, 999998, 9999998, 99999998, 999999998, 9999999998, 99999999998, 999999999998, 9999999999998, 99999999999998, 999999999999998, 9999999999999998, 99999999999999998, 999999999999999998, 9999999999999999998, 99999999999999999998
OFFSET
1,1
COMMENTS
By the definition, k*(k+3) = k*10^m+k. So k+3 = 10^m+1, that is k = 10^m-2. - Seiichi Manyama, Aug 31 2019
FORMULA
a(n) = A002283(n) - 1 = 10^n - 2. - Seiichi Manyama, Aug 31 2019
From Chai Wah Wu, Jun 15 2020: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2.
G.f.: x*(10*x + 8)/((x - 1)*(10*x - 1)). (End)
EXAMPLE
99998*(99998+3) = 9999899998 (concatenation of 99998 and 99998).
PROG
(PARI) for(k=1, 1e9, if(k*(k+3)==eval(Str(k, k)), print1(k", "))) \\ Seiichi Manyama, Aug 31 2019
(PARI) {a(n) = 10^n-2} \\ Seiichi Manyama, Aug 31 2019
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
John W. Layman, Sep 30 2004
EXTENSIONS
a(9)-a(20) from Seiichi Manyama, Aug 31 2019
STATUS
approved