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n modulo p repeated p times (where p=10).
50

%I #15 Apr 16 2023 22:14:09

%S 1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,

%T 4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,

%U 7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1

%N n modulo p repeated p times (where p=10).

%C Periodic with length p^2=100.

%C a(n) = A179051(n) for n < 90. - _Reinhard Zumkeller_, Jun 27 2010

%H <a href="/index/Rec#order_91">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

%F The following formulas are given for a general parameter p (p=10 for this sequence).

%F a(n)=(1+floor(n/p)) mod p.

%F a(n)=1+floor(n/p)-p*floor((n+p)/p^2).

%F a(n)=(((n+p) mod p^2)-(n mod p))/p.

%F a(n)=((n+p-(n mod p))/p) mod p.

%F G.f. g(x)=((p-1)x^(p^2)-px^(p(p-1))+1)/((1-x)(1-x^p)(1-x^(p^2))).

%F G.f. g(x)=(1-x^p)*sum{0<=k<(p-1), (k+1)*x^(k*p)}/((1-x)(1-x^(p^2))).

%t Flatten[Table[PadRight[{},10,Mod[n,10]],{n,11}]] (* _Harvey P. Dale_, May 10 2012 *)

%o (PARI) a(n)=(n\10+1)%10 \\ _Charles R Greathouse IV_, Nov 15 2022

%Y Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

%Y Cf. A133890, A133900, A133910.

%K nonn,easy

%O 0,11

%A _Hieronymus Fischer_, Oct 10 2007