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%I #44 Mar 19 2023 13:54:04
%S 0,0,1,19,279,3671,45431,540639,6260959,71068951,794428551,8773216559,
%T 95937737039,1040604153831,11210103801271,120060433858879,
%U 1279394234787519,13573881914016311,143459424905375591,1511020367139739599,15866744246492020399
%N Number of n-digit decimal numbers containing a fixed 2-digit integer with distinct digits as a substring.
%C First differences of A322052. - _Jon E. Schoenfield_, Jul 31 2021
%C See A138288 for the number of n-digit decimal numbers that do not contain a fixed 2-digit integer with distinct digits as a substring.
%H Colin Barker, <a href="/A322628/b322628.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (20,-101,10).
%F a(n) = 10*a(n-1) - a(n-2) + 9*10^(n-3) with a(0) = a(1) = 0, a(2) = 1.
%F G.f.: x^2*(x-1)/((10*x-1)*(x^2-10*x+1)). - _Alois P. Heinz_, Dec 20 2018
%F a(n) = (27*10^n + 5*(5-2*sqrt(6))^n*(-3+sqrt(6)) - 5*(3+sqrt(6))*(5+2*sqrt(6))^n) / 30 for n>0. - _Colin Barker_, Dec 21 2018
%p seq(coeff(series(x^2*(x-1)/((10*x-1)*(x^2-10*x+1)),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Dec 21 2018
%o (Python)
%o def find_int(i):
%o if i == 0: return (0)
%o intlist = [0,1,19]
%o for n in range(4,i+2):
%o if n > 3:
%o a = 10*(intlist[n-2])+(9*10**(n-3)-intlist[n-3])
%o intlist.append(a)
%o return (intlist[i-1])
%o for i in range(100):
%o print(find_int(i), end=', ')
%o (PARI) concat([0,0], Vec(x^2*(x-1)/((10*x-1)*(x^2-10*x+1)) + O(x^30))) \\ _Colin Barker_, Dec 21 2018
%o (GAP) a:=[0,1,19];; for n in [4..20] do a[n]:=20*a[n-1]-101*a[n-2]+10*a[n-3]; od; Concatenation([0],a); # _Muniru A Asiru_, Dec 21 2018
%Y Cf. A255372, A138288, A322052, A004189.
%K nonn,base,easy
%O 0,4
%A _Owen M Sheff_, Dec 20 2018
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