cp's OEIS Frontend

A232730 Number of n-digit numbers that yield an (n+1)-digit number after Reverse and Add.

%I A232730 #34 Feb 29 2024 18:02:48
%S A232730 5,45,495,4905,49500,494550,4950000,49495500,495000000,4949955000,
%T A232730 49500000000,494999550000,4950000000000,49499995500000,
%U A232730 495000000000000,4949999955000000,49500000000000000,494999999550000000,4950000000000000000,49499999995500000000
%N A232730 Number of n-digit numbers that yield an (n+1)-digit number after Reverse and Add.
%C A232730 A232729(n) + a(n) = 9*10^(n-1).
%H A232730 Robert Israel, <a href="/A232730/b232730.txt">Table of n, a(n) for n = 1..990</a>
%H A232730 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (10,10,-100).
%F A232730 a(1) = 5, a(3) = 495, a(2*k+1) = 100*a(2*k-1), k > 1.
%F A232730 a(2) = 45, a(4) = 4905, a(2*k) = 110*a(2*k-2) - 1000*a(2*k-4), k > 2.
%F A232730 G.f. = 5*x*(1+x)*(1-x)^2 / ((1-10*x)*(1-10*x^2)). - _M. F. Hasler_, Nov 30 2013
%F A232730 From _Colin Barker_, Mar 20 2017: (Start)
%F A232730 a(n) = -45*(10^(n/2-2) - 11*10^(n-3)) for n>2 even.
%F A232730 a(n) = 99*2^(n-3)*5^(n-2) for n>2 odd.
%F A232730 a(n) = 10*a(n-1) + 10*a(n-2) - 100*a(n-3) for n>4. (End)
%F A232730 E.g.f.: (99*cosh(10*x) - 90*cosh(sqrt(10)*x) + 99*sinh(10*x) + 10*x - 9)/200. - _Stefano Spezia_, Oct 27 2022
%e A232730 There are 5 1-digit numbers (5,6,7,8,9) that yield a 2-digit number (10,12,14,16,18), so a(1)=5.
%p A232730 a[1]:=5: t[0]:= 0: t[1]:= 5:
%p A232730 for n from 2 to 50 do
%p A232730 a[n]:= 45*10^(n-2) + 9*t[n-2];
%p A232730 t[n]:= a[n] + t[n-2];
%p A232730 od:
%p A232730 seq(a[n],n=1..50); # _Robert Israel_, Apr 21 2016
%t A232730 LinearRecurrence[{10,10,-100},{5,45,495,4905},20] (* _Harvey P. Dale_, Feb 29 2024 *)
%o A232730 (PARI) Vec(5*x*(1+x)*(1-x)^2 / ((1-10*x)*(1-10*x^2)) + O(x^30)) \\ _Colin Barker_, Mar 20 2017
%Y A232730 Cf. A232729, A232731.
%K A232730 nonn,base,easy
%O A232730 1,1
%A A232730 _Lars Blomberg_, Nov 29 2013
%E A232730 G.f. corrected and empirical formulas proved by _Robert Israel_, Apr 21 2016