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A232730
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Number of n-digit numbers that yield an (n+1)-digit number after Reverse and Add.
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3
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5, 45, 495, 4905, 49500, 494550, 4950000, 49495500, 495000000, 4949955000, 49500000000, 494999550000, 4950000000000, 49499995500000, 495000000000000, 4949999955000000, 49500000000000000, 494999999550000000, 4950000000000000000, 49499999995500000000
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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a(1) = 5, a(3) = 495, a(2*k+1) = 100*a(2*k-1), k > 1.
a(2) = 45, a(4) = 4905, a(2*k) = 110*a(2*k-2) - 1000*a(2*k-4), k > 2.
G.f. = 5*x*(1+x)*(1-x)^2 / ((1-10*x)*(1-10*x^2)). - M. F. Hasler, Nov 30 2013
a(n) = -45*(10^(n/2-2) - 11*10^(n-3)) for n>2 even.
a(n) = 99*2^(n-3)*5^(n-2) for n>2 odd.
a(n) = 10*a(n-1) + 10*a(n-2) - 100*a(n-3) for n>4. (End)
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
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EXAMPLE
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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.
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MAPLE
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a[1]:=5: t[0]:= 0: t[1]:= 5:
for n from 2 to 50 do
a[n]:= 45*10^(n-2) + 9*t[n-2];
t[n]:= a[n] + t[n-2];
od:
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MATHEMATICA
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LinearRecurrence[{10, 10, -100}, {5, 45, 495, 4905}, 20] (* Harvey P. Dale, Feb 29 2024 *)
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PROG
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(PARI) Vec(5*x*(1+x)*(1-x)^2 / ((1-10*x)*(1-10*x^2)) + O(x^30)) \\ Colin Barker, Mar 20 2017
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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EXTENSIONS
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G.f. corrected and empirical formulas proved by Robert Israel, Apr 21 2016
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STATUS
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approved
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