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A070199
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Number of palindromes of length <= n.
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6
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10, 19, 109, 199, 1099, 1999, 10999, 19999, 109999, 199999, 1099999, 1999999, 10999999, 19999999, 109999999, 199999999, 1099999999, 1999999999, 10999999999, 19999999999, 109999999999, 199999999999, 1099999999999, 1999999999999, 10999999999999, 19999999999999
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refs;
listen;
history;
text;
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3).
G.f.: x*(10 + 9*x - 10*x^2)/((1 - x)*(1 - 10*x^2)). (End)
a(n) = (-2*sqrt(10)+10^(n/2)*(11+2*sqrt(10)+(-1)^n*(-11+2*sqrt(10))))/(2*sqrt(10)). - Harvey P. Dale, Mar 18 2016
a(n) = 2^(n/2 + 1)*5^(n/2) - 1 for n even.
a(n) = 11*10^((n-1)/2) - 1 for n odd. (End)
E.g.f.: 2*cosh(sqrt(10)*x) - cosh(x) - 1 - sinh(x) + 11*sinh(sqrt(10)*x)/sqrt(10). - Stefano Spezia, Jul 01 2023
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MATHEMATICA
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LinearRecurrence[{1, 10, -10}, {10, 19, 109}, 30] (* Harvey P. Dale, Mar 18 2016 *)
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PROG
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(PARI) Vec(x*(10+9*x-10*x^2)/((1-x)*(1-10*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 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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STATUS
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approved
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