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A050250
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Number of nonzero palindromes less than 10^n.
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15
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9, 18, 108, 198, 1098, 1998, 10998, 19998, 109998, 199998, 1099998, 1999998, 10999998, 19999998, 109999998, 199999998, 1099999998, 1999999998, 10999999998, 19999999998, 109999999998, 199999999998, 1099999999998, 1999999999998, 10999999999998
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history;
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OFFSET
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1,1
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..1000
Dr. Math, Palindromic Numbers.
Dr. Math, Palindromic Numbers.
G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Palindromic Number.
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).
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FORMULA
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a(2*k) = 2*10^k - 2, a(2*k + 1) = 11*10^k - 2. - Sascha Kurz, Apr 14 2002
From Jonathan Vos Post, Jun 18 2008: (Start)
a(n) = Sum_{i=1..n} A050683(i).
a(n) = Sum_{i=1..n} 9*10^floor((i-1)/2).
a(n) = 9*Sum_{i=1..n} 10^floor((i-1)/2). (End)
From Bruno Berselli, Feb 15 2011: (Start)
G.f.: 9*x*(1+x)/((1-x)*(1-10*x^2)).
a(n) = (1/2)*10^((2*n + (-1)^n - 1)/4)*(13 - 9*(-1)^n) - 2. (End)
a(1)=9, a(2)=18, a(3)=108; for n>3, a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3). - Harvey P. Dale, Jan 29 2012
a(n) = 10*a(n-2) + 18. - R. J. Mathar, Nov 07 2015
E.g.f.: 2*cosh(sqrt(10)*x) - 2*(cosh(x) + sinh(x)) + 11*sinh(sqrt(10)*x)/sqrt(10). - Stefano Spezia, Jun 11 2022
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MAPLE
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A050250List := proc(len); local s, egf, ser; s:= 11/(2*sqrt(10));
egf := -2*exp(x) + (1-s)*exp(-sqrt(10)*x) + (1+s)*exp(sqrt(10)*x);
ser := series(egf, x, len+2): seq(simplify(n!*coeff(ser, x, n)), n = 1..len) end:
A050250List(25); # Peter Luschny, Jun 11 2022 after Stefano Spezia
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MATHEMATICA
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LinearRecurrence[{1, 10, -10}, {9, 18, 108}, 30] (* Harvey P. Dale, Jan 29 2012 *)
CoefficientList[Series[2Cosh[Sqrt[10]x]-2(Cosh[x]+Sinh[x])+11Sinh[Sqrt[10]x]/Sqrt[10], {x, 0, 25}], x]Table[n!, {n, 0, 25}] (* Stefano Spezia, Jun 11 2022 *)
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PROG
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(PARI) a(n)=10^(n\2)*(13-9*(-1)^n)/2-2 \\ Charles R Greathouse IV, Jun 25 2017
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CROSSREFS
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Cf. A002113, A002778, A050683.
Sequence in context: A186443 A066711 A033651 * A080453 A222811 A002169
Adjacent sequences: A050247 A050248 A050249 * A050251 A050252 A050253
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KEYWORD
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nonn,easy,base,nice
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AUTHOR
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Eric W. Weisstein, Dec 11 1999
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EXTENSIONS
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More terms from Patrick De Geest, Dec 15 1999
a(24)-a(25) from Jonathan Vos Post, Jun 18 2008
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STATUS
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approved
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