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A117855
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Number of nonzero palindromes of length n (in base 3).
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10
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2, 2, 6, 6, 18, 18, 54, 54, 162, 162, 486, 486, 1458, 1458, 4374, 4374, 13122, 13122, 39366, 39366, 118098, 118098, 354294, 354294, 1062882, 1062882, 3188646, 3188646, 9565938, 9565938, 28697814, 28697814, 86093442, 86093442, 258280326, 258280326, 774840978
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OFFSET
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1,1
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COMMENTS
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See A225367 for the sequence that counts all base 3 palindromes, including 0 (and thus also the number of n-digit terms in A006072). -- A nonzero palindrome of length L=2k-1 or of length L=2k is determined by the first k digits, which then determine the last k digits by symmetry. Since the first digit cannot be 0, there are 2*3^(k-1) possibilities. - M. F. Hasler, May 05 2013
Also the number of subsets of {1..n} with n not the sum of two subset elements (possibly the same). For example, the a(0) = 1 through a(4) = 6 subsets are:
{} {} {} {} {}
{1} {2} {1} {1}
{2} {3}
{3} {4}
{1,3} {1,4}
{2,3} {3,4}
For subsets with no subset summing to n we have A365377.
The complement is counted by A366131.
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LINKS
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FORMULA
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a(n) = 2*3^floor((n-1)/2).
a(n) = 3*a(n-2).
G.f.: -2*x*(x+1)/(3*x^2-1). (End)
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EXAMPLE
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The a(3)=6 palindromes of length 3 are: 101, 111, 121, 202, 212, and 222. - M. F. Hasler, May 05 2013
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MATHEMATICA
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With[{c=NestList[3#&, 2, 20]}, Riffle[c, c]] (* Harvey P. Dale, Mar 25 2018 *)
Table[Length[Select[Subsets[Range[n]], !MemberQ[Total/@Tuples[#, 2], n]&]], {n, 0, 10}] (* Gus Wiseman, Oct 18 2023 *)
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PROG
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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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STATUS
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approved
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