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A053599
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Number of nonempty subsequences {s(k)} of 1..n such that the difference sequence is palindromic.
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5
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1, 3, 7, 13, 23, 37, 59, 89, 135, 197, 291, 417, 607, 861, 1243, 1753, 2519, 3541, 5075, 7121, 10191, 14285, 20427, 28617, 40903, 57285, 81859, 114625, 163775, 229309, 327611, 458681, 655287, 917429, 1310643, 1834929, 2621359, 3669933, 5242795, 7339945
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of positive integers < 2^n such that the binary representation of the odd part is palindromic; i.e., palindromic without the final 0's. - Andrew Woods, May 19 2012
a(n) is the number of ideals of the quotient ring Z_{2^n}[u]/<u^2> for indeterminate u. - Fatih Temiz, Oct 11 2017
Conjecture: let b(n) be the number of subsets S of {1,2,...,n} having more than one element such that (sum of least two elements of S) > max(S). Then b(0) = b(1) = 0 and b(n+2) = a(n+1) for n >= 0. - Clark Kimberling Sep 27 2022
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LINKS
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FORMULA
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a(1)=1, a(2)=3 and, for n > 2, a(n) = 2*a(n-2) + 2*n - 1.
G.f.: x*(1+x)/((1-x)^2*(1-2*x^2)). - Colin Barker, Mar 28 2012
a(n) = 5*2^((n+1)/2) - 2*n - 7 for odd n, 7*2^(n/2) - 2*n - 7 for even n. - Andrew Woods, May 19 2012
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EXAMPLE
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For n=4 the 13 sequences are 1,2,3,4,12,13,14,23,24,34,123,234,1234.
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,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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