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A331471
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Consider the different ways to split the binary representation of n into palindromic parts; a(n) is the greatest possible sum of the parts of such a split.
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2
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0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 6, 3, 6, 7, 15, 1, 17, 9, 10, 5, 21, 7, 8, 3, 10, 6, 27, 7, 8, 15, 31, 1, 33, 17, 18, 9, 10, 10, 12, 5, 10, 21, 22, 7, 45, 15, 16, 3, 18, 10, 51, 6, 22, 27, 28, 7, 12, 9, 28, 15, 16, 31, 63, 1, 65, 33, 34, 17, 18, 18, 20, 9, 73
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OFFSET
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0,4
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COMMENTS
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Leading zeros are forbidden in the binary representation of n; however we allow leading zeros in the palindromic parts.
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LINKS
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FORMULA
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a(n) >= A000120(n) with equality iff n = 0 or n is a power of 2.
a(n) <= n with equality iff n belongs to A006995.
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EXAMPLE
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For n = 10:
- the binary representation of 10 is "1010",
- we can split it into "1" and "0" and "1" and "0" (1 and 0 and 1 and 0),
- or into "101" and "0" (5 and 0),
- or into "1" and "010" (1 and 2),
- hence a(n) = max(2, 5, 3) = 5.
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MATHEMATICA
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palQ[w_] := w == Reverse@w; ric[tg_, cr_] := Block[{m = Length@tg, t}, If[m == 0, Sow@ Total[ FromDigits[#, 2] & /@ cr], Do[ If[ palQ[t = Take[tg, k]], ric[Drop[tg, k], Join[cr, {t}]]], {k, m}]]]; a[n_] := Max[ Reap[ ric[ IntegerDigits[n, 2], {}]][[2, 1]]]; a /@ Range[0, 73] (* Giovanni Resta, Jan 19 2020 *)
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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