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A181385 Maximal number that can be obtained by reversing n in an integer base.

%I A181385 #26 Dec 26 2024 10:14:11
%S A181385 0,1,2,3,4,7,9,13,16,21,25,31,36,43,49,57,64,73,81,91,100,111,121,133,
%T A181385 144,157,169,183,196,211,225,241,256,273,289,307,324,343,361,381,400,
%U A181385 421,441,463,484,507,529,553,576,601,625,651,676,703,729,757,784,813,841
%N A181385 Maximal number that can be obtained by reversing n in an integer base.
%C A181385 The second differences of this sequence start with 2 zeros and then seem to alternate between 2 and -1 perpetually
%C A181385 The bases which yield the first five numbers on this list are {{2},{3},{2,4},{3,5},{3}}, where multiple items on the sublists indicate multiple bases yielding the same maximum. 3 and 4 seem to be the only 2 numbers with multiple bases that yield the same maximum. The numbers of the bases which yield numbers on this list for values n greater than 5 seem to be Floor((n+2)/2).
%C A181385 a(n) agrees with the lower matching number of the (n+1) X (n+1) white bishop graph up to at least n = 13. - _Eric W. Weisstein_, Dec 23 2024
%H A181385 Rémy Sigrist, <a href="/A181385/b181385.txt">Table of n, a(n) for n = 0..10000</a>
%H A181385 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WhiteBishopGraph.html">White Bishop Graph</a>.
%H A181385 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LowerMatchingNumber.html">Lower Matching Number</a>.
%H A181385 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A181385 a(n) = ceiling(n^2/2) - floor(n^2/4) for n != 2. - _Eric W. Weisstein_, Dec 23 2024
%F A181385 a(n) = (3 - 3 (-1)^n + 2 n^2)/8 for n != 2. - _Eric W. Weisstein_, Dec 23 2024
%F A181385 a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 6. - _Eric W. Weisstein_, Dec 23 2024
%F A181385 G.f.: x*(-1+x^2-2*x^4+x^5)/((-1+x)^3*(1+x)). - _Eric W. Weisstein_, Dec 23 2024
%t A181385 rev[x_, b_]:=FromDigits[Reverse[IntegerDigits[x, b]], b]
%t A181385 Max /@ Table[Table[rev[x, b], {b, 2, x + 1}], {x, STARTPOINT, ENDPOINT}]
%t A181385 Table[Piecewise[{{2, n == 2}}, 1/8 (3 - 3 (-1)^n + 2 n^2)], {n, 20}] (* _Eric W. Weisstein_, Dec 23 2024 *)
%t A181385 Table[Piecewise[{{2, n == 2}}, Ceiling[n^2/2] - Floor[n^2/4]], {n, 20}] (* _Eric W. Weisstein_, Dec 23 2024 *)
%t A181385 CoefficientList[Series[(-1 + x^2 - 2 x^4 + x^5)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 23 2024 *)
%t A181385 {1, 2} ~ Join ~ LinearRecurrence[{2, 0, -2, 1}, {3, 4, 7, 9}, 20] (* _Eric W. Weisstein_, Dec 23 2024 *)
%o A181385 (PARI) a(n) = vecmax(apply(b -> fromdigits(Vecrev(digits(n,b)),b), [2..max(2,n+1)])) \\ _Rémy Sigrist_, Jan 29 2020
%Y A181385 Cf. A091974.
%Y A181385 Cf. A000982 (ceiling(n^2/2)).
%Y A181385 Cf. A002620 (ceiling(n^2/4)).
%K A181385 base,nonn
%O A181385 0,3
%A A181385 _Dylan Hamilton_, Oct 16 2010
%E A181385 a(0) = 0 prepended by _Rémy Sigrist_, Jan 29 2020