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

0, 1, 2, 3, 4, 7, 9, 13, 16, 21, 25, 31, 36, 43, 49, 57, 64, 73, 81, 91, 100, 111, 121, 133, 144, 157, 169, 183, 196, 211, 225, 241, 256, 273, 289, 307, 324, 343, 361, 381, 400, 421, 441, 463, 484, 507, 529, 553, 576, 601, 625, 651, 676, 703, 729, 757, 784, 813, 841

Offset: 0

Author

Dylan Hamilton, Oct 16 2010

Keywords

• base
• nonn

Comments

The second differences of this sequence start with 2 zeros and then seem to alternate between 2 and -1 perpetually

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).

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

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, White Bishop Graph.
• Eric Weisstein's World of Mathematics, Lower Matching Number.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A091974.

Cf. A000982 (ceiling(n^2/2)).

Cf. A002620 (ceiling(n^2/4)).

Programs

Mathematica

rev[x_, b_]:=FromDigits[Reverse[IntegerDigits[x, b]], b]
Max /@ Table[Table[rev[x, b], {b, 2, x + 1}], {x, STARTPOINT, ENDPOINT}]
Table[Piecewise[{{2, n == 2}}, 1/8 (3 - 3 (-1)^n + 2 n^2)], {n, 20}] (* Eric W. Weisstein, Dec 23 2024 *)
Table[Piecewise[{{2, n == 2}}, Ceiling[n^2/2] - Floor[n^2/4]], {n, 20}] (* Eric W. Weisstein, Dec 23 2024 *)
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 *)
{1, 2} ~ Join ~ LinearRecurrence[{2, 0, -2, 1}, {3, 4, 7, 9}, 20] (* Eric W. Weisstein, Dec 23 2024 *)

PARI

a(n) = vecmax(apply(b -> fromdigits(Vecrev(digits(n,b)),b), [2..max(2,n+1)])) \\ Rémy Sigrist, Jan 29 2020

Formula

a(n) = ceiling(n^2/2) - floor(n^2/4) for n != 2. - Eric W. Weisstein, Dec 23 2024

a(n) = (3 - 3 (-1)^n + 2 n^2)/8 for n != 2. - Eric W. Weisstein, Dec 23 2024

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 6. - Eric W. Weisstein, Dec 23 2024

G.f.: x*(-1+x^2-2*x^4+x^5)/((-1+x)^3*(1+x)). - Eric W. Weisstein, Dec 23 2024

Extensions

a(0) = 0 prepended by Rémy Sigrist, Jan 29 2020