A048268 - OEIS

%I #39 Jun 30 2019 10:39:52

%S 6643,10,46,67,92,121,154,191,232,277,326,379,436,497,562,631,704,781,

%T 862,947,1036,1129,1226,1327,1432,1541,1654,1771,1892,2017,2146,2279,

%U 2416,2557,2702,2851,3004,3161,3322,3487,3656,3829,4006,4187,4372,4561

%N Smallest palindrome greater than n in bases n and n+1.

%C From _A.H.M. Smeets_, Jun 19 2019: (Start)

%C In the following, dig(expr) stands for the digit that represents the value of expression expr, and . stands for concatenation.

%C As for the naming of this sequence, the trivial 1 digit palindromes 0..dig(n-1) are excluded.

%C If a number m is palindromic in bases n and n+1, then m has an odd number of digits when represented in base n.

%C All three digit numbers in base n, that are palindromic in bases n and n+1 are given by:

%C 101_3                         22_4                      for n = 3,

%C 232_n                         1.dig(n).1_(n+1)

%C 343_n                         2.dig(n-1).2_(n+1)

%C up to and including

%C dig(n-2).dig(n-1).dig(n-2)_n  dig(n-3).4.dig(n-3)_(n+1) for n > 3, and

%C dig(n-1).0.dig(n-1)_n         dig(n-3).5.dig(n-3)_(n+1) for n > 4.

%C Let d_L(n) be the number of integers with L digits in base n (L being odd), being palindromic in bases n and n+1, then:

%C d_1(n) = n for n >= 2 (see above),

%C d_3(n) = n-2 for n >= 5 (see above),

%C d_5(n) = n-1 for n >= 7 and n == 1 (mod 3),

%C d_5(n) = n-4 for n >= 7 and n in {0, 2} (mod 3), and

%C it seems that d_7(n) is of order O(n^2*log(n)) for n large enough. (End)

%H Colin Barker, <a href="/A048268/b048268.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 2n^2 + 3n + 2 for n >= 4 (which is 232_n and 1n1_(n+1)).

%F a(n) = A130883(n+1) for n > 3. - _Robert G. Wilson v_, Oct 08 2014

%F From _Colin Barker_, Jun 30 2019: (Start)

%F G.f.: x^2*(6643 - 19919*x + 19945*x^2 - 6684*x^3 + 19*x^4) / (1 - x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6.

%F (End)

%e a(14) = 2*14^2 + 3*14 + 2 = 436, which is 232_14 and 1e1_15.

%t Do[ k = n + 2; While[ RealDigits[ k, n + 1 ][ [ 1 ] ] != Reverse[ RealDigits[ k, n + 1 ][ [ 1 ] ] ] || RealDigits[ k, n ][ [ 1 ] ] != Reverse[ RealDigits[ k, n ][ [ 1 ] ] ], k++ ]; Print[ k ], {n, 2, 75} ]

%t palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; f[n_] := Block[{k = n + 2}, While[ !palQ[k, n] || !palQ[k, n + 1], k++ ]; k]; Table[ f[n], {n, 2, 48}] (* _Robert G. Wilson v_, Sep 29 2004 *)

%o (PARI) isok(j, n) = my(da=digits(j,n), db=digits(j,n+1)); (Vecrev(da)==da) && (Vecrev(db)==db);

%o a(n) = {my(j = n); while(! isok(j, n), j++); j;} \\ _Michel Marcus_, Nov 16 2017

%o (PARI) Vec(x^2*(6643 - 19919*x + 19945*x^2 - 6684*x^3 + 19*x^4) / (1 - x)^3 + O(x^50)) \\ _Colin Barker_, Jun 30 2019

%Y Cf. A029965, A029966, A048269, A060792, A097928, A097929, A097930, A097931, A099145, A099146, A099147, A099153.

%K nonn,easy,base

%O 2,1

%A Ulrich Schimke (ulrschimke(AT)aol.com)

%E More terms from _Robert G. Wilson v_, Aug 14 2000