A048268 Smallest palindrome greater than n in bases n and n+1.

6643, 10, 46, 67, 92, 121, 154, 191, 232, 277, 326, 379, 436, 497, 562, 631, 704, 781, 862, 947, 1036, 1129, 1226, 1327, 1432, 1541, 1654, 1771, 1892, 2017, 2146, 2279, 2416, 2557, 2702, 2851, 3004, 3161, 3322, 3487, 3656, 3829, 4006, 4187, 4372, 4561

Offset: 2

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn
• easy
• base

Comments

From A.H.M. Smeets, Jun 19 2019: (Start)

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

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

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.

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

101_3 22_4 for n = 3,

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

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

up to and including

dig(n-2).dig(n-1).dig(n-2)n dig(n-3).4.dig(n-3)(n+1) for n > 3, and

dig(n-1).0.dig(n-1)n dig(n-3).5.dig(n-3)(n+1) for n > 4.

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:

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

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

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

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

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

Examples

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

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Programs

Mathematica

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} ]
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 *)

PARI

isok(j, n) = my(da=digits(j,n), db=digits(j,n+1)); (Vecrev(da)==da) && (Vecrev(db)==db);
a(n) = {my(j = n); while(! isok(j, n), j++); j;} \\ Michel Marcus, Nov 16 2017

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

Formula

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

a(n) = A130883(n+1) for n > 3. - Robert G. Wilson v, Oct 08 2014

From Colin Barker, Jun 30 2019: (Start)

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

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

(End)

Extensions

More terms from Robert G. Wilson v, Aug 14 2000