A048269 First palindrome greater than n+2 in bases n+2 and n.

5, 26, 21, 24, 154, 40, 121, 60, 181, 84, 253, 112, 337, 144, 433, 180, 541, 220, 661, 264, 793, 312, 937, 364, 1093, 420, 1261, 480, 1441, 544, 1633, 612, 1837, 684, 2053, 760, 2281, 840, 2521, 924, 2773, 1012, 3037, 1104, 3313, 1200, 3601, 1300, 3901

Offset: 2

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn
• easy
• base

Comments

a(2), a(3), a(4) and a(6) must be found explicitly.

Examples

			a(15)= (15+3)/2*15+(15+3)/2=144, which is (99) in base 15 and (88) in base 17.
		

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A048268.

Programs

Mathematica

Do[ k = n + 3; While[ RealDigits[ k, n + 2 ][[ 1 ] ] != Reverse[ RealDigits[ k, n + 2 ][[ 1 ] ] ] || RealDigits[ k, n ][[ 1 ] ] != Reverse[ RealDigits[ k, n ][[ 1 ] ] ], k++ ]; Print[ k ], {n, 2, 50} ]
LinearRecurrence[{0,3,0,-3,0,1},{5,26,21,24,154,40,121,60,181,84,253},50] (* Harvey P. Dale, May 12 2025 *)

PARI

Vec(x^2*(5 + 26*x + 6*x^2 - 54*x^3 + 106*x^4 + 46*x^5 - 283*x^6 - 14*x^7 + 259*x^8 - 81*x^10) / ((1 - x)^3*(1 + x)^3) + O(x^50)) \\ Colin Barker, Jun 30 2019

Formula

n even and n >= 8: a(n) = n^2+(n/2+3)*n+1 (which is (1 n/2+3 1) in base n and (1 n/2-2 1) in base n+2).

n odd and n >= 5: a(n) = (n+1)*(n+3)/2 (which is ((n+3)/2 (n+3)/2) in base n and ((n+1)/2 (n+1)/2) in base n+2).

From Colin Barker, Jun 30 2019: (Start)

G.f.: x^2*(5 + 26*x + 6*x^2 - 54*x^3 + 106*x^4 + 46*x^5 - 283*x^6 - 14*x^7 + 259*x^8 - 81*x^10) / ((1 - x)^3*(1 + x)^3).

a(n) = (5 + (-1)^(1 + n) + 2*(5 + (-1)^n)*n + 2*(2 + (-1)^n)*n^2) / 4 for n>6.

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>10.

(End)

Extensions

More terms from Robert G. Wilson v, Aug 15 2000