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

2,1

Comments

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

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).

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)

Example

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

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} ]

Prog

(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

Crossrefs

Cf. A048268.

Sequence in context: A057688 A259207 A300005 * A073069 A042281 A041625

Adjacent sequences: A048266 A048267 A048268 * A048270 A048271 A048272

Keyword

nonn,easy,base

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Extensions

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

Status

approved