A082461 Non-palindromic numbers whose decimal expansion is a concatenation of the form a_1 a_2 a_3 ... a_{k-1} a_k a_k a_{k-1} ... a_2 a_1 (k >= 1) or a_1 a_2 a_3 ... a_{k-1} a_k a_{k-1} ... a_2 a_1 (k >= 2) for positive integers a_1, ..., a_k. For i>1, a_i may have leading zeros.

1010, 1011, 1021, 1031, 1041, 1051, 1061, 1071, 1081, 1091, 1101, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191, 1201, 1211, 1212, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1301, 1311, 1313, 1321, 1341, 1351, 1361, 1371, 1381, 1391, 1401, 1411, 1414, 1421

Offset: 1

Author

K. Ramsharan (ramsharan(AT)indiainfo.com), Apr 26 2003

Keywords

• nonn
• base

Comments

Of course any number m can be written as m = a_1, but this trivial construction is excluded.

A palindromic number of four digits has the form abba, where a is in {1, 2, ..., 9} and b is in {0, 1, 2, ..., 9}. There are 9x10=90 possibilities. For example, 1551 or 2002, but not 3753. However, 3753 = 3(75)3 and 4646 = (46)(46) are terms of the present sequence. The 4-digit numbers in the present sequence therefore have the form ABA, where A is in {1, 2, ..., 9} and B is in {00, 01, 02, 03, ..., 99} \ {00, 11, 22, 33, ..., 99}; or CC, where C is in {10, 11, 12, ..., 99} \ {11, 22, 33, ..., 99}. In the first case there are 9x(100-10)=9x90=810 terms. In the second case, 90-9=81. Total: 810+81=891 4-digit non-palindromic terms.

Examples

			For example, 1235656312 is a term because we can group it as (12)(3)(56)(56)(3)(12), i.e. ABCCBA.
1010 = (10)(10), 1011 = 1(01)1, 1021 = 1(02)1, etc.
		

References

• M. Khoshnevisan, manuscript, March 2003.
• M. Khoshnevisan, "Generalized Smarandache Palindrome", Mathematics Magazine, Aurora, Canada, 10/2003.
• M. Khoshnevisan, Proposed problem 1062, The PME Journal, USA, Vol. 11, No. 9, p. 501, 2003.

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Charles Ashbacher, Lori Neirynck, The Density of Generalized Smarandache Palindromes.
• Charles Ashbacher, Lori Neirynck, The Density of Generalized Smarandache Palindromes [Cached copy, pdf file]

Extensions

Edited by N. J. A. Sloane, Jul 02 2017