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alternating n.  and  palindromic n.

First numbers that belong to both sets:

1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 121, 141, 161, 181, 212, 232, 252, 272, 292, 303, 323, 343, 363, 383, 414, 434, 454, 474, 494, 505, 525, 545, 565, 585, 616, 636, 656, 676, 696, 707, 727, 747, 767, 787, 818, 838, 858, 878, 898, 909, 929, 949, 969, 989, 10101, 10301, 10501, 10701, 10901, 12121, 12321, 12521, 12721, 12921, 14141, 14341, 14541, 14741, 14941, 16161, 16361, 16561, 16761, 16961, 18181, 18381, 18581, 18781, 18981, 21012, 21212, 21412, 21612, 21812, 23032, 23232, 23432, 23632, 23832, 25052, 25252, 25452, 25652, 25852, 27072, 27272, 27472, 27672, 27872, 29092, 29292, 29492, 29692, 29892, 30103, 30303, 30503, 30703, 30903, 32123, 32323, 32523, 32723, 32923, 34143, 34343, 34543, 34743, 34943, 36163, 36363, 36563, 36763, 36963, 38183, 38383, 38583, 38783, 38983, 41014, 41214, 41414, 41614, 41814, 43034, 43234, 43434, 43634, 43834, 45054, 45254, 45454, 45654, 45854, 47074, 47274, 47474, 47674, 47874, 49094, 49294, 49494, 49694, 49894, 50105, 50305, 50505, 50705, 50905, 52125, 52325, 52525, 52725, 52925, 54145, 54345, 54545, 54745, 54945, 56165, 56365, 56565, 56765, 56965, 58185, 58385, 58585, 58785, 58985, 61016, 61216, 61416, 61616, 61816, 63036, 63236, 63436, 63636, 63836, 65056, 65256, 65456, 65656, 65856, 67076, 67276, 67476, 67676, 67876, 69096, 69296, 69496, 69696, 69896, 70107, 70307, 70507, 70707, 70907, 72127, 72327, 72527, 72727, 72927, 74147, 74347, 74547, 74747, 74947, 76167, 76367, 76567, 76767, 76967, 78187, 78387, 78587, 78787, 78987, 81018, 81218, 81418, 81618, 81818, 83038, 83238, 83438, 83638, 83838, 85058, 85258, 85458, 85658, 85858, 87078, 87278, 87478, 87678, 87878, 89098, 89298, 89498, 89698, 89898, 90109, 90309, 90509, 90709, 90909, 92129, 92329, 92529, 92729, 92929, 94149, 94349, 94549, 94749, 94949, 96169, 96369, 96569, 96769, 96969, 98189, 98389, 98589, 98789, 98989, 1010101.

Some larger examples :