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undulating numbers
Numbers of the form abab...a or abab...b. more

The first 600 undulating numbers :
101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, 676, 686, 696, 707, 717, 727, 737, 747, 757, 767, 787, 797, 808, 818, 828, 838, 848, 858, 868, 878, 898, 909, 919, 929, 939, 949, 959, 969, 979, 989, 1010, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2323, 2424, 2525, 2626, 2727, 2828, 2929, 3030, 3131, 3232, 3434, 3535, 3636, 3737, 3838, 3939, 4040, 4141, 4242, 4343, 4545, 4646, 4747, 4848, 4949, 5050, 5151, 5252, 5353, 5454, 5656, 5757, 5858, 5959, 6060, 6161, 6262, 6363, 6464, 6565, 6767, 6868, 6969, 7070, 7171, 7272, 7373, 7474, 7575, 7676, 7878, 7979, 8080, 8181, 8282, 8383, 8484, 8585, 8686, 8787, 8989, 9090, 9191, 9292, 9393, 9494, 9595, 9696, 9797, 9898, 10101, 12121, 13131, 14141, 15151, 16161, 17171, 18181, 19191, 20202, 21212, 23232, 24242, 25252, 26262, 27272, 28282, 29292, 30303, 31313, 32323, 34343, 35353, 36363, 37373, 38383, 39393, 40404, 41414, 42424, 43434, 45454, 46464, 47474, 48484, 49494, 50505, 51515, 52525, 53535, 54545, 56565, 57575, 58585, 59595, 60606, 61616, 62626, 63636, 64646, 65656, 67676, 68686, 69696, 70707, 71717, 72727, 73737, 74747, 75757, 76767, 78787, 79797, 80808, 81818, 82828, 83838, 84848, 85858, 86868, 87878, 89898, 90909, 91919, 92929, 93939, 94949, 95959, 96969, 97979, 98989, 101010, 121212, 131313, 141414, 151515, 161616, 171717, 181818, 191919, 202020, 212121, 232323, 242424, 252525, 262626, 272727, 282828, 292929, 303030, 313131, 323232, 343434, 353535, 363636, 373737, 383838, 393939, 404040, 414141, 424242, 434343, 454545, 464646, 474747, 484848, 494949, 505050, 515151, 525252, 535353, 545454, 565656, 575757, 585858, 595959, 606060, 616161, 626262, 636363, 646464, 656565, 676767, 686868, 696969, 707070, 717171, 727272, 737373, 747474, 757575, 767676, 787878, 797979, 808080, 818181, 828282, 838383, 848484, 858585, 868686, 878787, 898989, 909090, 919191, 929292, 939393, 949494, 959595, 969696, 979797, 989898, 1010101, 1212121, 1313131, 1414141, 1515151, 1616161, 1717171, 1818181, 1919191, 2020202, 2121212, 2323232, 2424242, 2525252, 2626262, 2727272, 2828282, 2929292, 3030303, 3131313, 3232323, 3434343, 3535353, 3636363, 3737373, 3838383, 3939393, 4040404, 4141414, 4242424, 4343434, 4545454, 4646464, 4747474, 4848484, 4949494, 5050505, 5151515, 5252525, 5353535, 5454545, 5656565, 5757575, 5858585, 5959595, 6060606, 6161616, 6262626, 6363636, 6464646, 6565656, 6767676, 6868686, 6969696, 7070707, 7171717, 7272727, 7373737, 7474747, 7575757, 7676767, 7878787, 7979797, 8080808, 8181818, 8282828, 8383838, 8484848, 8585858, 8686868, 8787878, 8989898, 9090909, 9191919, 9292929, 9393939, 9494949, 9595959, 9696969, 9797979, 9898989, 10101010, 12121212, 13131313, 14141414, 15151515, 16161616, 17171717, 18181818, 19191919, 20202020, 21212121, 23232323, 24242424, 25252525, 26262626, 27272727, 28282828, 29292929, 30303030, 31313131, 32323232, 34343434, 35353535, 36363636, 37373737, 38383838, 39393939, 40404040, 41414141, 42424242, 43434343, 45454545, 46464646, 47474747, 48484848, 49494949, 50505050, 51515151, 52525252, 53535353, 54545454, 56565656, 57575757, 58585858, 59595959, 60606060, 61616161, 62626262, 63636363, 64646464, 65656565, 67676767, 68686868, 69696969, 70707070, 71717171, 72727272, 73737373, 74747474, 75757575, 76767676, 78787878, 79797979, 80808080, 81818181, 82828282, 83838383, 84848484, 85858585, 86868686, 87878787, 89898989, 90909090, 91919191, 92929292, 93939393, 94949494, 95959595, 96969696, 97979797, 98989898, 101010101, 121212121, 131313131, 141414141, 151515151, 161616161, 171717171, 181818181, 191919191, 202020202, 212121212, 232323232, 242424242, 252525252, 262626262, 272727272, 282828282, 292929292, 303030303, 313131313, 323232323, 343434343, 353535353, 363636363, 373737373, 383838383, 393939393, 404040404, 414141414, 424242424, 434343434, 454545454, 464646464, 474747474, 484848484, 494949494, 505050505, 515151515, 525252525, 535353535, 545454545, 565656565, 575757575, 585858585, 595959595, 606060606, 616161616, 626262626, 636363636, 646464646, 656565656, 676767676, 686868686, 696969696, 707070707, 717171717, 727272727, 737373737, 747474747, 757575757, 767676767, 787878787, 797979797, 808080808, 818181818, 828282828, 838383838, 848484848, 858585858, 868686868, 878787878, 898989898, 909090909, 919191919, 929292929, 939393939, 949494949, 959595959, 969696969, 979797979, 989898989, 1010101010, 1212121212, 1313131313, 1414141414, 1515151515, 1616161616, 1717171717, 1818181818, 1919191919, 2020202020, 2121212121, 2323232323, 2424242424, 2525252525, 2626262626, 2727272727, 2828282828, 2929292929, 3030303030, 3131313131, 3232323232, 3434343434, 3535353535, 3636363636, 3737373737, 3838383838, 3939393939, 4040404040, 4141414141, 4242424242, 4343434343, 4545454545, 4646464646.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1053 values, from 101 to 989898989898989).

n\r 0  1 
2498555 2 
3453300300 3 
4246281252274 4 
5165222222222222 5 
6213166151240134149 6 
7295131153116129113116 7 
8123137129137123144123137 8 
9145100100154100100154100100 9 
1054111111111111111111111111111 10 
1156989898999999999898111

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.