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alternating n.  and  pancake n.

First numbers that belong to both sets:

1, 2, 4, 7, 16, 29, 56, 67, 92, 121, 232, 254, 301, 436, 529, 781, 947, 1036, 1276, 1432, 1654, 1892, 2347, 2927, 3656, 4187, 4561, 5254, 5672, 6329, 6787, 7876, 8129, 9454, 12721, 14707, 18529, 23654, 23872, 25652, 27496, 34981, 43072, 45452, 47896, 49456, 52327, 54947, 61076, 70501, 85492, 89254, 90101, 96581, 98347, 101476, 169072, 169654, 236329, 250987, 294529, 303032, 309292, 361676, 367654, 416329, 430129, 492529, 496507, 614387, 654941, 690901, 703892, 707456, 747254, 781876, 785632, 814727, 818561, 870541, 943252, 1072381, 1410361, 1430587, 1478341, 1492129, 1890541, 1898327, 2141416, 2185096, 2381654, 2743654, 2905256, 3014741, 3214381, 3454507, 3810181, 3876721, 4163056, 4909412, 4925092, 5016529, 5070521, 5416987, 5430161, 5656567, 6561254, 6703292, 6721612, 7218101, 7498129, 7638187, 7696927, 7870529, 8183036, 8345656, 8509876, 8709052, 8965496, 9092981, 9454727, 9472129, 9854581, 9872347, 9890129, 10925476, 12189454, 12367852, 12587654, 14723452, 14985076, 16921654, 16927472, 18129232, 18547096, 21232387, 23014721, 23416747, 23498941, 25010129, 25236961, 25636381, 27294967, 27650767, 29272727, 29610361, 30909454, 32365036, 34101412, 34523896, 34565456, 34723612, 34765292, 38909432, 41254987, 41692147, 41810941, 43054561, 47234341, 50105056, 50909096, 50989852, 54189256, 54303832, 54387236, 54721492, 56365654, 58563254, 61654961, 61832761, 63298127, 67832129, 67878727, 69614101, 70501876, 70727672, 72547036, 74743652, 76749856, 78945896, 83456741, 87854141, 89238121, 90565612, 90781076, 92187832, 98147056, 98189092, 98301232, 101852129, 103054547, 103212529, 103456921, 109290721, 121298101, 121438321, 125270707, 129098347, 129870787, 143016329, 143270129, 143812321, 147498901, 149238727, 149290561, 161074327, 163434161, 165410767, 165610901, 167216329, 169878529, 181270321, 183696529, 187472567, 212561272, 214545256, 216101656, 216767432, 230985272, 232125832, 234523654, 234783616, 252169654, 254781452, 256703812, 258701632, 258769876, 274189654, 274587896, 278161492, 278727856, 278987632, 290561672, 296181292, 303478567, 305254987, 309494761, 309892961, 321856507, 343416529, 347016341, 369036529, 369498521, 385294921, 387212707, 410769454, 414101032, 414561616, 450165016, 458969254, 472181816, 472765876, 496581856, 498569254, 505254367, 505858529, 509076187, 509810347, 523212727, 525674101, 527458961, 541056961, 541616329, 543296167, 547258987, 549278941, 549676747, 585076529, 587816329, 614549212, 618763432, 636727456, 636905896, 638727412, 652525876, 672509476, 676329032, 694507816, 698389252, 698725654.

Some larger examples :