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centered pentagonal n.  and  balanced n.

First numbers that belong to both sets:

141, 226, 601, 681, 856, 2481, 10081, 10401, 13876, 15016, 28356, 40641, 41926, 50766, 58906, 59676, 60451, 133981, 144601, 145806, 153141, 154381, 163201, 177556, 182926, 185641, 198106, 199516, 206641, 208081, 209526, 215356, 219781, 233326, 234856, 241026, 248851, 252016, 279726, 305376, 307126, 341326, 379276, 444156, 487306, 527851, 530151, 532456, 537081, 546391, 560506, 562876, 567631, 577201, 586851, 599026, 621256, 626251, 654081, 664351, 666931, 669516, 674701, 685131, 703576, 708891, 711556, 716901, 727651, 735766, 752131, 754876, 782601, 788206, 799476, 813676, 819391, 822256, 825126, 830881, 845356, 848266, 931776, 965656, 975001, 981256, 984391, 987531, 1025601, 2136751, 2159926, 2187901, 2225481, 2296806, 2316016, 2364391, 2379001, 2388766, 2398551, 2442831, 2462641, 2467606, 2482531, 2517531, 2537641, 2542681, 2557831, 2567956, 2618881, 2665141, 2717016, 2764131, 2779926, 2801056, 2811651, 2843556, 2902516, 2924106, 2956641, 2972976, 2994826, 3122016, 3155631, 3217726, 3257556, 3303376, 3343731, 3355306, 3361101, 3378516, 3413481, 3454501, 3531331, 3561106, 3621031, 3639106, 3705766, 3724051, 3748501, 3760756, 3779176, 3803806, 3828516, 3940701, 3953266, 4016391, 4215256, 4267356, 4668306, 4833726, 4952641, 4959681, 5108676, 5267131, 5303481, 5435376, 5457516, 5479701, 5531641, 5673856, 5681391, 5749431, 5802631, 5817876, 5833141, 5902081, 5932851, 5955981, 6048951, 6260766, 6918081, 7018251, 7297431, 7573351, 7582056, 7634391, 7845531, 7987891, 7996831, 8023681, 8095501, 8167641, 8194776, 8212891, 8349391, 8422651, 8431831, 8579391, 8625766, 8662956, 8728231, 8793751, 8803131, 8831301, 8887776, 8897206, 9020251, 9029751, 9039256, 9096391, 9144141, 9153706, 9201606, 9259251, 9268876, 9355726, 9375081, 9423556, 9491631, 9579516, 9608901, 9628516, 9638331, 9648151, 9667806, 9677641, 9766381, 9845601, 9925141, 9965031, 10055076, 10105276, 10206051, 10246501, 10358151, 10368331, 10378516, 10480641, 10572981, 10583266, 10603851, 10645081, 10676056, 10727781, 10769251, 10842016, 10852431, 10925476, 11019751, 11072301, 11135526, 11230701, 11283751, 11315641, 11326281, 11347576, 11379556, 11390226, 11411581, 11422266, 11465056, 11497201, 11615451, 11626231, 11669401, 11701831, 11712651, 11723476, 11886451, 11995726, 12061531, 12105501, 12193681, 12215776, 12226831, 12293266, 12337656, 12359881, 12371001, 12393256, 12437826, 12460141, 12538401, 12594451, 12706926, 12819901, 12865231, 12899281, 13081641, 13127431.

Some larger examples :