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centered triangular n.  and  gapful n.

First numbers that belong to both sets:

4960, 9964, 10210, 12016, 12559, 14260, 15760, 28360, 38161, 49960, 103360, 115510, 119710, 124849, 130096, 132760, 137260, 151210, 156010, 170860, 175960, 179056, 191710, 197110, 213760, 219460, 261460, 267760, 318091, 326434, 375751, 428536, 433360, 486496, 601984, 605791, 626944, 727669, 826960, 895135, 927874, 1002460, 1014760, 1022176, 1052110, 1064710, 1087579, 1102960, 1115860, 1136656, 1155010, 1168210, 1186816, 1205569, 1208260, 1221760, 1238059, 1262710, 1276510, 1309936, 1318360, 1332460, 1375210, 1389610, 1433260, 1447960, 1456816, 1474609, 1492510, 1507510, 1510519, 1552960, 1568260, 1592896, 1614610, 1630210, 1649029, 1652176, 1677460, 1693360, 1741510, 1757710, 1796896, 1806760, 1823260, 1873210, 1890010, 1937449, 1940860, 1957960, 1961389, 1968256, 2079760, 2097460, 2223460, 2241760, 2371960, 2390860, 2525260, 2544760, 2683360, 2703460, 2846260, 2866960, 3018214, 3314524, 3713854, 4086226, 4120960, 4347760, 4491946, 4528360, 4675486, 4765960, 4954960, 6251584, 6611851, 7060096, 7773679, 8230960, 8443135, 8550235, 9214444, 9318835, 10003960, 10027216, 10120510, 10159510, 10276960, 10316260, 10379296, 10434610, 10474210, 10529776, 10585489, 10593460, 10633360, 10641349, 10753510, 10793710, 10890496, 10914760, 10955260, 11077210, 11118010, 11240860, 11281960, 11306656, 11356129, 11405710, 11447110, 11571760, 11613460, 11680336, 11730619, 11739010, 11781010, 11831509, 11839936, 11907460, 11949760, 12077110, 12119710, 12222256, 12247960, 12290860, 12420010, 12463210, 12541159, 12593260, 12636760, 12645469, 12662896, 12767710, 12811510, 12943360, 12987460, 13040479, 13058176, 13120210, 13164610, 13226896, 13298260, 13342960, 13477510, 13522510, 13630816, 13657960, 13703260, 13830499, 13839610, 13885210, 13894339, 14022460, 14068360, 14095936, 14206510, 14252710, 14391760, 14438260, 14512816, 14578210, 14625010, 14690656, 14709439, 14765860, 14812960, 14954710, 15002110, 15116176, 15135229, 15144760, 15192460, 15249799, 15336010, 15384010, 15528460, 15576760, 15605776, 15722110, 15770710, 15916960, 15965860, 16044256, 16054069, 16113010, 16162210, 16172059, 16231216, 16310260, 16359760, 16508710, 16558510, 16618369, 16678336, 16708360, 16758460, 16909210, 16959610, 17111260, 17161960, 17192416, 17314510, 17365510, 17508709, 17518960, 17570260, 17580529, 17652496, 17724610, 17776210, 17848576, 17931460, 17983360, 18139510, 18191710, 18317296, 18348760, 18401260, 18495949, 18559210, 18612010, 18770860, 18823960, 18855856, 18973039, 18983710, 19037110, 19101289, 19197760, 19251460, 19337536, 19413010, 19467010, 19542736, 19629460, 19683760, 19847110, 19901710, 20065960.

Some larger examples :