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alternating n.  and  centered octagonal n.

First numbers that belong to both sets:

1, 9, 25, 49, 81, 121, 169, 361, 529, 729, 961, 4761, 6561, 12321, 12769, 14161, 16129, 18769, 32761, 34969, 56169, 72361, 74529, 76729, 78961, 96721, 212521, 214369, 290521, 436921, 452929, 458329, 474721, 670761, 690561, 1038361, 1054729, 1234321, 1238769, 1252161, 1270129, 1292769, 1610361, 1630729, 1656369, 1852321, 1874161, 1896129, 3470769, 3876961, 3892729, 5230369, 5414929, 5470921, 5612161, 5650129, 5812921, 5870929, 5890329, 7230721, 7414729, 7436529, 7458361, 7634169, 9078169, 9418761, 9492561, 9678321, 9690769, 21058921, 21836929, 21874329, 25230529, 25270729, 25694761, 25816561, 27678121, 29278921, 29452329, 41254929, 41434969, 43072969, 43256929, 43414921, 45252529, 45414121, 47238129, 47430769, 47458321, 47872561, 49434961, 61418569, 63632529, 63696361, 69072721, 69272329, 81234169, 81414529, 83010321, 87216921, 89056969, 89094721, 103652761, 105452361, 105616729, 105698961, 109432521, 109474369, 121418361, 123454321, 123498769, 123632161, 123810129, 125014761, 129072321, 129254161, 129436129, 143256961, 161010721, 161874729, 165456769, 167210761, 169494361, 181252369, 183250369, 303212569, 307616521, 321090561, 329894569, 343694521, 345476569, 347412321, 349278721, 361418121, 361494169, 361874529, 365230321, 367450561, 381694369, 383258929, 383650569, 385690321, 387656721, 503418969, 521254561, 523814769, 525830761, 529874361, 541074121, 541632529, 541818729, 549292969, 567058969, 581436769, 583850569, 585010969, 587432169, 589858369, 709210161, 709636321, 723018321, 725278761, 741854169, 743816529, 745890721, 763030129, 763472161, 769452121, 783272169, 907636129, 927872521, 941078329, 941814721, 943656961, 949810761, 967894321, 987656329.