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centered octagonal n.  and  happy n.

First numbers that belong to both sets:

1, 49, 1521, 4225, 7225, 8281, 11025, 12321, 14641, 15625, 22201, 24025, 27225, 27889, 28561, 42025, 45369, 54289, 59049, 64009, 70225, 73441, 75625, 76729, 83521, 105625, 106929, 108241, 123201, 126025, 127449, 140625, 143641, 156025, 157609, 164025, 170569, 178929, 180625, 185761, 190969, 196249, 207025, 227529, 239121, 241081, 251001, 263169, 269361, 277729, 292681, 294849, 332929, 337561, 354025, 356409, 398161, 408321, 413449, 416025, 423801, 429025, 461041, 502681, 519841, 522729, 528529, 531441, 570025, 588289, 707281, 727609, 737881, 801025, 808201, 889249, 896809, 908209, 958441, 962361, 966289, 1010025, 1050625, 1054729, 1096209, 1100401, 1138489, 1177225, 1199025, 1229881, 1234321, 1320201, 1329409, 1380625, 1385329, 1394761, 1447209, 1456849, 1471369, 1500625, 1575025, 1600225, 1682209, 1687401, 1734489, 1739761, 1836025, 1934881, 1974025, 2007889, 2024929, 2093809, 2099601, 2223081, 2229049, 2393209, 2399401, 2405601, 2411809, 2505889, 2518569, 2563201, 2569609, 2666689, 2772225, 2812329, 2819041, 2920681, 2934369, 2948089, 2961841, 2968729, 2982529, 2989441, 3003289, 3045025, 3066001, 3087049, 3108169, 3122289, 3150625, 3193369, 3352561, 3404025, 3426201, 3523129, 3560769, 3606201, 3659569, 3674889, 3829849, 3940225, 3996001, 4020025, 4084441, 4100625, 4173849, 4247721, 4447881, 4498641, 4507129, 4549689, 4575321, 4592449, 4601025, 4695889, 4721929, 4730625, 4782969, 4809249, 4844401, 4959529, 4968441, 4995225, 5040025, 5049009, 5094049, 5103081, 5130225, 5157441, 5166529, 5303809, 5349969, 5424241, 5452225, 5470921, 5489649, 5574321, 5755201, 5774409, 5803281, 5822569, 6155361, 6205081, 6245001, 6255001, 6285049, 6589489, 6671889, 6734025, 6827769, 6985449, 6996025, 7027801, 7038409, 7070281, 7198489, 7209225, 7295401, 7469289, 7756225, 7856809, 7958041, 8003241, 8025889, 8094025, 8116801, 8219689, 8346321, 8381025, 8427409, 8508889, 8543929, 8579041, 8814961, 8874441, 8934121, 8946081, 8958049, 9272025, 9296401, 9320809, 9504889, 9628609, 9728161, 9778129, 9803161, 9853321, 9966649.