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centered square n.  and  balanced n.

First numbers that belong to both sets:

41, 613, 841, 2381, 2665, 3281, 8321, 9113, 10805, 12641, 12961, 14621, 18241, 21425, 27145, 36181, 39481, 42341, 42925, 45905, 49613, 53465, 55445, 57461, 136765, 138865, 140981, 144185, 150701, 151801, 157361, 158485, 161881, 167621, 168781, 171113, 173461, 178205, 180601, 181805, 186661, 194065, 197821, 200345, 201613, 206725, 208013, 209305, 219785, 223781, 226465, 230521, 231881, 238741, 248513, 546013, 554405, 562861, 582121, 584281, 597325, 599513, 601705, 617161, 635065, 637321, 639581, 669325, 692665, 699745, 702113, 706861, 742981, 760145, 800113, 805181, 810265, 812813, 830761, 841105, 859361, 864613, 899141, 904513, 923441, 945313, 1032485, 2152813, 2161121, 2215513, 2257813, 2270581, 2274845, 2313401, 2317705, 2352281, 2382745, 2395861, 2426605, 2466421, 2502085, 2506561, 2538005, 2628925, 2749513, 2758901, 2787161, 2839345, 2877601, 2911285, 2920945, 2925781, 2935465, 2945165, 2964613, 2974361, 3028261, 3057865, 3082645, 3092585, 3117505, 3167645, 3197921, 3238513, 3248701, 3258905, 3264013, 3315313, 3335945, 3351461, 3429581, 3450565, 3466345, 3476885, 3556445, 3567121, 3572465, 3588521, 3696481, 3701921, 3707365, 3723721, 3745585, 3773005, 3789505, 3795013, 3806041, 3811561, 3872545, 3883685, 3939625, 3945241, 4047013, 4064101, 4081225, 4098385, 4161613, 4814305, 4882813, 4951805, 5123201, 5148841, 5168113, 5187421, 5213221, 5271505, 5291005, 5297513, 5304025, 5474741, 5487985, 5654885, 5702065, 5715581, 5722345, 5749441, 5831113, 5892745, 6044765, 6093541, 6149525, 6212813, 6233981, 6390313, 6426113, 6490805, 6519661, 6534113, 6570313, 6796985, 6893185, 6982585, 7277113, 7307665, 7445941, 7632325, 7781513, 8028025, 8068145, 8116421, 8132545, 8156761, 8205301, 8278381, 8327281, 8359961, 8368141, 8376325, 8425513, 8549113, 8565661, 8640325, 8681945, 8706965, 8740381, 8799013, 8832605, 8866261, 8874685, 8883113, 8891545, 8916865, 8950681, 8984561, 9069541, 9086585, 9095113, 9103645, 9120721, 9146365, 9154921, 9283741, 9309613, 9335521, 9361465, 9370121, 9500441, 9526613, 9552821, 9596581, 9622885, 9675601, 9684401, 9719641, 9728461, 9772621, 9790313, 9799165, 9816881, 9843485, 9852361, 9887905, 9905701, 9932425, 9941341, 9950261, 9977045, 10003865, 10021765, 10066585, 10084541, 10102513, 10138505, 10165541, 10183585, 10210681, 10219721, 10237813, 10246865, 10255921, 10283113, 10310341, 10319425, 10328513, 10392241, 10492781, 10501945, 10529461, 10538641, 10566205.

Some larger examples :