Siamese method

Put number 1 in the middle of the top row. Put the numbers 2 up to n (= length of the square) each time one cell diagonal up and to the right. Put number n+1 below number n. Put the numbers n+2 up to 2n each time one cell diagonal up and to the right. Put number 2n+1 below number 2n. Etcetera ...

15x15 symmetric magic square

122	139	156	173	190	207	224	1	18	35	52	69	86	103	120
138	155	172	189	206	223	15	17	34	51	68	85	102	119	121
154	171	188	205	222	14	16	33	50	67	84	101	118	135	137
170	187	204	221	13	30	32	49	66	83	100	117	134	136	153
186	203	220	12	29	31	48	65	82	99	116	133	150	152	169
202	219	11	28	45	47	64	81	98	115	132	149	151	168	185
218	10	27	44	46	63	80	97	114	131	148	165	167	184	201
9	26	43	60	62	79	96	113	130	147	164	166	183	200	217
25	42	59	61	78	95	112	129	146	163	180	182	199	216	8
41	58	75	77	94	111	128	145	162	179	181	198	215	7	24
57	74	76	93	110	127	144	161	178	195	197	214	6	23	40
73	90	92	109	126	143	160	177	194	196	213	5	22	39	56
89	91	108	125	142	159	176	193	210	212	4	21	38	55	72
105	107	124	141	158	175	192	209	211	3	20	37	54	71	88
106	123	140	157	174	191	208	225	2	19	36	53	70	87	104

You can use this method to construct magic squares of odd order from 3x3 to infinite and you get a symmetric (but not pan)magic square.

See 3x3, 5x5, 7x7, 9x9, 11x11, 13x13, 15x15, 17x17, 19x19, 21x21, 23x23, 25x25, 27x27, 29x29 and 31x31

15x15, Siamese method.xls

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15x15 magic square