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 ...

19x19 symmetric magic square

192	213	234	255	276	297	318	339	360	1	22	43	64	85	106	127	148	169	190
212	233	254	275	296	317	338	359	19	21	42	63	84	105	126	147	168	189	191
232	253	274	295	316	337	358	18	20	41	62	83	104	125	146	167	188	209	211
252	273	294	315	336	357	17	38	40	61	82	103	124	145	166	187	208	210	231
272	293	314	335	356	16	37	39	60	81	102	123	144	165	186	207	228	230	251
292	313	334	355	15	36	57	59	80	101	122	143	164	185	206	227	229	250	271
312	333	354	14	35	56	58	79	100	121	142	163	184	205	226	247	249	270	291
332	353	13	34	55	76	78	99	120	141	162	183	204	225	246	248	269	290	311
352	12	33	54	75	77	98	119	140	161	182	203	224	245	266	268	289	310	331
11	32	53	74	95	97	118	139	160	181	202	223	244	265	267	288	309	330	351
31	52	73	94	96	117	138	159	180	201	222	243	264	285	287	308	329	350	10
51	72	93	114	116	137	158	179	200	221	242	263	284	286	307	328	349	9	30
71	92	113	115	136	157	178	199	220	241	262	283	304	306	327	348	8	29	50
91	112	133	135	156	177	198	219	240	261	282	303	305	326	347	7	28	49	70
111	132	134	155	176	197	218	239	260	281	302	323	325	346	6	27	48	69	90
131	152	154	175	196	217	238	259	280	301	322	324	345	5	26	47	68	89	110
151	153	174	195	216	237	258	279	300	321	342	344	4	25	46	67	88	109	130
171	173	194	215	236	257	278	299	320	341	343	3	24	45	66	87	108	129	150
172	193	214	235	256	277	298	319	340	361	2	23	44	65	86	107	128	149	170

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

19x19, Siamese method.xls

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19x19 magic square