Update

src/thue_morse3.v

@@ -204,3 +204,61 @@ Proof.
   apply LEMMA. rewrite H4 in H7. contradiction H7.
   reflexivity.
 Qed.
+
+Lemma tm_step_palindromic_even_center :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ (rev a) ++ tl
+  -> 0 < length a
+  -> even (length (hd ++ a)) = true.
+Proof.
+  intros n hd a tl. intros H I.
+  assert (J: a = removelast a ++ [ last a false ]).
+  apply app_removelast_last.
+  assert (K: {a=nil} + {~ a=nil}). apply list_eq_dec. apply bool_dec.
+  destruct K. rewrite e in I. inversion I. assumption.
+  rewrite J in H. rewrite rev_app_distr in H.
+  rewrite <- app_assoc in H. rewrite <- app_assoc in H.
+  rewrite app_assoc in H. 
+  replace
+    ([last a false] ++ rev [last a false] ++ rev (removelast a) ++ tl)
+    with ([last a false; last a false] ++ (rev (removelast a) ++ tl)) in H.
+  apply tm_step_consecutive_identical in H. rewrite J.
+  rewrite app_assoc. rewrite app_length. rewrite Nat.even_add.
+  rewrite <- Nat.negb_odd. rewrite H. reflexivity.
+  reflexivity.
+Qed.
+
+
+
+
+
+(*
+  TODO: les palindromes de longueur 16 sont centrés autour
+   des valeurs de la suite ci-dessous A056196
+   TODO: erratum 216 est un centre possible ! (216 = 2^3  3^3)
+
+A056196          Numbers n such that A055229(n) = 2.                                +30
+                                                                                      2
+ 8, 24, 32, 40, 56, 72, 88, 96, 104, 120, 128, 136, 152, 160, 168, 184, 200, 224, 232,
+ 248, 264, 280, 288, 296, 312, 328, 344, 352, 360, 376, 384, 392, 408, 416, 424, 440,
+ 456, 472, 480, 488, 504, 512, 520, 536, 544, 552, 568, 584, 600, 608, 616, 632, 640
+ (list; graph; refs; listen; history; text; internal format)
+ OFFSET      1,1
+ COMMENTS    By definition, the largest square divisor and squarefree part of these
+             numbers have GCD = 2.
+
+             Different from A036966. E.g., 81 is not here because A055229(81) = 1.
+
+             Numbers of the form 2^(2*k+1) * m, where k >= 1 and m is an odd number
+             whose prime factorization contains only exponents that are either 1 or
+             even. The asymptotic density of this sequence is (1/12) * Product_{p odd
+             prime} (1-1/(p^2*(p+1))) = A065465 / 11 = 0.08013762179319734335... -
+             Amiram Eldar, Dec 04 2020, Nov 25 2022
+ LINKS       Robert Israel, Table of n, a(n) for n = 1..10000
+ EXAMPLE     88 is here because 88 has squarefree part 22, largest square divisor 4,
+             and A055229(88) = gcd(22, 4) = 2.
+
+
+
+
+ *)