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A375477 Lexicographically earliest sequence of distinct nonnegative terms arranged in successive chunks whose digitsum = 10, said chunks being "linked" (see the Comments section for an explanation).

%I A375477 #13 Aug 17 2024 22:41:19
%S A375477 0,1,2,3,4,40,6,60,10,12,20,5,21,11,8,80,101,13,15,50,14,41,23,30,7,
%T A375477 70,100,1001,16,102,22,24,42,31,17,10001,19,91,103,33,32,104,43,111,
%U A375477 105,110,100001,106,201,107,1000001,109,901,112,51,113,120,10000001,114,121
%N A375477 Lexicographically earliest sequence of distinct nonnegative terms arranged in successive chunks whose digitsum = 10, said chunks being "linked" (see the Comments section for an explanation).
%C A375477 The 1st chunk with digitsum = 10 is (0, 1, 2, 3, 4), ending with a "4". The next chunk with digitsum = 10 must start with a "4" (this is the "link") and is thus (40, 6). As the next chunk with digitsum = 10 must start with a "6", we have (60, 10, 12). The next chunk with digitsum = 10 must start with a "2" and we have (20, 5, 21), etc.
%C A375477 No chunk is allowed to end with a zero.  Thus, no intermediate chunk digit sum can be 9, and anytime a chunk needs a digitsum of 2 to "complete", the next term must be of the form 10^k + 1 for k >= 1.
%C A375477 Infinite since there are an infinity of terms with digits sums <= 10.
%C A375477 a(5367) has 1001 digits.
%H A375477 Michael S. Branicky, <a href="/A375477/b375477.txt">Table of n, a(n) for n = 1..5366</a>
%H A375477 Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2024/08/tile-my-sequence.html">Tile my sequence</a>, personal blog of the author.
%o A375477 (Python)
%o A375477 from itertools import count, islice
%o A375477 def bgen(ds): # generator of terms with digital sum ds
%o A375477     def A051885(n): return ((n%9)+1)*10**(n//9)-1 # due to Chai Wah Wu
%o A375477     def A228915(n): # due to M. F. Hasler
%o A375477         p = r = 0
%o A375477         while True:
%o A375477             d = n % 10
%o A375477             if d < 9 and r: return (n+1)*10**p + A051885(r-1)
%o A375477             n //= 10; r += d; p += 1
%o A375477     k = A051885(ds)
%o A375477     while True: yield k; k = A228915(k)
%o A375477 def agen(): # generator of terms
%o A375477     an, ds_block, seen, link_set, min2 = 0, 0, set(), "123456789", 11
%o A375477     while True:
%o A375477         yield an
%o A375477         seen.add(an)
%o A375477         if ds_block == 8:
%o A375477             while min2 in seen: min2 = 10*min2 - 9
%o A375477             an, ds_an, link_an = min2, 2, "1"
%o A375477         else:
%o A375477             cand_ds = list(range(1, 9-ds_block)) + [10-ds_block]
%o A375477             dsg = [0] + [bgen(i) for i in range(1, 11-ds_block)]
%o A375477             dsi = [0] + [(next(dsg[i]), i) for i in range(1, 11-ds_block)]
%o A375477             while True:
%o A375477                 k, ds_k = min(dsi[j] for j in cand_ds)
%o A375477                 if k not in seen:
%o A375477                     sk, dst = str(k), ds_k + ds_block
%o A375477                     if sk[0] in link_set:
%o A375477                         if dst < 9 or (dst == 10 and k%10 != 0):
%o A375477                             an, ds_an, link_an = k, ds_k, sk[-1]
%o A375477                             break
%o A375477                 dsi[ds_k] = (next(dsg[ds_k]), ds_k)
%o A375477         ds_block = ds_block + ds_an
%o A375477         if ds_block == 10: ds_block, link_set = 0, link_an
%o A375477         else: link_set = "123456789"
%o A375477 print(list(islice(agen(), 60)))
%Y A375477 Cf. A375460.
%K A375477 base,nonn
%O A375477 1,3
%A A375477 _Eric Angelini_ and _Michael S. Branicky_, Aug 17 2024