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).

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, 70, 100, 1001, 16, 102, 22, 24, 42, 31, 17, 10001, 19, 91, 103, 33, 32, 104, 43, 111, 105, 110, 100001, 106, 201, 107, 1000001, 109, 901, 112, 51, 113, 120, 10000001, 114, 121

Offset: 1

Author

Eric Angelini and Michael S. Branicky, Aug 17 2024

Keywords

• base
• nonn

Comments

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.

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.

Infinite since there are an infinity of terms with digits sums <= 10.

a(5367) has 1001 digits.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..5366
• Eric Angelini, Tile my sequence, personal blog of the author.

Crossrefs

Cf. A375460.

Programs

Python

from itertools import count, islice
def bgen(ds): # generator of terms with digital sum ds
    def A051885(n): return ((n%9)+1)*10**(n//9)-1 # due to Chai Wah Wu
    def A228915(n): # due to M. F. Hasler
        p = r = 0
        while True:
            d = n % 10
            if d < 9 and r: return (n+1)*10**p + A051885(r-1)
            n //= 10; r += d; p += 1
    k = A051885(ds)
    while True: yield k; k = A228915(k)
def agen(): # generator of terms
    an, ds_block, seen, link_set, min2 = 0, 0, set(), "123456789", 11
    while True:
        yield an
        seen.add(an)
        if ds_block == 8:
            while min2 in seen: min2 = 10*min2 - 9
            an, ds_an, link_an = min2, 2, "1"
        else:
            cand_ds = list(range(1, 9-ds_block)) + [10-ds_block]
            dsg = [0] + [bgen(i) for i in range(1, 11-ds_block)]
            dsi = [0] + [(next(dsg[i]), i) for i in range(1, 11-ds_block)]
            while True:
                k, ds_k = min(dsi[j] for j in cand_ds)
                if k not in seen:
                    sk, dst = str(k), ds_k + ds_block
                    if sk[0] in link_set:
                        if dst < 9 or (dst == 10 and k%10 != 0):
                            an, ds_an, link_an = k, ds_k, sk[-1]
                            break
                dsi[ds_k] = (next(dsg[ds_k]), ds_k)
        ds_block = ds_block + ds_an
        if ds_block == 10: ds_block, link_set = 0, link_an
        else: link_set = "123456789"
print(list(islice(agen(), 60)))