A368477 a(n) is the number of integer tuples (b_1, b_2, ..., b_(k+1)) where 0 <= b_i <= 9, such that |b_i - b_(i+1)| = d_i for all i, where (d_1, d_2, ..., d_k) is the decimal expansion of n.

10, 18, 16, 14, 12, 10, 8, 6, 4, 2, 18, 34, 30, 26, 22, 18, 14, 10, 6, 2, 16, 30, 28, 24, 20, 16, 12, 8, 4, 2, 14, 26, 24, 22, 18, 14, 10, 6, 4, 2, 12, 22, 20, 18, 16, 12, 8, 6, 4, 2, 10, 18, 16, 14, 12, 10, 8, 6, 4, 2, 8, 14, 12, 10, 8, 8, 8, 6, 4, 2, 6, 10, 8, 6, 6, 6, 6, 6, 4, 2, 4, 6, 4, 4, 4, 4, 4, 4, 4

Offset

0,1

Comments

If n is (d_1, d_2, ..., d_(k-1), d_k) and m is (d_1, d_2, ..., d_(k-1), (10 - d_k) mod 10) then a(n) == a(m) (mod 4).

Sum_{i=0..10^k-1} a(10^k * n + i) = 10^k * a(n) for all n >= 0 and k >= 1.

Sum_{n=0..m * 10^k - 1} a(n) = 10^k * Sum_{n=0..m-1} a(n) for all m >= 1 and k >= 0.

Links

Table of n, a(n) for n=0..88.

Mathematics Stack Exchange, Discussion of concepts used to find nth term mathematically.

Math subreddit, Sequence mod 4 visualisation on a table.

Example

For n=0, there are 10 2-tuples (b_1, b_2) such that |b_1 - b_2| = 0; they are (0,0), (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7), (8,8), (9,9), so a(0)=10.
For n=1, there are 18 2-tuples (b_1, b_2) such that |b_1 - b_2| = 1; they are (0,1), (1,0), (1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5), (6,7), (7,6), (7,8), (8,7), (8,9), (9,8), so a(1)=18.
For n=42, there are 20 3-tuples (b_1, b_2, b_3) such that |b_1 - b_2| = 4 and |b_2 - b_3| = 2; they are (0,4,2), (0,4,6), (1,5,3), (1,5,7), (2,6,4), (2,6,8), (3,7,9), (3,7,5), (4,0,2), (4,8,6), (5,1,3), (5,9,7), (6,2,0), (6,2,4), (7,3,1), (7,3,5), (8,4,2), (8,4,6), (9,5,3), (9,5,7), so a(42)=20.

Prog

(Python)
def a(n):
  dp = [1] * 10
  for c in str(n):
    bdigit = int(c)
    new_dp = [0] * 10
    for j in range(10):
      if bdigit != 0 and j + bdigit <= 9:
        new_dp[j] += dp[j + bdigit]
      if j - bdigit >= 0:
        new_dp[j] += dp[j - bdigit]
    dp = new_dp
  return sum(dp)

Crossrefs

Sequence in context: A337325 A156382 A214894 * A084487 A055567 A373994

Adjacent sequences: A368474 A368475 A368476 * A368478 A368479 A368480

Keyword

base,easy,nonn

Author

Shahrukh Jamil, Dec 26 2023

Status

approved