A375723 Square array read by antidiagonals, where the top row is the powers of 2 (A000079) and the other numbers are the sum of the neighbors in the preceding row.

1, 2, 3, 4, 7, 10, 8, 14, 24, 34, 16, 28, 49, 83, 117, 32, 56, 98, 171, 288, 405, 64, 112, 196, 343, 597, 1002, 1407, 128, 224, 392, 686, 1200, 2085, 3492, 4899, 256, 448, 784, 1372, 2401, 4198, 7285, 12184, 17083, 512, 896, 1568, 2744, 4802, 8403, 14686, 25463, 42546, 59629

Offset

0,2

Comments

Each number in the top row of the array is determined by the pre-defined sequence (in this case, the powers of 2, A000079). Each number in lower rows is the sum of the numbers vertically or diagonally above it (so, the number at the left end of each row is the sum of two numbers, and all other numbers the sum of three). This is the same method as for constructing A217536, which has the top row be the nonnegative integers instead; other similar arrays are described in the comments of that sequence.

The main diagonal is the powers of 7 (A000420), and all numbers above or to the right of the main diagonal are multiples of powers of 2 and powers of 7. Specifically, the number in row m and column n, for n >= m, is 2^(n-m) * 7^m. Above the main diagonal, all numbers in the same column have the same final digit in base 10, and all numbers are 7/2 times the number immediately above.

More broadly, for any similarly constructed array with the powers of x as the top row, then the main diagonal will be the powers of (x^2 + x + 1) and the numbers above the main diagonal will be x^(n-m) * (x^2 + x + 1)^m (see also A062105, which can be interpreted as a similar array with the powers of 1 in the top row, and A020474 with the powers of 0).

Links

Table of n, a(n) for n=0..54.

Formula

T(m+1,n) = sum(T(m,k), |k-n| <= 1) (and T(0,n)=2^n), m, n >= 0.

Example

The array starts:
  1  2  4   8
  3  7  14  28
  10 24 49  98
  34 83 171 343

Crossrefs

The main diagonal gives A000420 (powers of 7). The first column gives A059738.

Cf. A217536, A062105, A020474.

Sequence in context: A098115 A182833 A182832 * A082673 A080724 A342780

Adjacent sequences: A375720 A375721 A375722 * A375724 A375725 A375726

Keyword

nonn,tabl

Author

WG Zeist, Aug 25 2024

Status

approved