Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence/Examples/490314, 817190, 1144066

Example of Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence

The integers:

$490 \, 314, 817 \, 190, 1 \, 144 \, 066$

are in arithmetic sequence and appear in row $23$ of Pascal's triangle.

Proof

We have:

\(\ds 817 \, 190 - 490 \, 314\)

\(=\)

\(\ds 326 \, 876\)

\(\ds 1 \, 144 \, 066 - 817 \, 190\)

\(=\)

\(\ds 326 \, 876\)

thus demonstrating the common difference of $326 \, 876$.

Then we have that row $23$ of Pascal's triangle is:

$1, 23, 253, 1771, 8855, 33 \, 649, 100 \, 947,$

$245 \, 157, \color {red} {490 \, 314, 817 \, 190, 1 \, 144 \, 066}, 1 \, 352 \, 078, 1 \, 352 \, 078,$

$1 \, 144 \, 066, 817 \, 190, 490 \, 314, 245 \, 157, 100 \, 947, 33 \, 649, 8855, 1771, 253, 23, 1$

This sequence is A010939 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $35$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $35$