The Virahanka–Fibonacci sequence

Long before Fibonacci wrote about it in Italy ($1202$ CE), the Indian poet–mathematician Virahanka ($\sim 6$th–$8$th century) had discovered the same sequence while counting metres in Sanskrit poetry.

Idea

Start with two $1$s. Each next term is the sum of the previous two: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots$

The rule in symbols: $F_1 = F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$.

A natural example: count the ways to climb a staircase taking $1$ or $2$ steps at a time. To reach step $1$: only $1$ way ($1$). To reach step $2$: $2$ ways ($1+1$ or $2$). Step $3$: $3$ ways ($1+1+1$, $1+2$, $2+1$). Step $4$: $5$ ways. Continuing, you get $1, 2, 3, 5, 8, \dots$ , exactly Virahanka's sequence!

The sequence has many surprising properties:

• The ratio $F_{n+1}/F_n$ approaches the golden ratio $\varphi \approx 1.618$.
• Sums show pretty patterns: $1 + 1 + 2 + 3 + 5 = 12 = 13 - 1 = F_7 - 1$.
• Flowers (sunflowers, daisies) often have a Virahanka number of petals.
• Pine cones and pineapple spirals come in Virahanka counts.

The deeper lesson: a simple rule can generate a vast, structured, beautiful world.

Worked examples

Example 1. Write the next two terms after $1, 1, 2, 3, 5, 8$.

Each is the sum of the previous two. $5 + 8 = 13$, then $8 + 13 = 21$. So the sequence continues $1, 1, 2, 3, 5, 8, 13, 21, \dots$.

Example 2. In how many ways can a child climb a $5$-step staircase taking $1$ or $2$ steps at a time?

The answer follows the Virahanka rule. $F_1 = 1, F_2 = 2, F_3 = 3, F_4 = 5, F_5 = 8$. So $8$ ways.

Example 3. Compute $F_{10}$.

Continuing: $F_6=8, F_7=13, F_8=21, F_9=34, F_{10}=55$.

Example 4. Add the first $6$ Virahanka numbers and compare with $F_8 - 1$.

$1+1+2+3+5+8 = 20$. And $F_8 = 21$, so $F_8 - 1 = 20$. ✓ (This is a general pattern: $F_1 + F_2 + \dots + F_n = F_{n+2} - 1$.)

Try it yourself

1. Continue the sequence to $F_{12}$.
2. How many ways to climb $6$ stairs taking $1$ or $2$ steps?
3. Compute $F_3 + F_4 + F_5$ and check it equals $F_7 - 1$.
4. Is $144$ a Virahanka number? Which position?
5. Find $F_{11}/F_{10}$ as a decimal. How close to $1.618$?
6. Adjacent Virahanka numbers , what is special about $\gcd(F_n, F_{n+1})$? Compute for $n=5,6,7$.
7. List Virahanka numbers less than $100$.
8. Start with $2, 5$ and apply the same rule. What sequence do you get?

Activity

Find a flower (sunflower, daisy, lily) and count its petals. Is the count a Virahanka number?

Draw squares of side $1, 1, 2, 3, 5$ in a spiral pattern. Connect their corners with quarter-circles. You will have drawn a Fibonacci spiral , like the swirl of a galaxy or a sea-shell.