# Can this sequence be expressed with a formula?

Can this sequence be expressed with a formula?

1 1 1

2 2

3 3 3

4 4 4

5 5

6 6 6

7 7

8 8 8

9 9 9

10 10

11 11 11

12 12 12

13 13

14 14 14

15 15

16 16 16

17 17 17

18 18

19 19 19

20 20

21 21 21

22 22 22

23 23

24 24 24

25 25 25

26 26

27 27 27

28 28

29 29 29

30 30 30

31 31

32 32 32

33 33 33

34 34

35 35 35

36 36

37 37 37

I don't know if such a formula exists, but à priori this one is good at evaluating the terms up to $\sim 30$ where it starts to fail: $$\left\lfloor\dfrac{n}{1+\phi}\right\rfloor=\left\lfloor\dfrac{n}{\phi^2}\right\rfloor\quad n\in\Bbb N$$ where $\phi$ is the golden ratio.

• Is this rounding down? What is the starting value for n?
– wvdz
Apr 20 '14 at 10:49
• @popovitsj Yes, the floor function outputs the first digit of a number. The starting value is 3. Apr 20 '14 at 10:51
• It seems your first answer was already working for me, I now tested it for values up to 200 and it holds up. I need it to work for 10^16 though... I don't understand your new answer, it's an equation?
– wvdz
Apr 20 '14 at 11:01
• @popovitsj My first answer didn't work, that's why I replaced it with this new one. (How did it work for you?) It is not an equation, it's just because you can express it in two different ways since $1+\phi=\phi^2$, where $\phi=\dfrac{1+\sqrt5}2$. Apr 20 '14 at 11:03
• No, I didn't mean the very first answer 8n/21, I didn't test that one. I meant n/(1+gr). Anyway... pretty amazing that you found it so quickly.. thx :)
– wvdz
Apr 20 '14 at 11:07