Today I shall share my most recent creation and a demonstration of its power.

## Decord sequences

The definition of a decord sequence is this:

`The first term of the sequence is called term 1. It is equal to whatever value you’re finding the decord sequence of.`

If the nth term of the sequence is 0, stop. The sequence is n terms long.

If the nth term of the sequence is a, and a is not a limit ordinal, the n+1th term is a-1.

If the nth term of the sequence is a, and a is a limit ordinal, the n+1th term is defined like so:

If you can define b and c so that b+c = a & b is the smallest ordinal >=w such that b+c = a, for some c, and b <= c,

term n+1 = c+b[n]+b[n-1]+b[n-2]…+b[1]

Else

term n+1 = a[n]+a[n-1],a[n-2]…+a[1]

Yes, I know you hate ellipses. So here's a definition that you'll like better:

`The first term of the sequence is called term 1. It is equal to whatever value you’re finding the decord sequence of.`

If the nth term of the sequence is 0, stop. The sequence is n terms long.

If the nth term of the sequence is a, and a is not a limit ordinal, the n+1th term is a-1.

If the nth term of the sequence is a, and a is a limit ordinal, the n+1th term is defined like so:

If you can define b and c so that b+c = a, AND b is the smallest ordinal >=w such that b+c = a, for some c, and b <= c,

term n+1 = c + expand(b,n)

Else

term n+1 = expand(a,n)

where expand(a,n) =

a[n] + expand(a,n-1), n > 1

a[n], n = 1

Here I am using the following definition of fundamental sequences:
`The first three rules are derived from the Wainer hierarchy; due to the fact that I'm only taking the fundamental sequences of 1-"term" ordinals I don't need all the Wainer hierarchy rules.`

Rule 1. w[n] = n

Rule 2. w^(a+1)[n] = (w^a)(n)

Rule 3. If a is a limit ordinal and is less than e_0, (w^a)[n] = w^(a[n])

The next rule is a modified version of the corresponding Wainer hierarchy rule, but rule 5 is from the Wainer hierarchy again.

Rule 4. e_0[1] = w

Rule 5. e_0[n+1] = w^e_0[n]

The next rules are stolen from the Veblen hierarchy.

Rule 6. All rules after rule 6 have a colon rather than a period after the number.

Rule 6a: phi_0(a+1)[n] = phi_0(a)n

Rule 6b: If a < phi_0(a) and a is a limit ordinal, phi_0(a)[n] = phi_0(a[n])

Rule 7: phi_(a+1)(0)[n] = phi^n_a(0) (which means phi_a(0) iterated n times, not phi^(n_a)(0)), for all a > 0

Rule 8: phi_(a+1)(b+1)[n] = phi^n_a(phi_(a+1)(b)+1)

Rule 9: if b is a limit ordinal smaller than phi_a(b) phi_a(b)[n] = phi_a(b[n])

Rule 10: if a is a limit ordinal smaller than gamma_0, phi_a(0)[n] = phi_(a[n])(0)

Rule 11: if a is a limit ordinal, phi_a(b+1)[n] = phi_a[n](phi_a(b)+1)

`Rule 12: gamma_(0)[n+1] = phi_(gamma_0[n])(0)`

Rule 13: gamma_(0)[1] = w

Rule 14: (a+b)[n] = a+(b[n]), a >=b>0, a & b both limit ordinals, a < a+b, b < a+b, where a is the minimal ordinal satisfying these conditions

And this is enough to construct decord sequences for everything up to gamma_0

Update:

Rule 15a: (gamma_(n+1))[0] = (gamma_n)+1

Rule 15b: (gamma_(n+1))[m+1] = phi_gamma_(n+1))[m](0)
Rule 16: (gamma_n)[m] = gamma_(n[m]) for lim ord n

Then I defined dco(a) = the length of a decord sequence starting at a.

then I made a demonstrative function, defined as dco(gamma_0+n). After some discussion, conwaylife.com forum user BlinkerSpawn agreed with me that it was comparable to f_gamma_0(n). (specifically, that it was at least comparable to f_gamma_0(n-a) for small a).

As a demonstration, here's the beginning of the decord sequence of gamma_0+1:
`gamma_0+1`

gamma_0

phi_w(0)+w

phi_w(0)+6

phi_w(0)+5

phi_w(0)+4

phi_w(0)+3

phi_w(0)+2

phi_w(0)+1

phi_w(0)

phi_10(0) + phi_9(0) + phi_8(0) ... + z_0 + e_0

phi_10(0) + phi_9(0) + phi_8(0) ... + z_0 + w^^11 + w^^10 +... w^w + w

...

...

...

and so forth. Presumably, dco(gamma_0+n) exceeds things like graham's number or even hydra(any small value) at even rather low n, since f_gamma_0(n) is of course considerably more powerful than f_e_0(n)