Gilbreth Sequences

Supporting material for the definition of Gilbreth Sequences
in the On-Line Encyclopedia of Integer Sequences (OEIS).

Rationale

I have been fascinated with brick patterns for a long time. This led me to study the 1909 book Bricklaying System by Frank B. Gilbreth. Bond Chart No 25 on page 265 of the book shows 10 successive units of a basic module used in decorative brick patterns.

I have identified the underlying algorithm which allows to recreate the 10 first units shown on Bond Chart No 25 and to continue the sequence indefinitely based on the same logic.

Each successive one is a half brick wider and two courses higher than the one that precedes it.

A course of bricks is a sequence of bricks on the same row.

Definition of Top-Down and Bottom-Up Gilbreth Sequences

This infinite sequence of Gilbreth Brick Units can be described as the conjunction of two separate sequences:

• a top-down sequence which generates the upper part of the unit, from the uppermost course with a single header brick down to the course in the middle of the unit, included.

• a bottom-up sequence which generates the lower part of the unit, from the lowermost course with a single header brick up to the course in the middle of the unit, excluded.

A header brick is a brick seen from the short side, as a half brick; it is heading towards us. A stretcher brick is a brick seen from the long side, as a whole brick; it is stretching along its whole length in the facade.

I have devised a numerical transcription of bricks and brick courses which forms the basis for the mathematical definition of the top-down and the bottom-up sequences for their inclusion in the OEIS database.

• a header brick is represented as digit 1 (half length)
• a stretcher brick is represented as digit 2 (full length)
• a course of bricks is represented as a number in which successive digits, from left to right, represent the corresponding bricks found in the course in the same left to right order.

The first ten terms of the top-down sequence can be transcribed from the visual descriptions of the ten units in Bond Chart 25 by creating term #i from a transcription of the middle course in unit #i:

1. 1
2. 2
3. 12
4. 22
5. 212
6. 222
7. 2122
8. 2222
9. 22122
10. 22222

The first nine terms of the bottom-up sequence can be transcribed similarly by creating term #i as a transcription of course just below the middle course in unit #i+1:

1. 1
2. 2
3. 21
4. 22
5. 212
6. 222
7. 2212
8. 2222
9. 22122

Each term in the bottom-up sequence is a mirror of the corresponding term in the top-down sequence, with the same digits in reversed order.

Related Sequence in the OEIS

Sequence A094626 in the OEIS is related to the two Gilbreth sequences:

1. 1
2. 2
3. 12
4. 22
5. 122
6. 222
7. 1222
8. 2222
9. 12222
10. 22222

Starting with term #1, each term in sequence A094626 has the same digits as the corresponding terms in the Top-Down and Bottom-Up Gilbreth sequences, except that the digit 1 is found in the first position in A094626 rather than in the middle position, rounded left in the Top-Down Gilbreth sequence or rounded right in the Bottom-Up Gilbreth sequence:

Term	A094626	Top-Down	Bottom-Up
1	1	1	1
2	2	2	2
3	12	12	21
4	22	22	22
5	122	212	212
6	222	222	222
7	1222	2122	2212
8	2222	2222	2222
9	12222	22122	22122

Recursive Definition of the Sequences

Starting from the fact that Each successive [unit] is a half brick wider (…) than the one that precedes it, the length of the middle course increases by:

• a half brick after 1 step,
• a whole brick after 2 steps,
• a brick and a half after 3 steps,
• two whole bricks after 4 steps.

Due to the balanced design of the units, the middle course in unit #n+4 is actually identical to the middle course in unit #n with a whole stretcher brick added on each side.

This leads to a recursive definition of Gilbreth Sequences in two stages: initial conditions for the first four steps followed with a recursive definition of step n+4 based on step n.

Initial Conditions

Term	Top-Down	Bottom-Up
1	1	1
2	2	2
3	12	21
4	22	22

Term #3 embeds the idiosyncratic preference for center header positions rounded to the left in the Top-Down Gilbreth sequence and rounded to the right in the Bottom-Up Gilbreth sequence.

Recursive Definition

The recursive process at play is identical in both Gilbreth sequences, a concatenation, noted here || of a whole stretcher brick 2 on the left and on the right of the term found 4 steps before in the sequence:

u(n+4) = "2" || u(n) || "2"

This concatenation of digits can be expressed in numerical terms by the addition of three terms shifted to the expected positions through a multiplication by the corresponding powers of 10:

• 2, the stretcher on the right, not shifted
• u(n), shifted by one,
• 2, the stretcher on the left, shifted by the number of digits in u(n) plus one for the number of digits in the stretcher on the right.

The shift required for the stretcher digit 2 on the left is:

• the number of digits in item u(n)
• plus one for the stretcher digit 2 on the right.

The number of digits in item u(n) is known for the first four items:

Term	Top-Down	Bottom-Up	Digits
1	1	1	1
2	2	2	1
3	12	21	2
4	22	22	2

The length can be computed recursively for the following items, given that the term at step #n+4 grows by two stretcher digits compared to the term at step #n:

digits( u(n+4) ) = digits( u(n) ) + 2

In the first four terms, the number of digits increases by one in every other step. By recursion, this remains the same in the next four terms, indefinitely. We can thus compute the number of digits of the term #n as:

digits( u(n) ) = 1 + floor(n/2)

We can now rewrite the concatenation:

u(n+4) = "2" || u(n) || "2"

using the computed shift for the number of digits in u(n) as:

u(n+4) = 2 * 10^( 1 + floor(n/2) + 1 ) + 10 * u(n) + 2

or:

u(n+4) = 2 * 10^( 2 + floor(n/2) ) + 10 * u(n) + 2

Relationship with Sequence A094626

Sequence A094626 may be defined in the same way for comparison. It shares the same initial conditions as the Top-Down Gilbreth sequence:

Term	A094626	Top-Down	Bottom-Up
1	1	1	1
2	2	2	2
3	12	12	21
4	22	22	22

but the following terms are produced by a recursive process which differs from the one in the two Gilbreth sequences:

u(n+4) = u(n) || "22"

Stretcher digits 2 are always added to the right in A094626, which makes it a more distant relative of the two tight-knit Gilbreth sequences.

References

Bricklaying System

Internet Archive

• Bricklaying System (copy from Cornell University)
• Bricklaying System (copy from Getty Research Institute)
• Bricklaying System (copy from University of California)

Cheaper by the Dozen

Internet Archive

• Cheaper by the Dozen

Frank and Lillian Gilbreth

New England Historical Society

• Lillian Gilbreth, Genius in the Art of Living

Wikipedia

• Frank Bunker Gilbreth
• Lillian Moller Gilbreth

License

CC-BY Eric Bréchemier