A135137: missed terms(?)

Jonathan Post jvospost3 at gmail.com
Fri Feb 15 17:03:54 CET 2008

"A penny for your thoughts."

"In for a penny, in for a pound."

Though Maximilian Hasler wrote: "... American bank notes [are]...
really not scientific at all" --  I point out that Sir Isaac Newton
himself, the greatest scientist of all time, was profoundly involved
in the Mint, and England's great recoining.

As wikipedia points out, in summarizing his later life:

"Newton was also a member of the Parliament of England from 1689 to
1690 and in 1701, but his only recorded comments were to complain
about a cold draft in the chamber and request that the window be

"Newton moved to London to take up the post of warden of the Royal
Mint in 1696, a position that he had obtained through the patronage of
Charles Montagu, 1st Earl of Halifax, then Chancellor of the
Exchequer. He took charge of England's great recoining, somewhat
treading on the toes of Master Lucas (and securing the job of deputy
comptroller of the temporary Chester branch for Edmond Halley). Newton
became perhaps the best-known Master of the Mint upon Lucas' death in
1699, a position Newton held until his death. These appointments were
intended as sinecures, but Newton took them seriously, retiring from
his Cambridge duties in 1701, and exercising his power to reform the
currency and punish clippers and counterfeiters. As Master of the Mint
in 1717 Newton unofficially moved the Pound Sterling from the silver
standard to the gold standard by creating a relationship between gold
coins and the silver penny in the 'Law of Queen Anne'; these were all
great reforms at the time, adding considerably to the wealth and
stability of England. It was his work at the Mint, rather than his
earlier contributions to science, that earned him a knighthood from
Queen Anne in 1705."

In fact, he not only made policy for the Mint; he enforced it,
including managing the intelligence operatives who arrested
counterfeiters, but personally interrogated the major ones, and
ordered their executions.

Not only Science, but literally life or death!

As to the more narrow interpretation of A135137.

First, it is evident that "bank notes in common circulation" must be a
subset of:

A124146  U.S.A. currency denominations in dollars.

Second, note the comment: " As the four other largest denominations
are no longer distributed (although still legal tender)" in A108536
Maximal "digit" in position n from right of numbers in "base American
money" representation (A080897).

I admit that  "bank notes in common circulation" could well be
clarified with the comment:  "$1, $5, $10, $20, $50, and $100 are the
most common paper money in the united states (with $1 coins as well,
that being the largest current coinage, although in the past there was
a $20 "gold eagle" coin).

$2 bills are legal tender, but relatively rare.

There are many $2 banknotes, bills or coins, including:
    * Australian two dollar note
    * Australian two dollar coin
    * Toonie, the Canadian two-dollar coin
    * United States two-dollar bill
    * One of the coins of Hong Kong
    * One of the withdrawn Canadian banknotes
Other currencies that issue $2 banknotes, bills or coins are:
    * Bahamian dollar
    * Barbadian dollar
    * Belize dollar
    * Bermudian dollar
    * Cook Islands dollar
    * Fijian dollar
    * New Zealand dollar
    * Samoan tala
    * Singapore dollar
    * Solomon Islands dollar
    * Tuvaluan dollar

As to obsolete U.S. Coinage:


"What denominations of coins are no longer being produced?

"There are quite a few denominations of coins that the United States
Mint does not produce any longer for general circulation. They are the
half-cent coin, the two-cent coin, the three-cent coin, the half-dime
coin (although it was replaced by the five-cent coin), a twenty-cent
coins, and the various denominations of gold coins. Although the Mint
does produce a series of gold bullion coins, these are not intended
for circulation."

Some OEIS seqs already hotlink to:

U.S. Treasury,  FAQs: Currency Denominations

Question: What denominations of currency are in circulation today?
Will any new denominations be produced?

Answer: The present denominations of our currency in production are
$1, $2, 5$, $10, $20, $50 and $100.

and this should be added as a hotlink to A135137 Numbers representing
the sum of the denominations of any three USA bank notes in common

when I use the OEIS search engine and Search: currency
Displaying 1-7 of 7 results found
I see Euro currency, Norwegian currency, and currency of the obscure
Italian city state Fibonacci.

"Dollar" occurs indirectly as well in the likes of:
A110617  The decimal expansion of 1/64532 (related to an optimal mixed
strategy for Hofstadter's million dollar game).
A101111  Second Beale cipher.
and more directly in the likes of:

A130734 List of numbers of cents you can have in US coins without
having change for a dollar.

A108536  Maximal "digit" in position n from right of numbers in "base
American money" representation (A080897).

I grew up reading, and being puzzled by math problems for British
consumption, which depended on the mostly duodecimal system.

There was no Wikipedia (as excerpted below) for me to reference, and
the Encyclopedia Brittanica helped only incompletely.

In 1816, a new silver coinage was introduced in denominations of
sixpence, 1, 2½ and 5 shillings. The crown was only issued
intermittently until 1900. It was followed by a new gold coinage in
1817 consisting of 10 shillings and 1 pound coins, known as the half
sovereign and sovereign. The silver fourpenny coin was reintroduced in
1836, followed by the threepence in 1838, with the fourpenny coin only
issued for colonial use after 1855. In 1848, the 2 shilling florin was
introduced, followed by the short-lived double florin in 1887. In
1860, copper was replaced by bronze in the farthing, halfpenny and

During the First World War, production of the half sovereign and
sovereign was suspended and, although the gold standard was restored,
the coins saw little circulation again. In 1920, the silver standard,
maintained at .925 since 1552, was reduced to .500. In 1937, a
nickel-brass threepenny coin was introduced, with the last silver
threepenny coins issued seven years later. In 1947, the remaining
silver coins were replaced with cupro-nickel. Inflation caused the
farthing to cease production in 1956 and be demonetized in 1960. In
the run up to decimalization, the halfpenny and half crown were
demonetized in 1969.

On 15 February 1971, the U.K. decimalized, replacing the shilling and
penny with a single subdivision, the "new penny". The word new was
used on coins until 1981.

The first decimal coins were introduced in 1968. These were
cupro-nickel 5 and 10 new pence coins which were equivalent to and
circulated alongside the 1 and 2 shillings coins. 1 shilling coins
were demonetized in 1991 following the reduction in size of the 5 new
pence coin, and 2 shilling coins were similarly demonetized in 1993.
The curved equilateral heptagonal, cupro-nickel 50 new pence coin
replaced the 10 shillings note in 1969. The decimal coinage was
completed in 1971 with the introduction of the bronze ½, 1 and 2 new
pence coins and the withdrawal of the 1 and 3 "old" pence coins. 6
pence coins circulated at a value of 2½ new pence until 1980. In 1982,
the word "new" was dropped from the coinage and a 20 pence coin was
introduced, followed by a 1 pound coin in 1983. The ½ penny coin was
last produced in 1983 and demonetized in 1984. The 1990s saw the
replacement of bronze with copper-plated steel and the reduction in
size of the 5, 10 and 50 pence coins. The bi-metallic two pounds coin
was introduced in 1998.

At present, the oldest circulating coins in the U.K. are the 1 and 2
new pence copper coins introduced in 1971. Before decimalisation, it
was possible to get in your change coins aged one hundred years or
more, with one of five different monarch's heads on the obverse.

    Banknotes of the pound sterling

The first sterling notes were issued by the Bank of England shortly
after its foundation in 1694. Denominations were initially written on
the notes at the time of issue. From 1745, the notes were printed in
denominations between 20 and 1000 pounds, with any odd shillings added
in hand. 10 pound notes were added in 1759, followed by 5 pounds in
1793 and 1 and 2 pounds in 1797. The lowest two denominations were
withdrawn following the end of the Napoleonic wars. In 1855, the notes
were converted to being entirely printed, with denominations of 5, 10,
20, 50, 100, 200, 300, 500 and 1000 pounds issued.

The Bank of Scotland began issuing notes in 1695. Although the pound
scots was still the currency of Scotland, these notes were denominated
in sterling in values up to 100 pounds. From 1727, the Royal Bank of
Scotland also issued notes. Both banks issued some notes denominated
in guineas as well as pounds. In the 19th century, regulations limited
the smallest note issued by Scottish banks to be the 1 pound
denomination, a note not permitted in England.

With the extension of sterling to Ireland in 1825, the Bank of Ireland
began issuing sterling notes, later followed by other Irish banks.
These notes included the unusual denominations of 30 shillings and 3
pounds. The highest denomination issued by the Irish banks was 100

In 1826, banks at least 65 miles from London were given permission to
issue their own paper money. From 1844, new banks were excluded from
issuing notes in England and Wales but not in Scotland and Ireland.
Consequently, the number of private banknotes dwindled in England and
Wales but proliferated in Scotland and Ireland. The last English
private banknotes were issued in 1921.

In 1914, the Treasury introduced notes for 10 shillings and 1 pound to
replace gold coins. These circulated until 1928, when they were
replaced by Bank of England notes. Irish independence reduced the
number of Irish banks issuing sterling notes to five operating in
Northern Ireland. The Second World War had a drastic effect on the
note production of the Bank of England. Fearful of mass forgery by the
Nazis (see Operation Bernhard), all notes for 10 pounds and above
ceased production, leaving the bank to issue only 10 shillings, 1 and
5 pounds notes. Scottish and Northern Irish issues were unaffected,
with issues in denominations of 1, 5, 10, 20, 50 and 100 pounds.

The Bank of England reintroduced 10 pound notes in 1964. In 1969, the
10 shilling note was replaced by the 50 new pence coin as part of the
preparation for decimalization. 20 pound Bank of England notes were
reintroduced in 1970, followed by 50 pounds in 1982. Following the
introduction of the 1 pound coin in 1983, Bank of England 1 pound
notes were withdrawn in 1988. Scottish and Northern Irish banks
followed, with only the Royal Bank of Scotland continuing to issue
this denomination.

In conclusion, currency and coins may be mere theory for ivory tower
mathematicians, but they are of great importance to the working men
and women of the world for their direct value, and of curiously
greater importance in Recreational Mathematics. Because of the latter,
I strongly agree with njas that these belong in the OEIS.


Prof. Jonathan Vos Post

p.s. speaking of British money: Lyrics for the Beatles'
"You Never Give Me Your Money" [on the Abbey Road album, 1969]

You never give me your money
You only give me your funny paper
And in the middle of negotiations
You break down

I never give you my number
I only give you my situation
And in the middle of investigation
I break down

Out of college, money spent
See no future, pay no rent
All the money's gone, nowhere to go
Any jobber got the sack
Monday morning, turning back
Yellow lorrie, slow, nowhere to go
But oh, that magic feeling, nowhere to go
Oh, that magic feeling, nowhere to go
Nowhere to go

One sweet dream
Pick up the bags and get in the limousine...

This song is about The Beatles' business problems. When their manager
Brian Epstein died in 1967, they were burdened with handling their own

"Funny Paper" is how The Beatles felt they were paid. They got
frustrated when their accountants would tell them how much they were
worth "on paper," without actually telling them how much money they

p.p.s. "A Penny for Your Thoughts" is also one of my favorite episodes
of the television series The Twilight Zone. Opening narration:
"Mr. Hector B. Poole, resident of the Twilight Zone. Flip a coin and
keep flipping it. What are the odds? Half the time it will come up
heads, half the time tails. But in one freakish chance in a million,
it'll land on its edge. Mr. Hector B. Poole, a bright human coin, on
his way to the bank."
Synopsis: Hector B. Poole, a timid bank clerk, gains telepathic powers
after tossing a coin that miraculously stands on its edge. He
discovers that he is able to "read" other people's thoughts, and is
surprised to hear the things people are thinking around him. He first
"hears" a businessman, Mr. Sykes, trying to take out a large loan to
pay for a run at the horse track to win back money he has embezzled
from his company. [truncated]

On 2/15/08, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> Neil,
>  The difference is that here the amounts are given in the %N.
>  So it is well defined, and the word "cents" is just a nice, motivating add-on.
>  In addition, "cents" is a generic term, not only used for US$ but also
>  for Euros etc.
>  None of my previous critics applies to this sequence.
>  Best regards,
> Maximilian
> PS: one main problem of the other sequence is that manifestly it is
>  not clear what "bank notes in common circulation" means.
>  >  %N A001299 Number of ways of making change for n cents using coins of 1, 5, 10, 25 cents.

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