Math Is Fun Forum

the fifth term is really weird (1) and last term
cos other than the fifth and last term, the sequence is generally increasing

there are also a few perfect squares hidden in the sequence, eg
1, 25, 121, 49, 1156

here is a nice mnemonic for pi:

Poe, E.: Near a Raven
The poem below was composed in 1995 and bears an uncanny similarity to a certain famous poem by Edgar Allen Poe. This is one of my longer pieces of constrained writing. Constrained writing (an old idea, but one which greatly increased in popularity in the latter half of the 20th century as a result of the work of the mostly French group Oulipo) is the art of constructing a work of prose or poetry that obeys one or more artificially-imposed rules. For example, there are two published novels from which the letter 'e' is absent - Gadsby, by Ernest Vincent Wright (1938), and La disparition by George Perec (still in print, and even available in a recent English translation (A Void, translated by Gilbert Adair) that also obeys the constraint!).

Your mission, should you decide to accept it, is to figure out the constraint imposed on this poem. The answer is given after the end, so if you want to try to figure it out, just look at the beginning of the poem, which is probably both necessary and sufficient to discover the rule. If you'd like a veiled hint, here's one: this poem could not have been written prior to 1947, since all the details of the rule it uses were not known before then.

            Poe, E.
      Near a Raven

Midnights so dreary, tired and weary.
    Silently pondering volumes extolling all by-now obsolete lore.
During my rather long nap - the weirdest tap!
    An ominous vibrating sound disturbing my chamber's antedoor.
        "This", I whispered quietly, "I ignore".

Perfectly, the intellect remembers: the ghostly fires, a glittering ember.
    Inflamed by lightning's outbursts, windows cast penumbras upon this floor.
Sorrowful, as one mistreated, unhappy thoughts I heeded:
    That inimitable lesson in elegance - Lenore -
        Is delighting, exciting...nevermore.

Ominously, curtains parted (my serenity outsmarted),
    And fear overcame my being - the fear of "forevermore".
Fearful foreboding abided, selfish sentiment confided,
    As I said, "Methinks mysterious traveler knocks afore.
        A man is visiting, of age threescore."

Taking little time, briskly addressing something: "Sir," (robustly)
    "Tell what source originates clamorous noise afore?
Disturbing sleep unkindly, is it you a-tapping, so slyly?
    Why, devil incarnate!--" Here completely unveiled I my antedoor--
        Just darkness, I ascertained - nothing more.

While surrounded by darkness then, I persevered to clearly comprehend.
    I perceived the weirdest dream...of everlasting "nevermores".
Quite, quite, quick nocturnal doubts fled - such relief! - as my intellect said,
    (Desiring, imagining still) that perchance the apparition was uttering a whispered "Lenore".
        This only, as evermore.

Silently, I reinforced, remaining anxious, quite scared, afraid,
    While intrusive tap did then come thrice - O, so stronger than sounded afore.
"Surely" (said silently) "it was the banging, clanging window lattice."
    Glancing out, I quaked, upset by horrors hereinbefore,
        Perceiving: a "nevermore".

Completely disturbed, I said, "Utter, please, what prevails ahead.
    Repose, relief, cessation, or but more dreary 'nevermores'?"
The bird intruded thence - O, irritation ever since! -
    Then sat on Pallas' pallid bust, watching me (I sat not, therefore),
        And stated "nevermores".

Bemused by raven's dissonance, my soul exclaimed, "I seek intelligence;
    Explain thy purpose, or soon cease intoning forlorn 'nevermores'!"
"Nevermores", winged corvus proclaimed - thusly was a raven named?
    Actually maintain a surname, upon Pluvious seashore?
        I heard an oppressive "nevermore".

My sentiments extremely pained, to perceive an utterance so plain,
    Most interested, mystified, a meaning I hoped for.
"Surely," said the raven's watcher, "separate discourse is wiser.
    Therefore, liberation I'll obtain, retreating heretofore -
        Eliminating all the 'nevermores' ".

Still, the detestable raven just remained, unmoving, on sculptured bust.
    Always saying "never" (by a red chamber's door).
A poor, tender heartache maven - a sorrowful bird - a raven!
    O, I wished thoroughly, forthwith, that he'd fly heretofore.
        Still sitting, he recited "nevermores".

The raven's dirge induced alarm - "nevermore" quite wearisome.
    I meditated: "Might its utterances summarize of a calamity before?"
O, a sadness was manifest - a sorrowful cry of unrest;
    "O," I thought sincerely, "it's a melancholy great - furthermore,
        Removing doubt, this explains 'nevermores' ".

Seizing just that moment to sit - closely, carefully, advancing beside it,
    Sinking down, intrigued, where velvet cushion lay afore.
A creature, midnight-black, watched there - it studied my soul, unawares.
    Wherefore, explanations my insight entreated for.
        Silently, I pondered the "nevermores".

"Disentangle, nefarious bird! Disengage - I am disturbed!"
    Intently its eye burned, raising the cry within my core.
"That delectable Lenore - whose velvet pillow this was, heretofore,
    Departed thence, unsettling my consciousness therefore.
        She's returning - that maiden - aye, nevermore."

Since, to me, that thought was madness, I renounced continuing sadness.
    Continuing on, I soundly, adamantly forswore:
"Wretch," (addressing blackbird only) "fly swiftly - emancipate me!"
    "Respite, respite, detestable raven - and discharge me, I implore!"
        A ghostly answer of: "nevermore".

" 'Tis a prophet? Wraith? Strange devil? Or the ultimate evil?"
    "Answer, tempter-sent creature!", I inquired, like before.
"Forlorn, though firmly undaunted, with 'nevermores' quite indoctrinated,
    Is everything depressing, generating great sorrow evermore?
        I am subdued!", I then swore.

In answer, the raven turned - relentless distress it spurned.
    "Comfort, surcease, quiet, silence!" - pleaded I for.
"Will my (abusive raven!) sorrows persist unabated?
    Nevermore Lenore respondeth?", adamantly I encored.
        The appeal was ignored.

"O, satanic inferno's denizen -- go!", I said boldly, standing then.
    "Take henceforth loathsome "nevermores" - O, to an ugly Plutonian shore!
Let nary one expression, O bird, remain still here, replacing mirth.
    Promptly leave and retreat!", I resolutely swore.
        Blackbird's riposte: "nevermore".

So he sitteth, observing always, perching ominously on these doorways.
    Squatting on the stony bust so untroubled, O therefore.
Suffering stark raven's conversings, so I am condemned, subserving,
    To a nightmare cursed, containing miseries galore.
        Thus henceforth, I'll rise (from a darkness, a grave) -- nevermore!

                        -- Original: E. Poe
                        -- Redone by measuring circles.

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Solution:

Despite the rather difficult constraint (to be revealed shortly), this pastiche of "The Raven" attempts to follow the story, tone, and rhyme scheme of the original as closely as possible - even, for instance, including the internal rhymes in the first and third line of each stanza. The first major structural concession in our version is that the original has six lines per stanza, with the fourth and fifth lines usually being very similar. Due to the nature of the constraint I imposed (revealed in the next paragraph), this would have been nearly impossible to do. Therefore, this version eliminates the similar line in each stanza. The second major shortcoming is that most of the highly regular meter in Poe's original is lost. Although it's not impossible to capture meter using this constraint, it's hard, so I decided not to attempt it but to concentrate on other aspects of the poem.

To reveal the constraint under which I labored, start at the very beginning (with the word 'Poe') and write next to each word the number of letters it contains. Put a decimal point after the first digit. Look at the first few digits (or more if, like me, you know the first several hundred by heart). Now do you see?!

Even given the difficult constraint, strong echoes of the original are still present. The very first line, although its meter is wrong, is surprisingly close. Others which are very close, even to the point of using many of the same words, are stanza 4 line 5, stanza 6 line 3, stanza 7 line 4, and stanza 15, line 1. Observe how most of the stanzas are engineered to end with some form of the word "nevermore", as they should.

Note the use of the term "blackbird" a couple of times. Though not, strictly speaking, correct (a raven is a black bird, not a blackbird), the term is particularly appropriate. It is a reference to George Perec's La disparition, and in particular its English translation (A Void) by Gilbert Adair. A Void contains another written-with-constraints version of "The Raven" - in this case the constraint being "write it without using the letter 'e'". Of course, this means the title has to be different, too, since "The Raven" contains the letter 'e'. The title of Adair's poem is (wait for it...) Black Bird!

"Near a Raven" encodes the first 740 decimals of pi. To be explicit, the encoding rule is this: a word of N letters represents

- the digit N if N<10
- the digit 0 if N=10, and
- two adjacent digits if N>10
    (e.g., a 12-letter word represents the digit '1' followed by '2').

Word-length pi mnemonics have been around a long time. An old chestnut, but still one of the best, is:

How I need a drink, alcoholic in nature, after the heavy lectures involving quantum mechanics!

A much less well-known example is this nice poem by Joseph Shipley (1960):

But a time I spent wandering in bloomy night;
Yon tower, tinkling chimewise, loftily opportune.
Out, up, and together came sudden to Sunday rite,
The one solemnly off to correct plenilune.

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