## Tuesday, August 28, 2012

### Flagpole

oneseventhreesixfiveeleveneightninetwothirteenfourtentwelvesixteenfifteentwentysixfourteeneighteenseventeennineteentwentyonehundredeleventwentyonetwentyfivetwentytwotwentythreetwentyfourtwentyeightthirtyeightthirtytwentyseventwentyninethirtyonethirtyseventhirtythreethirtytwofortytwothirtyfourthirtyfivethirtysixthirtyninefortyfortyonefortythreefortyfivefortysevenfortyfourfortysixfiftyfourfortyeightfortyninefiftyonefiftysevenfiftyfiftytwofiftythreeonehundredtwelvefiftyfivefiftyeightfiftysixfiftyninesixtysixtyonesixtythreesixtyfivesixtysevensixtyfoursixtytwoseventyfoursixtysixsixtyeightsixtynineseventyseventyoneseventytwoseventythreeseventyfiveseventysixseventysevenseventyeightseventynineeightysixeightyfoureightyoneninetyfoureightythreeeightyeightytwoeightyfiveeightyseveneightyeighteightynineninetyeightninetythreeninetyninetyoneninetytwoninetyfiveninetysixninetysevenonehundredfiveninetynineonehundredonehundredtwoonehundredoneonehundredthreeonehundredfouronehundredsixonehundredeightonehundredsevenonehundrednineonehundredtenonehundredthirteen...

This is the top of the flagpole of Eric Angelini's English flagpole sequence. If we subtract the indices from the sequence, we get: 0, 5, 0, 2, 0, 5, 1, 1, -7, 3, -7, -2, -1, 2, 0, 10, -3, 0, -2, -1, -1, 89, -2, 1, -3, -3, -3, 0, 9, 0, -4, -3, -2, 3, -2, -4, 5, -4, -4, -4, -2, -2, -2, -1, 0, 1, -3, -2, 5, -2, -2, -1, 4, -4, -3, -3, 55, -3, -1, -4, -2, -2, -2, -1, 0, 1, -3, -6, 5, -4, -3, -3, -3, -3, -3, -3, -2, -2, -2, -2, -2, 4, 1, -3, 9, -3, -7, -6, -4, -3, -3, -3, 5, -1, -5, -5, -5, -3, -3, -3, 4, -3, -3, -2, -4, -3, -3, -2, -1, -3, -2, -2, 0, ... Herein, the positions of the zeros are: 1, 3, 5, 15, 18, 28, 30, 45, 65, 113, ... In effect, these are the indices of the flagpole sequence where a(n) = n. It turns out that the majority of flagpole sequence numbers lie on that line — of the first 1000000 terms, 688153. Within this region, the largest spacing between consecutive on-the-line terms is between term #321947 (=321947) and term #411273 (=411273). This large gap is due to the demand for the letter L exceeding the local supply. A much larger gap will begin ~10^9 when demand for the letter M exceeds its local supply.