A049525 -id:A049525 - cp's OEIS frontend

A005224 T is the first, fourth, eleventh, ... letter in this sentence, not counting spaces or commas (Aronson's sequence).

1, 4, 11, 16, 24, 29, 33, 35, 39, 45, 47, 51, 56, 58, 62, 64, 69, 73, 78, 80, 84, 89, 94, 99, 104, 111, 116, 122, 126, 131, 136, 142, 147, 158, 164, 169, 174, 181, 183, 193, 199, 205, 208, 214, 220, 226, 231, 237, 243, 249, 254, 270, 288, 303, 307, 319, 323, 341

Offset: 1

N. J. A. Sloane

a(10^9) = 11281384554. - Hans Havermann, Apr 21 2017
First differences start: 3, 7, 5, 8, 5, 4, 2, 4, 6, 2, 4, 5, 2, 4, 2, 5, 4, 5, 2, 4, 5, 5, 5, 5, 7, 5, 6, 4, 5, 5, 6, 5, 11, 6, 5, 5, 7, 2, 10, 6, ... - Daniel Forgues, Sep 11 2019
Named after the British clinical pharmacologist Jeffrey Kenneth Aronson (b. 1947). - Amiram Eldar, Jun 23 2021

			The sentence begins
1234567890 1234567890 1234567890 1234567890 1234567890
Tisthefirs tfourthele venthsixte enthtwenty fourthtwen
tyninththi rtythirdth irtyfiftht hirtyninth fortyfifth
fortyseven thfiftyfir stfiftysix thfiftyeig hthsixtyse
condsixtyf ourthsixty ninthseven tythirdsev entyeighth
eightiethe ightyfourt heightynin thninetyfo urthninety
ninthonehu ndredfourt honehundre deleventho nehundreds
ixteenthon ehundredtw entysecond onehundred twentysixt
honehundre dthirtyfir stonehundr edthirtysi xthonehund
redfortyse cond...

J. K. Aronson, quoted by D. R. Hofstadter in Metamagical Themas, Basic Books, NY, 1985, p. 44.
James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Benoit Cloitre, N. J. A. Sloane and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seq., Vol. 6 (2003), Article 03.2.2; arXiv preprint, arXiv:math/0305308 [math.NT], 2003.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 1.
Hans Havermann, Aronson's Sequence.
N. J. A. Sloane, Seven Staggering Sequences, 2006.
Eric Weisstein's World of Mathematics, Aronson's Sequence.
Wikipedia, Aronson's sequence.

Cf. A049525, A055508, A081023, A081024, A079000, A097395, A284744.

seed="tisthe"; s[1]=1;s[2]=4;
name[n_]:=StringReplace[IntegerName[n,{"English","Ordinal"}],{"-"->""," "->""}];
s[n_]:=seed=StringJoin[seed<>name[StringPosition[seed,"t"][[n-2,1]]]];
l=s/@Range[58]; Table[StringPosition[Last[l],"t"][[i,1]],{i,1,Length[l]}] (* Ivan N. Ianakiev, Mar 25 2020 *)

from num2words import num2words
from itertools import islice
def n2w(n):
    os = num2words(n, ordinal=True).replace(" and", "")
    return os.replace(" ", "").replace("-", "").replace(chr(44), "")
def agen(): # generator of terms
    s, idx = "tisthe", 0
    while True:
        idx_rel = 1 + s.index("t")
        idx += idx_rel
        yield idx
        s = s[idx_rel:] + n2w(idx)
print(list(islice(agen(), 58))) # Michael S. Branicky, Mar 18 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Oct 31 2000

A055508 H is the first, fifth, sixteenth, ..., letter in the sequence not counting spaces and commas.

Original entry on oeis.org

1, 5, 16, 25, 36, 38, 47, 49, 57, 59, 71, 81, 93, 103, 119, 134, 141, 149, 156, 172, 176, 184, 194, 198, 218, 234, 238, 254, 258, 281, 299, 303, 313, 321, 325, 343, 347, 363, 365, 369, 379, 385, 389, 397, 407, 411, 419, 427, 429, 433, 450, 454, 469, 471, 475

Offset: 1

Robert G. Wilson v, Jun 29 2000

Along with the letters "T" and "I", this completes the series, i.e. no other letters in the alphabet can substain a sequence with such a definition.

Cf. A005224 and A049525.

A097963 ["comma"], is the first, fifteenth, twenty-sixth, fortieth, ... character in this sentence, including spaces and punctuations.

Original entry on oeis.org

1, 15, 26, 40, 50, 60, 70, 82, 97, 113, 137, 165, 190, 213, 237, 265, 290, 313, 339, 367, 396, 424, 452, 479, 507, 529, 556, 582, 610, 629, 655, 680, 703, 724, 753, 780, 805, 826, 854, 882, 911, 934, 962, 989, 1016, 1040, 1063, 1089, 1116, 1152, 1191, 1230

Offset: 1

Ray G. Opao, Sep 21 2004

Yes, the definition begins with a comma!

			Label the coordinates:
00000000011111111112222222222333333333344444444445...
12345678901234567890123456789012345678901234567890... Then (comma)
["Comma"], is the first, fifteenth, twenty-sixth, fortieth, fiftieth, fifty-ninth, seventy-second, eighty-eighth, one hundred third, one hundred twenty-second, ... character.

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000

Cf. A055508, A005224, A049525.

from itertools import accumulate, repeat
from num2words import num2words
A097963_list = [1]+list(accumulate(repeat(15,100), lambda x, : x+2+len(num2words(x,to='ordinal').replace(' and ',' ').replace(', ',' ')))) # _Chai Wah Wu, Aug 30 2021

After a(2) = 15, the remaining terms are given by the recursion a(n+1) = a(n) + 2 + length(OrdinalName(a(n))); e.g., a(3) = a(2) + 2 + length(OrdinalName(a(2))) = 15 + 2 + length("fifteenth") = 15 + 2 + 9 = 26. - Jon E. Schoenfield, Aug 13 2007

Corrected and extended by Jon E. Schoenfield, Aug 13 2007

A094110 Start with the word "One". The next word is the number of letters written previously. Convert this infinite sequence of words into an infinite sequence of numbers.

Original entry on oeis.org

1, 3, 8, 13, 21, 30, 36, 45, 54, 63, 73, 85, 95, 105, 119, 137, 158, 178, 200, 210, 223, 244, 263, 283, 304, 320, 338, 361, 381, 402, 416, 434, 455, 475, 497, 519, 538, 560, 576, 597, 619, 637, 658, 678, 700, 712, 730, 748, 770, 789, 811, 829, 851, 871, 893

Offset: 1

Eric Angelini, May 03 2004

Variant of A060403. [From R. J. Mathar, Dec 15 2008]

			The sentence begins
1234567890 1234567890 1234567890 1234567890 1234567890
OneThreeEi ghtThirtee nTwentyone ThirtyThir tysixForty
fiveFiftyf ourSixtyth reeSeventy threeEight yfiveNinet
yfiveOnehu ndredfiveO nehundredn ineteenOne hundredthi
rtysevenOn ehundredfi ftyeightOn ehundredse ventyeight
Twohundred Twohundred tenTwohund redtwentyt hreeTwohun
dredfortyf ourTwohund redsixtyth reeTwohund redeightyt
hreeThreeh undredfour Threehundr edtwentyTh reehundred
thirtyeigh tThreehund redsixtyon eThreehund redeightyo
neFourhund redtwoFour hundredsix teenFourhu ndredthirt
yfourFourh undredfift yfiveFourh undredseve ntyfiveFou
rhundredni netysevenF ivehundred nineteenFi vehundredt
hirtyeight Fivehundre dsixtyFive hundredsev entysixFiv
ehundredni netysevenS ixhundredn ineteenSix hundredthi
rtysevenSi xhundredfi ftyeightSi xhundredse ventyeight
Sevenhundr edSevenhun dredtwelve Sevenhundr edthirtySe
venhundred fortyeight Sevenhundr edseventyS evenhundre
deightynin eEighthund redelevenE ighthundre dtwentynin
eEighthund redfiftyon eEighthund redseventy oneEighthu
ndredninet ythreeNine hundredsix teenNinehu ndredthirt

Landon Curt Noll, The English name of a number.

Cf. A005224, A049525 & A055508.

Edited and extended by Robert G. Wilson v, May 14 2004

A261545 a(n) indicates which letter of the English alphabet is the n-th most common.

Original entry on oeis.org

5, 20, 1, 15, 9, 14, 19, 18, 8, 12, 4, 3, 21, 13, 6, 16, 7, 23, 25, 2, 22, 11, 24, 10, 17, 26

Offset: 1

Peter Norvig and Robert G. Wilson v, Aug 24 2015

This is according to Mayzner revisited (Peter Norvig), Cornell, British National Corpus and the Brown corpus.

			The most common letter of the English alphabet is the fifth letter 'E', so a(1)=5; the second most common letter is the twentieth letter 'T', so a(2)=20; the third most common letter is 'A', so a(3)=1; etc.

Martin Gardner, "Codes, Ciphers, and Secret Writing." Dover Publications, Inc. New York 1984.
M. S. Mayzner, Margaret Elizabeth Tresselt, "Tables of Single-letter and Digram Frequency Counts for Various Word-length and Letter-position Combinations," Psychonomic Press, 1965.
A. Young," Mathematical Ciphers: From Caesar to RSA." Mathematical World, Volume 25. American Mathematical Society 2006.

Cornell, Department of Mathematics, Math Explorer's Club, English Letter Frequency.
Gifted Mathematics, Learn How to be Successful in Mathematics Competitions Worldwide, The Undecipherable Cipher: Middle Secondary Mathematics.
Brian Kelk, Some examples of letter frequency rankings in English.
Peter Norvig, English Letter Frequency Counts: Mayzner Revisited or ETAOIN SRHLDCU.
Prooffreader.com, Graphing the distribution of English letters towards the beginning, middle or end of words. [Via Wayback Machine]
Wikipedia, Letter frequency.
Wikipedia, Gadsby (novel).

Cf. A005224, A049525, A055508, A126810.

ToCharacterCode["etaoinsrhldcumfpgwybvkxjqz"] - 96

cp's OEIS Frontend

A005224 T is the first, fourth, eleventh, ... letter in this sentence, not counting spaces or commas (Aronson's sequence).

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A055508 H is the first, fifth, sixteenth, ..., letter in the sequence not counting spaces and commas.

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A097963 ["comma"], is the first, fifteenth, twenty-sixth, fortieth, ... character in this sentence, including spaces and punctuations.

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A094110 Start with the word "One". The next word is the number of letters written previously. Convert this infinite sequence of words into an infinite sequence of numbers.

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A261545 a(n) indicates which letter of the English alphabet is the n-th most common.

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