A097614 -id:A097614 - cp's OEIS frontend

A098326 Recurrence derived from the decimal places of sqrt(2). a(0)=0, a(i+1)=position of first occurrence of a(i) in decimal places of sqrt(2).

0, 13, 5, 7, 11, 186, 239, 336, 1284, 5889, 11708, 70286, 19276, 35435, 22479, 42202, 28785, 107081, 973876, 1187108

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 13 2004

			sqrt(2)=1.4142135623730950488...
So for example a(2)=13 because 13th decimal place of sqrt(2) is 0; then a(3)=5 because 13 is found starting at the 5th decimal place; a(4)=7 because 5 is at the 7th decimal place and so on.

Other recurrence sequences: A097614 for Pi, A098266 for e, A098289 for log(2), A098290 for Zeta(3), A098319 for 1/Pi, A098320 for 1/e, A098321 for gamma, A098322 for G, A098323 for 1/G, A098324 for Golden Ratio (phi), A098325 for sqrt(Pi), A120482 for sqrt(3), A189893 for sqrt(5). A002193 for digits of sqrt(2).

with(StringTools): Digits:=10000: G:=convert(evalf(sqrt(2)),string): a[0]:=0: for n from 1 to 10 do a[n]:=Search(convert(a[n-1],string), G)-2:printf("%d, ",a[n-1]):od: # Nathaniel Johnston, Apr 30 2011

a(18)-a(19) from Nathaniel Johnston, Apr 30 2011

A098327 Recurrence sequence derived from the decimal places of sqrt(e).

Original entry on oeis.org

0, 9, 60, 79, 59, 137, 479, 2897, 1397, 24474, 63515, 71287, 191542, 1432289, 1766633, 1380465, 2894629, 1464385, 10676561

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 13 2004

a(20) > 5*10^7.

			sqrt(e)=1.6487212707001281468...
So for example, with a(1)=0, a(2)=9 because 9th decimal place is 0; a(3)=60 because 9 appears at decimal place number 60 and so on.

Other recurrence sequences: A097614 for Pi, A098266 for e, A098289 for log(2), A098290 for Zeta(3), A098319 for 1/Pi, A098320 for 1/e, A098321 for gamma, A098322 for G, A098323 for 1/G, A098324 for Golden Ratio (phi), A098325 for sqrt(Pi), A098326 for sqrt(2).

a(1)=0, p(i)=position of first occurrence of a(i) in decimal places of sqrt(e), a(i+1)=p(i).

A098328 Recurrence sequence derived from the digits of the cube root of 2 after its decimal point.

Original entry on oeis.org

0, 7, 14, 42, 147, 321, 473, 322, 785, 1779, 3039, 1957, 16446, 274134, 374781, 110639, 248175, 385504, 2359264, 5108010, 3822244, 3812946, 9896631

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 14 2004

			2^(1/3)=1.259921049894873164767210607...
So for example, with a(1)=0, a(2)=7 because the 7th digit after the decimal point is 0; a(3)=14 because the 14th digit after the decimal point is 7 and so on.

Other recurrence sequences: A097614 for Pi, A098266 for e, A098289 for log(2), A098290 for Zeta(3), A098319 for 1/Pi, A098320 for 1/e, A098321 for gamma, A098322 for G, A098323 for 1/G, A098324 for Golden Ratio (phi), A098325 for sqrt(Pi), A098326 for sqrt(2), A098327 for sqrt(e). A002580 for digits of 2^(1/3).

with(StringTools): Digits:=10000: G:=convert(evalf(root(2,3)),string): a[0]:=0: for n from 1 to 12 do a[n]:=Search(convert(a[n-1],string), G)-2:printf("%d, ",a[n-1]):od: # Nathaniel Johnston, Apr 30 2011

a(1)=0. a(1)=0, p(i)=position of first occurrence of a(i) in decimal places of 2^(1/3), a(i+1)=p(i).

More terms from Ryan Propper, Jul 21 2006

A120482 Recurrence sequence derived from the digits of the square root of 3 after its decimal point.

Original entry on oeis.org

0, 4, 22, 215, 2737, 8636, 20805, 38867, 1868, 6505, 5767, 1004, 1216, 11702, 55995, 43202, 314308, 2100749, 2420235, 7750204, 5141127, 2950527, 3113789, 42198, 119161, 96031, 77643, 10695, 105061, 37099, 176209, 3390478, 4549989, 9038843

Offset: 0

Ryan Propper, Jul 21 2006

			sqrt(3) = 1.73205080756887729352744634151...
So for example, with a(0) = 0, a(1) = 4 because the 4th digit after the decimal point is 0; a(2) = 22 because the 22nd digit after the decimal point is 4 and so on.

Other recurrence sequences: A097614 for Pi, A098266 for e, A098289 for log(2), A098290 for Zeta(3), A098319 for 1/Pi, A098320 for 1/e, A098321 for gamma, A098322 for G, A098323 for 1/G, A098324 for Golden Ratio (phi), A098325 for sqrt(Pi), A098326 for sqrt(2), A189893 for sqrt(5), A098327 for sqrt(e), A098328 for 2^(1/3).

with(StringTools): Digits:=10000: G:=convert(evalf(sqrt(3)),string): a[0]:=0: for n from 1 to 6 do a[n]:=Search(convert(a[n-1],string), G)-2:printf("%d, ",a[n-1]):od: # Nathaniel Johnston, Apr 30 2011

a(0) = 0; for i >= 0, a(i+1) = position of first occurrence of a(i) in decimal places of sqrt(3).

A189893 Recurrence sequence derived from the digits of the square root of 5 after its decimal point.

Original entry on oeis.org

0, 4, 10, 65, 173, 22, 96, 15, 48, 78, 13, 201, 487, 594, 2719, 5146, 8719, 11530, 15308, 76411, 76016, 42220, 67129, 45349, 170266, 255576, 457846, 865810, 1131083, 8045547, 7669757

Offset: 0

Nathaniel Johnston, Apr 30 2011

			sqrt(5) = 2.2360679774997896964091736687...
So for example, with a(0) = 0, a(1) = 4 because the 4th digit after the decimal point is 0; a(2) = 10 because the 10th digit after the decimal point is 4 and so on.

Other recurrence sequences: A097614 for Pi, A098266 for e, A098289 for log(2), A098290 for Zeta(3), A098319 for 1/Pi, A098320 for 1/e, A098321 for gamma, A098322 for G, A098323 for 1/G, A098324 for Golden Ratio (phi), A098325 for sqrt(Pi), A098326 for sqrt(2), A120482 for sqrt(3), A098327 for sqrt(e), A098328 for 2^(1/3).

with(StringTools): Digits:=10000: G:=convert(evalf(sqrt(5)),string): a[0]:=0: for n from 1 to 17 do a[n]:=Search(convert(a[n-1],string), G)-2:printf("%d, ",a[n-1]):od:

a(0) = 0; for i >= 0, a(i+1) = position of first occurrence of a(i) in decimal places of sqrt(5).

A247345 a(1) = 16. For n > 1, a(n) is the position of the first occurrence of a(n-1) after the decimal point in the decimal expansion of Pi.

Original entry on oeis.org

16, 40, 70, 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, 40, 70, 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169, 40, 70, 96

Offset: 1

Felix Fröhlich, Jul 26 2015

After the first term, the sequence enters a repeating cycle of length 20.
According to the information on the website of Anderson, this sequence was discovered earlier by Dan Sikorsky.
First element a(1) = 61 leads after seven steps 61, 219, 716, 39, 43, 23, 16, ... into the same repeating cycle, same with a(1) = 71 after five steps 71, 39, 43, 23, 16, ... - Martin Renner, Feb 22 2020

D. Andersen, Loop Sequences within Pi, The Pi-Search Page.

Cf. A000796, A097614.

L={16}; pi=ToString[ FromDigits@ RealDigits[Pi-3, 10, 10^5 ][[1]]]; Do[
AppendTo[L, StringPosition[ pi, ToString@ Last@ L, 1][[1, 1]]], {23}]; L (* Giovanni Resta, Jul 26 2015 *)

A374405 The nonnegative terms followed by their first position in the concatenation of all terms of the sequence.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 3, 7, 4, 6, 5, 11, 6, 10, 7, 8, 8, 18, 9, 22, 10, 15, 11, 2, 12, 3, 13, 35, 14, 39, 15, 27, 16, 13, 17, 51, 18, 20, 19, 59, 20, 57, 21, 24, 22, 4, 23, 33, 24, 5, 25, 81, 26, 85, 27, 45, 28, 93, 29, 97, 30, 101, 31, 34, 32, 77, 33, 36, 34, 108

Offset: 1

Eric Angelini, Jul 07 2024

The nonnegative terms are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ... They are odd-indexed in the above Data section; the even-indexed terms of the Data section form the last column of the Example section hereunder.

			The 1st nonnegative term is 0 and 0 is followed by 1 in the sequence as the earliest 0 of the concatenation of all terms of the sequence is at position 1;
the 2nd nonnegative term is 1 and 1 is followed by 2 in the sequence as the earliest 1 of the concatenation of all terms of the sequence is at position 2;
the 3rd nonnegative term is 2 and 2 is followed by 4 in the sequence as the earliest 2 of the concatenation of all terms of the sequence is at position 4;
the 4th nonnegative term is 3 and 3 is followed by 7 in the sequence as the earliest 3 of the concatenation of all terms of the sequence is at position 7;
the 5th nonnegative term is 4 and 4 is followed by 6 in the sequence as the earliest 4 of the concatenation of all terms of the sequence is at position 6;
the 6th nonnegative term is 5 and 5 is followed by 11 in the sequence as the earliest 5 of the concatenation of all terms of the sequence is at position 11;
the 7th nonnegative term is 6 and 6 is followed by 10 in the sequence as the earliest 6 of the concatenation of all terms of the sequence is at position 10;
the 8th nonnegative term is 7 and 7 is followed by 8 in the sequence as the earliest 7 of the concatenation of all terms of the sequence is at position 8;
the 9th nonnegative term is 8 and 8 is followed by 18 in the sequence as the earliest 8 of the concatenation of all terms of the sequence is at position 18;
the 10th nonnegative term is 9 and 9 is followed by 22 in the sequence as the earliest 9 of the concatenation of all terms of the sequence is at position 22;
the 11th nonnegative term is 10 and 10 is followed by 15 in the sequence as the earliest 10 of the concatenation of all terms of the sequence is at position 15;
the 12th nonnegative term is 11 and 11 is followed by 2 in the sequence as the earliest 11 of the concatenation of all terms of the sequence is at position 2;
the 13th nonnegative term is 12 and 12 is followed by 3 in the sequence as the earliest 12 of the concatenation of all terms of the sequence is at position 3; etc.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A001477, A097614.

from itertools import count, islice
def agen(): # generator of terms
    s = ""
    for n in count(0):
        sn = str(n)
        s += sn
        locn = 1 + s.index(sn)
        s += str(locn)
        yield from [n, locn]
print(list(islice(agen(), 70))) # Michael S. Branicky, Jul 07 2024

a(63) and beyond from Michael S. Branicky, Jul 07 2024

A119744 a(n) is the position of a(n-1) in the decimal expansion of Pi, starting with a(1)=13.

Original entry on oeis.org

13, 111, 154, 3645, 2699, 1394, 526, 613, 971, 39, 44, 60, 128, 149, 2607, 23047, 21876, 361554, 403706, 1674698, 19210577

Offset: 1

Zak Seidov, Jun 16 2006

			In the decimal expansion of Pi (A000796) written as a string
3141592653589793238462643383279502884197169399375105820974944592307816...,
the string "13" is found at position 111, the string "111" is at position 154, the string "154" at position 3645, etc.,
hence the sequence starting with a(1)=13 is
13,111,154,3645,2699,...
In general, the sequence may end in a cycle, e.g. sequence s1
starting with a(1)=1 is
s1: 1,2,7,14,2,7,14,2,7,14,2 (cycle is 2,7,14),
Also s0, s2, s3, s4, s7, s10, s11, s14, s15, s16, s25 end with the same cycle:
s0: 0,33,25,90,248,480,105,50,32,16,41,3,1,2,7,14,2
s2: 2,7,14,2,7,14,2,
s3: 3,1,2,7,14,2,7,14,2
s4: 4,3,1, 2,7,14,2,7,14,2
s5: 5,5,5,5 (simplest cycle!)
s6: 6, 8, 12,149,2607,23047,21876,361554,403706,1674698,19210577,next term>2*10^8
s7: 7,14,2,7, 14, (see s1)
s8: 8,12,149, (see s6)
s9: 9,6, 8, (see s6)
s10: 10,50,32,16,41,3,1,2, (see s1)
s11: 11,95,31,1, (see s1)
s12: 12,149, (see s6)
s13: this sequence, is there cycle or not? next term>2*10^8
s14: 14,2, (see s1)
s15: 15,4,3,1, (see s1)
s16: 16,41,3,1, (see s1)
s17: 17,96,181,729,771,626,21,94,59,5,5,5,(see s5)
s18: 18,425,822,135,2728,11023,12721,54517,102917,183252,410024,613425,1525497,
3426169,3591590,10748112, is there cycle or not? next term>2*10^8
s19: 19,38,18, (see s18), is there cycle or not? next term>2*10^8
s20: 20,54,192,976,1808,26035,43352,93226,3603,9736,10514,54423,140517,1549413,
20801035, is there cycle or not? next term>2*10^8
s21: 21,94,59,5,5,5,(see s5)
s22: 22,136,735,469,387,864,722,2140,8434,9666,4000,14637,85171,3538,5037,37934,
62186,6529,37803,68887,5871,22098,172393,591481,14933,51852, 5762,7347,11749,
12529,61828,268516,657761,531469,1246616,6755774,22119206,83934772,128149562,
is there cycle or not? next term>2*10^8
s23: 23,17,96,181,729,771,626,21,94,59,5,5,5, (see s17, s5)
s24: 24,293,572,405,596,180,3665,10143,63892,465223,522194,1637321,10980764,
184160876,65620598,35543320,97248583,109914084,40782089,
48875829,77976212,182755461,114041877, is there cycle or not? next term>2*10^8
s25: 25,90,248,480,105,50,32,16,41,3,1,2,7,14,2 (see s0, s1)

Dave Andersen, The Pi-Search Page

Cf. A000796 = Decimal expansion of Pi, A097614 = sequence based on positions of digits in decimal digits of Pi.

Edited by N. J. A. Sloane, Dec 09 2017

A140478 In the decimal expansion of Pi, the string "8" is found at position 11 counting from the first digit after the decimal point. The string "11" is found at position 94, the string "94" at position 58, the string "58" at position 10, etc.

Original entry on oeis.org

8, 11, 94, 58, 10, 49, 57, 404, 1272, 8699, 3292, 3332, 48033, 90311, 112817, 1149731, 24909936, 1838500, 5264650, 29232231, 76236585, 64535680, 49758988, 191873638, 1718045848, 4235464084, 955635320, 2807341746

Offset: 1

Leonid Ianoushevitch (leonid163(AT)mail.ru), Jun 29 2008

Here the number 8 was chosen because it is the smallest number (except 1) that does not appear in known part of A097614. 1 is uninteresting because it produces a sequence of 1's. The smallest number (except 1) that does not appear neither in known part of this sequence nor of A097614 is 12.

Dave Andersen, The Pi-Search Page.

Cf. A097614.

Corrected and extended by Hans Havermann, Feb 21 2020

A332756 A loop sequence within Pi. Let a(1) = 19. For n > 1, a(n+1) is the position of the first occurrence of a(n) after the decimal point in the decimal expansion of Pi.

Original entry on oeis.org

19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37, 46, 19, 37

Offset: 1

Martin Renner, Feb 22 2020

The sequence has a repeating cycle of length 3.

David G. Andersen, The Pi-Search Page.
Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000796, A057680, A097614, A247345.

PadRight[{}, 100, {19, 37, 46}] (* Paolo Xausa, Apr 27 2024 *)

From Stefano Spezia, Feb 22 2020: (Start)
O.g.f.: x*(19 + 37*x + 46*x^2)/(1 - x^3).
E.g.f.: 34*(cosh(x) + sinh(x)) - 46 - 6*exp(-x/2)*(sqrt(3)*sin(sqrt(3)*x/2) - 2*cos(sqrt(3)*x/2)).
a(n) = a(n-3) for n > 3. (End)

cp's OEIS Frontend

A098326 Recurrence derived from the decimal places of sqrt(2). a(0)=0, a(i+1)=position of first occurrence of a(i) in decimal places of sqrt(2).

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Maple

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A098327 Recurrence sequence derived from the decimal places of sqrt(e).

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A098328 Recurrence sequence derived from the digits of the cube root of 2 after its decimal point.

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Maple

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A120482 Recurrence sequence derived from the digits of the square root of 3 after its decimal point.

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Maple

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A189893 Recurrence sequence derived from the digits of the square root of 5 after its decimal point.

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Maple

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A247345 a(1) = 16. For n > 1, a(n) is the position of the first occurrence of a(n-1) after the decimal point in the decimal expansion of Pi.

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Mathematica

A374405 The nonnegative terms followed by their first position in the concatenation of all terms of the sequence.

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Python

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A119744 a(n) is the position of a(n-1) in the decimal expansion of Pi, starting with a(1)=13.

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A140478 In the decimal expansion of Pi, the string "8" is found at position 11 counting from the first digit after the decimal point. The string "11" is found at position 94, the string "94" at position 58, the string "58" at position 10, etc.

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