cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A098290 Recurrence sequence based on positions of digits in decimal places of Zeta(3) (Apery's constant).

Original entry on oeis.org

0, 2, 1, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10, 208, 380, 394, 159, 10
Offset: 0

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 02 2004

Keywords

Comments

This recurrence sequence starts to repeat quite quickly because 1 appears at the 10th digit of Zeta(3), which is also where 159 starts.
Can the transcendental numbers such that recurrence relations of this kind eventually repeat be characterized? - Nathaniel Johnston, Apr 30 2011

Examples

			Zeta(3) = 1.2020569031595942853997...
a(0)=0, a(1)=2 because 2nd decimal = 0, a(2)=1 because first digit = 2, etc.

Crossrefs

Cf. A002117 for digits of Zeta(3). 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), A189893 for sqrt(5), A098327 for sqrt(e), A098328 for 2^(1/3).

Programs

  • Maple
    with(StringTools): Digits:=400: G:=convert(evalf(Zeta(3)-1), string): a[0]:=0: for n from 1 to 50 do a[n]:=Search(convert(a[n-1], string), G)-1:printf("%d, ", a[n-1]):od: # Nathaniel Johnston, Apr 30 2011

Formula

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

A098322 Recurrence sequence based on positions of digits in decimal places of Catalan's constant, G (often also called K).

Original entry on oeis.org

0, 16, 48, 101, 421, 2374, 7728, 9449, 17685, 83666, 71168, 128130, 555251, 412816, 271385, 1111695, 1910101, 11633401, 14851698, 9668058, 43227391, 159078942
Offset: 0

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 03 2004

Keywords

Examples

			So for example, a(2)=16 because 16th digit of G is 0.
a(3)=48 because 16 appears at the 48th-49th digits of G, a(4)=101 because the 101st to 102nd digits of G form "48" and so on.

Crossrefs

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. A006752 for digits of Catalan's constant.

Formula

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

Extensions

14 more terms. 159078942 does not occur within first 2 billion digits of Catalan's constant. Sean A. Irvine, Sep 02 2009

A098323 Recurrence sequence based on positions of digits in decimal places of 1/G, where G is Catalan's constant (also often called K).

Original entry on oeis.org

0, 1, 3, 9, 2, 33, 27, 82, 48, 162, 279, 1140, 5727, 20729, 717726, 430977, 1112328
Offset: 0

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 03 2004

Keywords

Examples

			1/G=1.091744063703906101454159473...
So for example, a(2)=1 because first decimal place of 1/G is 0.
a(3)=3 because 3rd decimal place of 1/G is 1, a(4)=9 because the 9th decimal place of 1/G is 3 and so on.

Crossrefs

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.

Formula

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

Extensions

a(13) from Nathaniel Johnston, Apr 30 2011
a(14)-a(16) from D. S. McNeil, Oct 01 2011

A098324 Recurrence sequence based on positions of digits in decimal places of phi, the Golden Ratio = (1+sqrt(5))/2.

Original entry on oeis.org

0, 4, 11, 34, 26, 67, 150, 1485, 2497, 8001, 2773, 16668, 39567, 80705, 15643, 19267, 29310, 223602, 2318795, 9376463, 7972671, 2412975, 3754694, 9560425, 1910435
Offset: 0

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 03 2004

Keywords

Examples

			phi=1.61803398874989484820...
So for example, a(2)=4 because 4th decimal place of phi is 0.
a(3)=11 because 11th decimal place of phi is 4, a(4)=34 because 11 appears at the 34th to 35th decimal places and so on.

Crossrefs

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.

Programs

  • Maple
    with(StringTools): Digits:=100000: G:=convert(evalf((1+sqrt(5))/2),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: # Nathaniel Johnston, Apr 30 2011

Formula

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

Extensions

a(17)-a(24) from Nathaniel Johnston, Apr 30 2011

A098325 Recurrence sequence based on positions of digits in decimal places of sqrt(Pi).

Original entry on oeis.org

0, 9, 10, 75, 39, 218, 78, 61, 45, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
Offset: 0

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 03 2004

Keywords

Examples

			sqrt(Pi)=1.7724538509055...
So for example, a(2)=9 because 9th decimal place of sqrt(Pi) is 0.
a(3)=10 because 10th decimal place of sqrt(Pi) is 9, a(4)=75 because 10 appears at the 75th to 76th decimal places and so on.
This sequence, like the one for Zeta(3) (A098290), repeats after just a few terms once the sequence hits 4 at position 4.

Crossrefs

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. A002161 for digits of sqrt(Pi).

Programs

  • Maple
    with(StringTools): Digits:=1000: G:=convert(evalf(sqrt(Pi)),string): a[0]:=0: for n from 1 to 15 do a[n]:=Search(convert(a[n-1],string), G)-2:printf("%d, ",a[n-1]):od: # Nathaniel Johnston, Apr 30 2011

Formula

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

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).

Original entry on oeis.org

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

Views

Author

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

Keywords

Examples

			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.

Crossrefs

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).

Programs

  • Maple
    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

Extensions

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

Views

Author

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

Keywords

Comments

a(20) > 5*10^7.

Examples

			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.

Crossrefs

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).

Formula

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

Views

Author

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

Keywords

Examples

			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.

Crossrefs

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).

Programs

  • Maple
    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

Formula

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).

Extensions

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

Views

Author

Ryan Propper, Jul 21 2006

Keywords

Examples

			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.

Crossrefs

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).

Programs

  • Maple
    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

Formula

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

Views

Author

Nathaniel Johnston, Apr 30 2011

Keywords

Examples

			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.

Crossrefs

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).

Programs

  • Maple
    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:

Formula

a(0) = 0; for i >= 0, a(i+1) = position of first occurrence of a(i) in decimal places of sqrt(5).
Showing 1-10 of 10 results.

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