A098289 -id:A098289 - cp's OEIS frontend

A195913 The denominator in a fraction expansion of log(2)-Pi/8.

2, 3, 12, 30, 35, 56, 90, 99, 132, 182, 195, 240, 306, 323, 380, 462, 483, 552, 650, 675, 756, 870, 899, 992, 1122, 1155, 1260, 1406, 1443, 1560, 1722, 1763, 1892, 2070, 2115, 2256, 2450, 2499, 2652, 2862, 2915

Offset: 1

Mohammad K. Azarian, Sep 25 2011

The minus sign in front of a fraction is considered the sign of the numerator and hence the sign of the fraction does not appear in this sequence.

			1/2 - 1/3 + 1/12 + 1/30 - 1/35 + 1/56 + 1/90 - 1/99 + 1/132 + 1/182 - 1/195 + 1/240 + ... = [(1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + (1/9 - 1/10) + (1/11 - 1/12) + ...] - (1/2)*[(1 - 1/3) + (1/5 - 1/7) + (1/9 - 1/11) + (1/13 - 1/15) + ... ] = log(2) - Pi/8.

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

Cf. A195909, A195697, A195947, A164833, A118324, A098289, A075549, A016655, A019675, A161685, A144981, A168056, A004772.

log(2) - Pi/8 = Sum_{n>=1} (-1)^(n+1)*(1/n) + (-1/2)*Sum_{n>=0} (-1)^n*(1/(2*n+1)).
Empirical g.f.: x*(2+x+9*x^2+14*x^3+3*x^4+3*x^5) / ((1-x)^3*(1+x+x^2)^2). - Colin Barker, Dec 17 2015
From Bernard Schott, Aug 11 2019: (Start)
k >= 1, a(3*k) = (4*k-1) * 4*k,
k >= 0, a(3*k+1) = (4*k+1) * (4*k+2),
k >= 0, a(3*k+2) = (4*k+1) * (4*k+3).
The even terms a(3*k) and a(3*k+1) come from log(2) and the odd terms a(3*k+2) come from - Pi/8. (End)

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

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

A097291 Contains exactly once every pair (i,j) of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 2, 3, 1, 4, 4, 2, 4, 3, 4, 1, 5, 5, 2, 5, 3, 5, 4, 5, 1, 6, 6, 2, 6, 3, 6, 4, 6, 5, 6, 1, 7, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 1, 8, 8, 2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 1, 9, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 1, 10, 10, 2, 10, 3, 10, 4, 10, 5, 10, 6, 10, 7, 10, 8, 10, 9, 10

Offset: 1

Clark Kimberling, Aug 05 2004

nonn

The pairs (i,j) are ordered in segments thus: segment 1: 1 segment 2: 1 2 2 segment 3: 1 3 3 2 3 segment 4: 1 4 4 2 4 3 4 segment 5: 1 5 5 2 5 3 5 4 5 and so on.

Cf. A098289.

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).

A097290 Rectangular array T, by antidiagonals: T(n,k) = rank of k-th n in A097289.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 11, 7, 9, 13, 16, 12, 10, 14, 18, 21, 23, 17, 15, 19, 25, 28, 30, 24, 22, 20, 26, 32, 35, 37, 39, 31, 29, 27, 33, 41, 44, 46, 48, 40, 38, 36, 34, 42, 50, 53, 55, 57, 59, 49, 47, 45, 43, 51, 61, 64, 66, 68, 70, 60, 58, 56, 54, 52, 62, 72, 75, 77, 79, 81, 83

Offset: 1

Clark Kimberling, Aug 05 2004

As a sequence this is a permutation of the natural numbers.

			Northwest corner:
1 2 5 8
3 4 11 16
6 7 12 23
9 10 17 24

Cf. A098289.

cp's OEIS Frontend

A195913 The denominator in a fraction expansion of log(2)-Pi/8.

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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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A097291 Contains exactly once every pair (i,j) of positive integers.

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

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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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A097290 Rectangular array T, by antidiagonals: T(n,k) = rank of k-th n in A097289.

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