A101272 -id:A101272 - cp's OEIS frontend

A020793 Decimal expansion of 1/6.

1, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

N. J. A. Sloane

Except for the first term identical to A010722, A040006 and A021019. Except for the first terms the same as A021028, A021100, A021388, A071279, A101272, A168608, A177057,... - M. F. Hasler, Oct 24 2011

Decimal expansion of gamma(1) = 5/3 (with offset 1), where gamma(n) = Cp(n)/Cv(n) = is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Jul 10 2016

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Springer, 2013, see p. 224.

Wikipedia, Poisson's constant.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010722, A021019, A021028, A021100, A021388, A040006, A070064, A071279, A081822, A101272, A168608, A177057, A272001, A272002.

RealDigits[1/6,10,120][[1]] (* or *) PadRight[{1},120,{6}] (* Harvey P. Dale, Dec 30 2018 *)

a(n)=6-5*!n  \\ M. F. Hasler, Oct 24 2011

a(n) = 6^n mod 10. - Zerinvary Lajos, Nov 26 2009
Equals Sum_{k>=1} 1/7^k. - Bruno Berselli, Jan 03 2014
10 * 1/6 = 5/3 = (5/2 R)/(3/2 R) = Cp(1)/Cv(1) = A272002/A272001, with R = A081822 (or A070064). - Natan Arie Consigli, Jul 10 2016
G.f.: (1 + 5*x)/(1 - x). - Ilya Gutkovskiy, Jul 10 2016
Equals Sum_{k>=1} 1/(k*Pi)^2. - Maciej Kaniewski, Sep 14 2017
Equals Sum_{k>=1} (zeta(2*k)-1)/4^k. - Amiram Eldar, Jun 08 2021
K_{n>=2} 2*n/(2*n - 3) = 5/3. (see Clawson at p. 224). - Stefano Spezia, Jul 01 2024
E.g.f.: 6*exp(x) - 5. - Elmo R. Oliveira, Aug 05 2024

A234305 Irregular triangle read by rows. Theoretical distribution of electrons based on the Janet's sequence A167268.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 5, 2, 2, 6, 2, 2, 6, 1, 2, 2, 6, 2, 2, 2, 6, 2, 1, 2, 2, 6, 2, 2, 2, 2, 6, 2, 3, 2, 2, 6, 2, 4, 2, 2, 6, 2, 5, 2, 2, 6, 2, 6, 2, 2, 6, 2, 6, 1, 2, 2, 6, 2, 6, 2, 2, 2, 6, 2, 6, 2, 1, 2, 2, 6, 2, 6, 2, 2, 2, 2, 6, 2, 6, 2, 3, 2, 2, 6, 2, 6, 2, 4

Offset: 1

Paul Curtz, Jan 02 2014

a(n) is not A173642, a compact Bohr-Stoner model (1924), modified by Charles Janet in 1930. The good distribution is A168208.
Only sequences N16(n) in A234398 are used:
N16(1)= 1 followed by 2's              = A040000,
N16(2)= 1, 2, 3, 4, 5, followed by 6's = A101272,
N16(3)= 1 to  9, followed by 10's,
N16(4)= 1 to 13, followed by 14's, etc.
The distribution by rows are in the example.
The N16(n)'s are respectively on columns (hence triangle T)
1, 2, 4,  6,  9, 12, 16, 20, 25, 30, 36,   A002620(n+2)
   3, 5,  8, 11, 15, 19, 24, 29, 35,       A024206(n+2)
      7, 10, 14, 18, 23, 28, 34,           A014616(n+3)
         13, 17, 22, 27, 33,               A004116(n+4)
             21, 26, 32,
                 31, etc.
See A163255.
Antidiagonals give the natural numbers A000027, like rows sums in the example.
A033638=1, 1, 2, 3, 5, 7,... is upon the triangle T.

			1,      H
2,       He
2, 1,    Li
2, 2,    Be
2, 2, 1,
2, 2, 2,
2, 2, 3,
2, 2, 4,
2, 2, 5,
2, 2, 6,
2, 2, 6, 1,
2, 2, 6, 2,
2, 2, 6, 2, 1,
2, 2, 6, 2, 2,
2, 2, 6, 2, 3,
2, 2, 6, 2, 4,
2, 2, 6, 2, 5,
2, 2, 6, 2, 6,
2, 2, 6, 2, 6, 1,
2, 2, 6, 2, 6, 2,
2, 2, 6, 2, 6, 2, 1,
2, 2, 6, 2, 6, 2, 2,
2, 2, 6, 2, 6, 2, 3, etc.

Cf. A002061, A002522 (or A160457), A014206, A059100, diagonals of the triangle T. A004526.

A234398 Distribution of the natural numbers using the sequences family mentioned in the comments.

Original entry on oeis.org

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

Offset: 1

Paul Curtz, Dec 25 2013

Based on A016825=2,6,10,..., the family is
N16(1)=1, followed by 2's                  =A040000,
N16(2)=1,2,3,4,5, followed by 6's          =A101272,
N16(3)=1,2,3,4,5,6,7,8,9, followed by 10's, not in the OEIS,
N16(4)=1,2,3,4,5,6,7,8,9,10,11,12,13, followed by 14's, idem.
The N16(n) gives the successive columns beginning at row 1, 3, 9, 19, ... =A058331.
Sum of every row: n =A000027.
Note that with only N16(1),
1,
2,
2, 1,
2, 2,
2, 2, 1,
2, 2, 2,
2, 2, 2, 1,
2, 2, 2, 2,
2, 2, 2, 2, 1, etc
is A169695(n+1).
A169695(n) corresponds to A028310.

			1,
2,
2, 1,
2, 2,
2, 3,
2, 4,
2, 5,
2, 6,
2, 6, 1,
2, 6, 2,
2, 6, 3,
2, 6, 4, etc.

Cf. A173642.

A210032 a(n)=n for n=1,2,3 and 4; a(n)=5 for n >= 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

A. Timothy Royappa, Mar 16 2012

In atomic spectroscopy, a(n) is the number of D term symbols with spin multiplicity equal to n, i.e., there is one singlet-D term (n=1), and there are two doublet-D terms (n=2), three triple-D terms (n=3), four quartet-D terms (n=4) and five terms for every other D term of multiplicity 5 or higher (n >= 5).

Decimal expansion of 11111/9000. - Arkadiusz Wesolowski, Mar 29 2012

Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010716, A101272, A158411, A168092, A128999.

Join[Range@4, Table[5, {83}]] (* Arkadiusz Wesolowski, Mar 29 2012 *)
PadRight[{1,2,3,4},120,{5}] (* Harvey P. Dale, Sep 23 2017 *)

a(n) = min(n,5). - Wesley Ivan Hurt, Apr 16 2014
From Elmo R. Oliveira, Jun 26 2024: (Start)
G.f.: x*(1+x+x^2+x^3+x^4)/(1-x) = x*(1-x^5)/(1-x)^2.
a(n) = 1 + A158411(n-1) = A101272(n+1) - 1 = A168093(n-1) - 2. (End)

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A020793 Decimal expansion of 1/6.

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A234305 Irregular triangle read by rows. Theoretical distribution of electrons based on the Janet's sequence A167268.

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A234398 Distribution of the natural numbers using the sequences family mentioned in the comments.

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A210032 a(n)=n for n=1,2,3 and 4; a(n)=5 for n >= 5.

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