A016631 -id:A016631 - cp's OEIS frontend

A171647 a(1) = 1; for n > 1, a(n) = 2a(n-1) if n is even, a(n) = ((n+1)/(n-1))a(n-1) if n is odd.

1, 2, 4, 8, 12, 24, 32, 64, 80, 160, 192, 384, 448, 896, 1024, 2048, 2304, 4608, 5120, 10240, 11264, 22528, 24576, 49152, 53248, 106496, 114688, 229376, 245760, 491520, 524288, 1048576, 1114112, 2228224, 2359296, 4718592, 4980736, 9961472

Offset: 1

Gary W. Adamson, Dec 13 2009

a(n) is the number of subsets of {1,2,...,n} that contain exactly one odd number. For example, for n=5, a(5)=12 and the 12 subsets are {1}, {3}, {5}, {1,2}, {1,4}, {2,3}, {2,5}, {3,4}, {4,5}, {1,2,4}, {2,3,4}, {2,4,5}. - Enrique Navarrete, Dec 15 2019

2*a(n-1) is the number of subsets of {1,2,...,n} that contain exactly one even number. For example, for n=5, 2*a(4)=16 and the 16 subsets are {2}, {4}, {1,2}, {1,4}, {2,3}, {2,5}, {3,4}, {4,5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}, {1,2,3,5}, {1,3,4,5}. - Enrique Navarrete, Dec 16 2019

			a(6) = 2*a(5) = 2*12 = 24;
a(7) = (8/6)*a(6) = (4/3)*24 = 32.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

Cf. A001787, A036289 (bisections).

Cf. A016631.

[ n eq 1 select 1 else IsEven(n) select 2*Self(n-1) else ((n+1)/(n-1))*Self(n-1): n in [1..40] ];

a[n_] := If[ OddQ@ n, (n + 1)/(n - 1) a[n - 1] , 2 a[n - 1]]; a[1] = 1; Array[a, 38]
LinearRecurrence[{0,4,0,-4},{1,2,4,8},40] (* Harvey P. Dale, Jan 14 2015 *)

From R. J. Mathar, Dec 06 2010: (Start)
a(n) = 4*a(n-2) - 4*a(n-4).
G.f.: x*(1+2*x)/(-1+2*x^2)^2. (End)
a(n) = (2*n - (-1)^n+1)*2^((2*n + (-1)^n - 9)/4). - Bruno Berselli, Dec 07 2010
G.f.: G(0), where G(k) = 1 + 2*x*(k+1)/(k + 1 - x*(k+1)*(k+2)/(x*(k+2) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 27 2013
Sum_{n>=1} 1/a(n) = 3*log(2) (A016631). - Amiram Eldar, Aug 27 2022

A229857 Round(2^(m-n-2)/(m*log(8))), where m = 2^n - n - 2.

Original entry on oeis.org

5043, 2417158053779, 5245728941618725066052704993134, 215872416866954281715178071724040762825421437510476267629647193878371

Offset: 5

Arkadiusz Wesolowski, Oct 01 2013

a(9) has 145 digits and is too large to include.
Conjecture: a(n) < f(n) = number of primes of the form k*2^(n+2) + 1 with k odd that exist between a = 2^(n+2) + 1 and b = floor((2^(2^n) + 1)/(3*2^(n+2) + 1)).
For comparison, f(5) = 5746.
If the extended Riemann hypothesis is true, then for every fixed epsilon > 0, f(n) = Li(b)/(a - 1) + O(b^(1/2 + epsilon)), where Li(b) = integral(2..b, dt/log(t)).

P. Borwein, S. Choi, B. Rooney and A. Weirathmueller, The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike, Springer, Berlin, 2008, pp. 57-58.

Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A016631, A023394, A046052.

A069975 a(n) = n(16n^2 - 1).

Original entry on oeis.org

15, 126, 429, 1020, 1995, 3450, 5481, 8184, 11655, 15990, 21285, 27636, 35139, 43890, 53985, 65520, 78591, 93294, 109725, 127980, 148155, 170346, 194649, 221160, 249975, 281190, 314901, 351204, 390195, 431970, 476625, 524256, 574959, 628830, 685965, 746460

Offset: 1

Benoit Cloitre, Apr 30 2002

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A016631, A069140.

Table[n(16n^2-1),{n,40}] (* Harvey P. Dale, Dec 17 2018 *)

a(n) = n*(16*n^2-1); \\ Michel Marcus, Nov 25 2013

my(x='x+O('x^37)); Vec(3*x*(5+22*x+5*x^2)/(1-x)^4) \\ Elmo R. Oliveira, Sep 05 2025

Sum_{n>=1} 1/a(n) = 3*log(2) - 2 = A016631 - 2. (Ramanujan)
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - log(2) + sqrt(2)*log(sqrt(2)-1). - Amiram Eldar, Jun 24 2022
From Elmo R. Oliveira, Sep 05 2025: (Start)
G.f.: 3*x*(5 + 22*x + 5*x^2)/(x-1)^4.
E.g.f.: x*(15 + 48*x + 16*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
a(n) = A069140(n)/4. (End)

More terms from Elmo R. Oliveira, Sep 05 2025

A161174 Decimal expansion of the natural logarithm of 63/8.

Original entry on oeis.org

2, 0, 6, 3, 6, 9, 3, 1, 8, 4, 7, 1, 1, 6, 9, 6, 7, 5, 9, 6, 4, 4, 1, 4, 6, 8, 5, 2, 9, 1, 3, 7, 0, 1, 4, 3, 4, 7, 0, 5, 5, 6, 5, 4, 4, 2, 1, 4, 6, 5, 9, 4, 3, 2, 9, 5, 6, 6, 7, 3, 8, 7, 8, 8, 7, 3, 2, 3, 8, 7, 5, 8, 3, 2, 8, 0, 2, 0, 3, 0, 5, 4, 7, 1, 7, 3, 0, 0, 0, 2, 7, 2, 2, 6, 0, 7, 9, 6, 4, 1, 9, 4, 2, 9, 9

Offset: 1

Brock Ericson (narutofan909(AT)cox.net), Jun 04 2009

			2.0636931...

RealDigits[Log[63/8],10,120][[1]] (* Harvey P. Dale, Jan 16 2012 *)

PARI
```
log(63/8) \\ Michel Marcus, Jul 06 2015
```

Log(63/8).

Equals A016686 minus A016631. - R. J. Mathar, Jun 16 2009

Leading 2 inserted, and more digits and keyword:cons added by R. J. Mathar, Jun 26 2009

A382821 Decimal expansion of (3/2) * (log(3) - 1).

Original entry on oeis.org

1, 4, 7, 9, 1, 8, 4, 3, 3, 0, 0, 2, 1, 6, 4, 5, 3, 7, 0, 9, 2, 8, 6, 7, 8, 5, 5, 3, 8, 3, 7, 8, 8, 5, 5, 6, 9, 7, 1, 2, 3, 5, 8, 3, 6, 7, 3, 4, 1, 2, 4, 1, 7, 7, 6, 0, 2, 0, 4, 1, 5, 0, 0, 4, 5, 6, 2, 4, 1, 4, 3, 9, 8, 2, 7, 9, 1, 3, 4, 5, 0, 3, 1, 0, 4, 2, 3

Offset: 0

Sean A. Irvine, Apr 05 2025

			0.147918433002164537092867855...

I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.2).

Cf. A016627, A016631, A145425.

Essentially the same as A093064.

RealDigits[(3/2) * (Log[3] - 1), 10, 120][[1]] (* Amiram Eldar, Aug 08 2025 *)

Equals Sum_{k>=1} 1/(k*(9*k^2-1)).

Equals Sum_{k>=1} zeta(2*k+1)/9^k. - Amiram Eldar, Aug 08 2025

cp's OEIS Frontend

A171647 a(1) = 1; for n > 1, a(n) = 2*a(n-1) if n is even, a(n) = ((n+1)/(n-1))*a(n-1) if n is odd.

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A229857 Round(2^(m-n-2)/(m*log(8))), where m = 2^n - n - 2.

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A069975 a(n) = n*(16*n^2 - 1).

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A161174 Decimal expansion of the natural logarithm of 63/8.

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Mathematica

PARI

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A382821 Decimal expansion of (3/2) * (log(3) - 1).

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Programs

Mathematica

Formula

A171647 a(1) = 1; for n > 1, a(n) = 2a(n-1) if n is even, a(n) = ((n+1)/(n-1))a(n-1) if n is odd.

A069975 a(n) = n(16n^2 - 1).