A010767 -id:A010767 - cp's OEIS frontend

A093406 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3) + a(n-4).

1, 3, 11, 31, 71, 145, 289, 601, 1321, 2979, 6683, 14743, 32111, 69697, 151777, 332113, 728689, 1598883, 3503627, 7668079, 16774775, 36704017, 80343361, 175916521, 385196761, 843365379, 1846290395, 4041672871, 8847607391, 19368919297, 42403014721, 92830645537

Offset: 1

Gary W. Adamson, Mar 28 2004

a(n)/a(n-1) tends to 2.189207115... = 1 + 2^(1/4) = 1 + A010767.

			a(4) = 31, since M^4 * [1,1,1,1] = [3, 11, 31, 71].

E. J. Barbeau, Polynomials, Springer-Verlag NY Inc, 1989, p. 136.

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

Cf. A010767, A052101.

LinearRecurrence[{4,-6,4,1},{1,3,11,31},40] (* Harvey P. Dale, Jul 22 2013 *)

We use a 4 X 4 matrix corresponding to the characteristic polynomial (x - 1)^4 - 2 = 0 = x^4 - 4x^3 + 6x^2 - 4x - 1 = 0, being [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 4 -6 4]. Let the matrix = M. Perform M^n * [1, 1, 1, 1]. a(n) = the third term from the left, (the other 3 terms being offset members of the series).

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)+a(n-4). G.f.: -x*(x^3+5*x^2-x+1)/ (x^4+4*x^3-6*x^2+4*x-1). [Colin Barker, Oct 21 2012]

Corrected by T. D. Noe, Nov 08 2006

New name using recurrence from Colin Barker, Joerg Arndt, Apr 15 2021

A329216 Decimal expansion of 2^(5/12).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(5/12) is the ratio of the frequencies of the pitches in a perfect fourth (e.g., D4-G4) in 12-tone equal temperament.

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (this sequence)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (A329219)
Major seventh:   2^(11/12) = 1.8877486253...  (A329220)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(5/12), 10, 100]] (* Paolo Xausa, Apr 28 2024 *)

PARI
```
default(realprecision, 100); 2^(5/12)
```

Equals 2/A328229.

A329220 Decimal expansion of 2^(11/12).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(11/12) is the ratio of the frequencies of the pitches in a major seventh (e.g., D4-C#5) in 12-tone equal temperament.

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (A329216)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (A329219)
Major seventh:   2^(11/12) = 1.8877486253...  (this sequence)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(11/12), 10, 100]] (* Paolo Xausa, Apr 28 2024 *)

PARI
```
default(realprecision, 100); 2^(11/12)
```

Equals 2/A010774.

Equals Product_{k>=0} (1 + (-1)^k/(12*k + 1)). - Amiram Eldar, Jul 29 2020

A159824 Continued fraction for Pi^Pi (cf. A073233).

Original entry on oeis.org

36, 2, 6, 9, 2, 1, 2, 5, 1, 1, 6, 2, 1, 291, 1, 38, 50, 1, 2, 5, 4, 1, 2, 2, 1, 5, 1, 4, 13, 2, 1, 4, 3, 3, 1, 2, 25, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 1, 43, 1, 2, 7, 3, 1, 1, 1, 2, 4, 2, 1, 1, 3, 1, 3, 3, 2, 2, 16, 3, 5, 2, 1, 5, 2, 1, 10, 1, 1, 3, 1, 13, 1, 1, 3, 1, 10, 4, 1, 1, 1, 38, 1, 2, 2, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 30 2009

nonn

			36.4621596072079117709908260... = 36 + 1/(2 + 1/(6 + 1/(9 + 1/(2 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A001076, A001203, A001204, A002945, A010767, A011002, A011003, A073233, A073238, A179613, A179615, A179616, A179617.

ContinuedFraction[Pi^Pi,200] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2010 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^Pi); for (n=1, 20001, write("b159824.txt", n-1, " ", x[n])); }

Edited by N. J. A. Sloane, Jul 22 2010

A196535 Decimal expansion of Sum_{j=0..oo} exp(-Pi(2j+1)^2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.04321391826429779829201838202725...

Jolley, Summation of Series, Dover (1961) eq (114) on page 22.
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986), p. 729, formula 14.

Michael I. Shamos, A catalog of the real numbers, (2007), p. 78.

Cf. A093580, A068466, A010767.

(root[4](2)-1)*GAMMA(1/4)/2^(11/4)/Pi^(3/4) ; evalf(%) ;

RealDigits[ EllipticTheta[2, 0, Exp[-4*Pi]]/2, 10, 105] // First // Prepend[#, 0]&  (* Jean-François Alcover, Feb 12 2013 *)

Equals (2^(1/4)-1) * Gamma(1/4) / ( 2^(11/4) * Pi^(3/4) ).

Equals theta2(exp(-4*Pi))/2.

12 more digits from Jean-François Alcover, Feb 12 2013

A269430 Decimal expansion of (1 + Pi)/2.

Original entry on oeis.org

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

Offset: 1

Jani Melik, Feb 26 2016

			2.0707963267948966192313216916397514420985846996875529...

Index entries for transcendental numbers

Cf. A000796, A002193, A002194, A002163, A174968, A001622, A002161, A010767, A011002, A011003, A191502, A053004.

PARI
```
(1 + Pi)/2 \\ Altug Alkan, Apr 07 2016
```
Sage
```
N((1+pi)/2, digits=110)
```

Equals A096444 + 1 or A019669 + 1/2.

Equals Sum_{k>=0} 2^k/binomial(2*k+2,k). - Amiram Eldar, Jun 30 2020

A308320 Decimal expansion of 2^(-7/4); exact length of the A4 paper size measured in meters according to the ISO 216 standard.

Original entry on oeis.org

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

Offset: 0

Jianing Song, May 20 2019

Also exact width of the A3 paper size measured in meters.

According to the ISO 216 standard, the A0 paper size is defined to have an area of 1 square meter where the ratio of the length to the width is sqrt(2), so the length is 2^(1/4) m and the width is 2^(-1/4) m. For each n >= 0, the length of the size A(n+1) is equal to the width of the size A(n) and the width of the size A(n+1) is equal to half of the length of the size A(n), so the area of the size A(n+1) is half of that of A(n). Equivalently, the length of the A(n) size is 2^(-n/2 + 1/4) m and the width is 2^(-n/2 - 1/4) m. For the A4 size, the exact length and width are 2^(-7/4) m = 297.301... mm and 2^(-9/4) m = 210.224... mm (A308321), and the actual length and width are 297 mm and 210 mm.

			0.29730177...
The exact lengths and widths (rounded to the nearest 1/10 mm) and areas of the A-series are as follows:
.
  size |       exact length      |       exact width      | exact area (mm^2)
   A0  | 2^(  1/4) m = 1189.2 mm | 2^(- 1/4) m = 840.9 mm |  1000000
   A1  | 2^(- 1/4) m =  840.9 mm | 2^(- 3/4) m = 594.6 mm |   500000
   A2  | 2^(- 3/4) m =  594.6 mm | 2^(- 5/4) m = 420.4 mm |   250000
   A3  | 2^(- 5/4) m =  420.4 mm | 2^(- 7/4) m = 297.3 mm |   125000
   A4  | 2^(- 7/4) m =  297.3 mm | 2^(- 9/4) m = 210.2 mm |    62500
   A5  | 2^(- 9/4) m =  210.2 mm | 2^(-11/4) m = 148.7 mm |    31250
   A6  | 2^(-11/4) m =  148.7 mm | 2^(-13/4) m = 105.1 mm |    15625
   A7  | 2^(-13/4) m =  105.1 mm | 2^(-15/4) m =  74.3 mm |     7812.5
   A8  | 2^(-15/4) m =   74.3 mm | 2^(-17/4) m =  52.6 mm |     3906.25
   A9  | 2^(-17/4) m =   52.6 mm | 2^(-19/4) m =  37.2 mm |     1953.125
   A10 | 2^(-19/4) m =   37.2 mm | 2^(-21/4) m =  26.3 mm |      976.5625
.
And the actual lengths, widths and areas (note that the actual areas are always smaller than the exact areas) are as follows:
.
  size | actual length (mm) | actual width (mm) | actual area (mm^2)
   A0  |        1189        |        841        |  999949 (99.9949%)
   A1  |         841        |        594        |  499554 (99.9108%)
   A2  |         594        |        420        |  249480 (99.7920%)
   A3  |         420        |        297        |  124740 (99.7920%)
   A4  |         297        |        210        |   62370 (99.7920%)
   A5  |         210        |        148        |   31080 (99.4560%)
   A6  |         148        |        105        |   15540 (99.4560%)
   A7  |         105        |         74        |    7770 (99.4560%)
   A8  |          74        |         52        |    3848 (98.5088%)
   A9  |          52        |         37        |    1924 (98.5088%)
   A10 |          37        |         26        |     962 (98.5088%)

Prepressure, A4.
Wikipedia, ISO 216.

Cf. A010767 (2^(1/4)), A228497 (2^(-1/4)), A308321 (2^(-9/4)).

RealDigits[2^(-7/4),10,88][[1]] (* James C. McMahon, Feb 26 2024 *)

PARI
```
default(realprecision, 100); 2^(-7/4)
```

Equals square root of A222066. - R. J. Mathar, Jan 27 2021

Edited by Jon E. Schoenfield, Feb 25 2024

A308321 Decimal expansion of 2^(-9/4); exact width of the A4 paper size measured in meters according to the ISO 216 standard.

Original entry on oeis.org

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

Offset: 0

Jianing Song, May 20 2019

Also exact length of the A5 paper size measured in meters.

According to the ISO 216 standard, the A0 paper size is defined to have an area of 1 square meter where the ratio of the length to the width is sqrt(2), so the length is 2^(1/4) m and the width is 2^(-1/4) m. For each n >= 0, the length of the size A(n+1) is equal to the width of the size A(n) and the width of the size A(n+1) is equal to half of the length of the size A(n), so the area of the size A(n+1) is half of that of A(n). Equivalently, the length of the A(n) size is 2^(-n/2 + 1/4) m and the width is 2^(-n/2 - 1/4) m. For the A4 size, the exact length and width are 2^(-7/4) m = 297.301... mm (A308320) and 2^(-9/4) m = 210.224... mm, and the actual length and width are 297 mm and 210 mm.

			The exact lengths and widths (rounded to the nearest 1/10 mm) and areas of the A-series are as follows:
.
  size |       exact length      |       exact width      | exact area (mm^2)
   A0  | 2^(  1/4) m = 1189.2 mm | 2^(- 1/4) m = 840.9 mm |  1000000
   A1  | 2^(- 1/4) m =  840.9 mm | 2^(- 3/4) m = 594.6 mm |   500000
   A2  | 2^(- 3/4) m =  594.6 mm | 2^(- 5/4) m = 420.4 mm |   250000
   A3  | 2^(- 5/4) m =  420.4 mm | 2^(- 7/4) m = 297.3 mm |   125000
   A4  | 2^(- 7/4) m =  297.3 mm | 2^(- 9/4) m = 210.2 mm |    62500
   A5  | 2^(- 9/4) m =  210.2 mm | 2^(-11/4) m = 148.7 mm |    31250
   A6  | 2^(-11/4) m =  148.7 mm | 2^(-13/4) m = 105.1 mm |    15625
   A7  | 2^(-13/4) m =  105.1 mm | 2^(-15/4) m =  74.3 mm |     7812.5
   A8  | 2^(-15/4) m =   74.3 mm | 2^(-17/4) m =  52.6 mm |     3906.25
   A9  | 2^(-17/4) m =   52.6 mm | 2^(-19/4) m =  37.2 mm |     1953.125
   A10 | 2^(-19/4) m =   37.2 mm | 2^(-21/4) m =  26.3 mm |      976.5625
.
And the actual lengths, widths and areas (note that the actual areas are always smaller than the exact areas) are as follows:
.
  size | actual length (mm) | actual width (mm) | actual area (mm^2)
   A0  |        1189        |        841        |  999949 (99.9949%)
   A1  |         841        |        594        |  499554 (99.9108%)
   A2  |         594        |        420        |  249480 (99.7920%)
   A3  |         420        |        297        |  124740 (99.7920%)
   A4  |         297        |        210        |   62370 (99.7920%)
   A5  |         210        |        148        |   31080 (99.4560%)
   A6  |         148        |        105        |   15540 (99.4560%)
   A7  |         105        |         74        |    7770 (99.4560%)
   A8  |          74        |         52        |    3848 (98.5088%)
   A9  |          52        |         37        |    1924 (98.5088%)
   A10 |          37        |         26        |     962 (98.5088%)

Prepressure, A4.
Wikipedia, ISO 216.

Cf. A010767 (2^(1/4)), A228497 (2^(-1/4)), A308320 (2^(-7/4)).

RealDigits[2^(-9/4),10,88][[1]] (* James C. McMahon, Feb 26 2024 *)

PARI
```
default(realprecision, 100); 2^(-9/4)
```

Edited by Jon E. Schoenfield, Feb 25 2024

A379101 Decimal expansion of log(2)/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 15 2024

			0.17328679513998632735430803036454414201887503359006...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24.2, p. 414.

Cf. A001505, A002162, A010767, A016655.

Mathematica
```
RealDigits[Log[2]/4, 10, 100][[1]]
```
PARI
```
log(2)/4 \\ Amiram Eldar, Aug 19 2025
```

Equals log(A010767) = A016655/20. - Hugo Pfoertner, Dec 15 2024
From Amiram Eldar, Aug 19 2025: (Start)
Equals -Sum_{k>=0} zeta(2*k)/(2^(2*k+1)*(2*k+1)).
Equals Sum_{k>=0} 1/((4*k + 1)*(4*k + 2)*(4*k + 3)) = Sum_{k>=0} 1/A001505(k). (End)

A380907 Decimal expansion of 1/(2^(1/4)*sqrt(1+Pi/4)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Feb 08 2025

			0.62932496342101931026228634377882172549266644...

Piotr Garbaczewski and Vladimir A. Stephanovich, Dynamics of Confined Lévy Flights in Terms of (Lévy) Semigroups, Acta Phys. Pol. B 43, 977-999 (2012). See p. 992.

Cf. A003881, A010767, A019704, A228497.

RealDigits[1/(2^(1/4)Sqrt[1+Pi/4]),10,100][[1]]

cp's OEIS Frontend

A093406 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) + a(n-4).

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A329216 Decimal expansion of 2^(5/12).

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Mathematica

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Formula

A329220 Decimal expansion of 2^(11/12).

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Mathematica

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A159824 Continued fraction for Pi^Pi (cf. A073233).

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Mathematica

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Extensions

A196535 Decimal expansion of Sum_{j=0..oo} exp(-Pi*(2*j+1)^2).

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Maple

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A269430 Decimal expansion of (1 + Pi)/2.

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PARI

Sage

Formula

A308320 Decimal expansion of 2^(-7/4); exact length of the A4 paper size measured in meters according to the ISO 216 standard.

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A093406 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3) + a(n-4).

A196535 Decimal expansion of Sum_{j=0..oo} exp(-Pi(2j+1)^2).