A250129 -id:A250129 - cp's OEIS frontend

A256781 Decimal expansion of the generalized Euler constant gamma(1,8).

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.788631390202002367443880819838976661978118204921...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), p. 134.

Cf. A001620 (EulerGamma), A016631, A228725 (gamma(1,2)), A250129, A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/8 + (1/8)*(Pi(R)/2*(Sqrt(2)+1) + Log(2) + Sqrt(2)*Log(Sqrt(2) + 1)); // G. C. Greubel, Aug 28 2018

RealDigits[-3/8*Log[2] - PolyGamma[1/8]/8, 10, 105] // First

Euler/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)) \\ Michel Marcus, Apr 10 2015

Equals EulerGamma/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)).
Equals Sum_{n>=0} (1/(8n+1) - 1/4*arctanh(4/(8n+5))).
Equals -(psi(1/8) + log(8))/8 = -(A250129 + A016631)/8. - Amiram Eldar, Jan 07 2024

A306716 Decimal expansion of the negated value of the digamma function at 1/10.

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Aug 22 2019

			Equals 10.4237549404110767951682162190100254042916425624441889203263920841...

Eric Weisstein's World of Mathematics, Digamma Function
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200135, A222457, A250129.

Maple
```
evalf(-Psi(1/10), 102);
```

RealDigits[-PolyGamma[1/10], 10, 105][[1]]

PARI
```
-psi(1/10)
```

Psi(1/10) = -gamma - Pi*5^(1/4)*(sqrt(2 + sqrt(5))/2) - 2*log(2) - 5*log(5)/4 - 3*sqrt(5)*log((1 + sqrt(5))/2)/2, where gamma is the Euler-Mascheroni constant A001620.

Equals gamma - H(-9/10), H(z) the harmonic number. - Peter Luschny, Aug 22 2019

A001533 a(n) = (8n+1)(8*n+7).

Original entry on oeis.org

7, 135, 391, 775, 1287, 1927, 2695, 3591, 4615, 5767, 7047, 8455, 9991, 11655, 13447, 15367, 17415, 19591, 21895, 24327, 26887, 29575, 32391, 35335, 38407, 41607, 44935, 48391, 51975, 55687, 59527, 63495, 67591, 71815, 76167, 80647, 85255, 89991, 94855, 99847, 104967

Offset: 0

N. J. A. Sloane

From Klaus Purath, Aug 18 2022: (Start)
This is A028560(8*n+1), and thus a(n) + 9 is a square. (See formulas)
7 is the only prime number of this sequence in which all odd prime factors occur.
Each prime factor p appears exactly twice in any interval of p consecutive terms. If a(m) and a(n) are within such an interval containing p, then m + n == -1 (mod p). (End)

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001539, A004771, A017077, A017113, A028560, A250129.

a[n_] := (8 n + 1)*(8 n + 7); Array[a, 40, 0] (* Amiram Eldar, Sep 08 2022 *)
LinearRecurrence[{3,-3,1},{7,135,391},40] (* Harvey P. Dale, Jan 07 2023 *)

a(n)=(8*n+1)*(8*n+7) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*A001539(n) - 5.
a(n) = 128*n + a(n-1) with a(0)=7. - Vincenzo Librandi, Nov 12 2010
Sum_{n>=0} 1/a(n) = (Psi(7/8)-Psi(1/8))/48 = 0.1580099..., see A250129. - R. J. Mathar, May 30 2022 [ = (sqrt(2)+1)*Pi/48. - Amiram Eldar, Sep 08 2022]
From Klaus Purath, Aug 18 2022: (Start)
a(n) = A028560(8*n+1).
a(n) + 9 = ((a(n+1) - a(n-1))/32)^2 = A017113(n)^2.
a(2*n) = (a(n+1) - a(n-1))*n + 7. (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017077(n)*A004771(n).
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(cot(Pi/16)) + sin(Pi/8) * log(cot(3*Pi/16)))/12.
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/8)*cos(sqrt(5/2)*Pi/4).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/8)*cos(sqrt(2)*Pi/4). (End)
G.f.: -(7+114*x+7*x^2)/(x-1)^3. - R. J. Mathar, Apr 23 2024
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(7 + 64*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A005570 Number of walks on cubic lattice.

Original entry on oeis.org

17, 50, 99, 164, 245, 342, 455, 584, 729, 890, 1067, 1260, 1469, 1694, 1935, 2192, 2465, 2754, 3059, 3380, 3717, 4070, 4439, 4824, 5225, 5642, 6075, 6524, 6989, 7470, 7967, 8480, 9009, 9554, 10115, 10692, 11285, 11894, 12519, 13160, 13817, 14490, 15179, 15884, 16605

Offset: 1

N. J. A. Sloane

Partial sums of A158057.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeremy Gardiner, Table of n, a(n) for n = 1..999
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6 (see figure 7).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A001620, A158057, A250129.

[8*n^2 + 9*n : n in [1..40]]; // Vincenzo Librandi, Nov 05 2014

CoefficientList[Series[(17 - x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 05 2014 *)

Vec((17-x)/(1-x)^3 + O(x^50)) \\ Michel Marcus, Nov 05 2014

a(n) = 8*n^2 + 9*n.
G.f.: (17-x)/(1-x)^3. Simon Plouffe in his 1992 dissertation.
a(n) = 16*A000217(n) + n. - Jon Perry, Nov 05 2014
Sum_{n>=1} 1/a(n) = 80/81 +Psi(1/8)/9+gamma/9 = 0.11973.. see A001620 and A250129. - R. J. Mathar, May 30 2022
Sum_{n>=1} 1/a(n) = 80/81 - (sqrt(2)+1)*Pi/18 - log(1+sqrt(2))*sqrt(2)/9 -4*log(2)/9. - Amiram Eldar, Sep 10 2022
From Elmo R. Oliveira, Jan 28 2025: (Start)
E.g.f.: exp(x)*x*(17 + 8*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Formula and more terms from Jeffrey Shallit, Aug 15 1995

cp's OEIS Frontend

A256781 Decimal expansion of the generalized Euler constant gamma(1,8).

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A306716 Decimal expansion of the negated value of the digamma function at 1/10.

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A001533 a(n) = (8*n+1)*(8*n+7).

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A005570 Number of walks on cubic lattice.

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A001533 a(n) = (8n+1)(8*n+7).