A200135 -id:A200135 - cp's OEIS frontend

A005475 a(n) = n(5n+1)/2.

0, 3, 11, 24, 42, 65, 93, 126, 164, 207, 255, 308, 366, 429, 497, 570, 648, 731, 819, 912, 1010, 1113, 1221, 1334, 1452, 1575, 1703, 1836, 1974, 2117, 2265, 2418, 2576, 2739, 2907, 3080, 3258, 3441, 3629

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 0, in the direction 0, 11, ..., and the line from 3, in the direction 3, 24, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. - Omar E. Pol, Sep 26 2011

For n >= 3, a(n) is the sum of the numbers appearing in the 3rd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

G. C. Greubel, Table of n, a(n) for n = 0..5000
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
Leo Tavares, Illustration: Hexagonal Trapeziums.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A110449, A130520, A162147, A202803.

Cf. similar sequences listed in A022289.

seq(binomial(5*n+1,2)/5, n=0..34); # Zerinvary Lajos, Jan 21 2007
a:=n->sum(2*n+j, j=1..n): seq(a(n), n=0..38); # Zerinvary Lajos, Apr 29 2007

Table[n (5 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 13 2016 *)

a(n)=n*(5*n+1)/2; \\ Joerg Arndt, Mar 27 2013

a(n) = A110449(n, 2) for n>1.
a(n) = a(n-1) + 5*n - 2 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A130520(5*n+2). - Philippe Deléham, Mar 26 2013
a(n) = A202803(n)/2. - Philippe Deléham, Mar 27 2013
a(n) = A162147(n) - A162147(n-1). - J. M. Bergot, Jun 21 2013
a(n) = A000217(3*n) - A000217(2*n). - Bruno Berselli, Oct 13 2016
From G. C. Greubel, Aug 23 2017: (Start)
G.f.: x*(2*x + 3)/(1-x)^3.
E.g.f.: (x/2)*(5*x+6)*exp(x). (End)
Sum_{n>=1} 1/a(n) = 10+2*gamma+2*Psi(1/5) = 0.57635... see A001620 and A200135. - R. J. Mathar, May 30 2022
Sum_{n>=1} 1/a(n) = 10 - sqrt(1+2/sqrt(5))*Pi - sqrt(5)*log(phi) - 5*log(5)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 10 2022

Incorrect comment deleted and minor errors corrected by Johannes W. Meijer, Feb 04 2010

A256779 Decimal expansion of the generalized Euler constant gamma(1,5).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.735920396831617584189289725844752893059997383987625...

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), A016628, A200135, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 + Pi(R)/10*Sqrt(1 + 2/Sqrt(5)) + Log(5)/20 + Sqrt(5)/10*Log((1 + Sqrt(5))/2); // G. C. Greubel, Aug 28 2018

RealDigits[-Log[5]/5 - PolyGamma[1/5]/5, 10, 105] // First

Euler/5 + Pi/10*sqrt(1 + 2/sqrt(5)) + log(5)/20 + sqrt(5)/10*log((1 + sqrt(5))/2) \\ Michel Marcus, Apr 10 2015

Equals EulerGamma/5 + Pi/10*sqrt(1 + 2/sqrt(5)) + log(5)/20 + sqrt(5)/10*log((1 + sqrt(5))/2).
Equals Sum_{n>=0} (1/(5n+1) - 2/5*arctanh(5/(10n+7))).
Equals -(psi(1/5) + log(5))/5 = (A200135 - A016628)/5. - Amiram Eldar, Jan 07 2024

A083912 Number of divisors of n that are congruent to 2 modulo 10.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0

Offset: 1

Reinhard Zumkeller, May 08 2003

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A000005, A001227, A010879.
Cf. A001620, A002392, A200135 (psi(1/5)).
Cf. A083910, A083911, A083913, A083914, A083915, A083916, A083917, A083918, A083919.

a[n_] := Sum[If[Mod[d, 10] == 2, 1, 0], {d, Divisors[n]}];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 2 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

A083912(n) = sumdiv(n,d,2==(d%10)); \\ Antti Karttunen, Jan 22 2020

a(n) = A000005(n) - A083910(n) - A083911(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(2*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,10) - (1 - gamma)/10 = 0.256367..., gamma(2,10) = -(psi(1/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

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

A200136 Decimal expansion of the negated value of the digamma function at 2/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(2/5) = -2.5613845445851161457306754820475...

Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A200135-A200138.

-gamma-Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)*log((3+sqrt(5))/2)/4 ; evalf(%) ;

RealDigits[ PolyGamma[2/5], 10, 84] // First (* Jean-François Alcover, Feb 21 2013 *)

-psi(2/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(2/5) = -gamma -Pi*sqrt(1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

A262246 Decimal expansion of Sum_{k>=0} (-1)^k/(5k+2).

Original entry on oeis.org

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

Offset: 0

Gheorghe Coserea, Oct 06 2015

			0.4069016342...

Gheorghe Coserea, Table of n, a(n) for n = 0..2015
H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.

Cf. A003881, A024396, A113476, A181048, A181049, A181122, A193534.

N[(1/5)*((Sqrt[5]-1)*Log[2] + Sqrt[5]*Log[Sin[3*Pi/10]] + (Pi/2)*Sec[Pi/10]), 100] (* G. C. Greubel, Oct 07 2015 *) (* fixed by Vaclav Kotesovec, Dec 11 2017 *)

default(realprecision, 87);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(5*n+2)))), "3..-2"))

Sum_{n>=0} (-1)^n/(5n+2) = Integral_{x=0..1} x/(1+x^5)dx.
From G. C. Greubel, Oct 07 2015: (Start)
Sum_{n>=0} (-1)^n/(5n+2) = (1/5)*(sqrt(5)*log(phi) - log(2) + Pi*(5*phi^2)^(-1/4)), where 2*phi=1+sqrt(5).
Sum_{n>=0} (-1)^n/(5n+2) = (1/5)*(sqrt(5)*log(2*sin(3*Pi/10)) - log(2) + (Pi/2)*sec(Pi/10)).
(End)
Sum_{n>=0} (-1)^n/(5n+2) = (Psi(1/5) - Psi(7/10))/10 , see A200135 and A354643. - Robert Israel, Oct 08 2015
From Peter Bala, Feb 19 2024: (Start)
Equals (1/2)*Sum_{n >= 0} n!*(5/2)^n/(Product_{k = 0..n} 5*k + 2) = (1/2)*Sum_{n >= 0} n!*(5/2)^n/A047055(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(5*k + 2)).
Continued fraction: 1/(2 + 2^2/(5 + 7^2/(5 + 12^2/(5 + ... + (5*n + 2)^2/(5 + ... ))))).
The slowly converging series representation Sum_{n >= 0} (-1)^n/(5*n + 2) for the constant can be accelerated to give the following faster converging series
1/4 + (5/2)*Sum_{n >= 0} (-1)^n/((5*n + 2)*(5*n + 7)) and
19/56 + (5^2/2)*Sum_{n >= 0} (-1)^n/((5*n + 2)*(5*n + 7)*(5*n + 12)).
These two series are the cases r = 1 and r = 2 of the general result:
for r >= 0, the constant equals C(r) + ((5/2)^r)*r!*Sum_{n >= 0} (-1)^n/((5*n + 2)*(5*n + 7)*...*(5*n + 5*r + 2)), where C(r) is the rational number (1/2)*Sum_{k = 0..r-1} (5/2)^k*k!/(2*7*12*...*(5*k + 2)). The general result can be proved by the WZ method as described in Wilf. (End)
From Peter Bala, Mar 03 2024: (Start)
Equals (1/2)*hypergeom([2/5, 1], [7/5], -1).
Gauss's continued fraction: 1/(2 + 2^2/(7 + 5^2/(12 + 7^2/(17 + 10^2/(22 + 12^2/(27 + 15^2/(32 + 17^2/(37 + 20^2/(42 + 22^2/(47 + ... )))))))))). (End)

A001545 a(n) = (5n+1)(5*n+4).

Original entry on oeis.org

4, 54, 154, 304, 504, 754, 1054, 1404, 1804, 2254, 2754, 3304, 3904, 4554, 5254, 6004, 6804, 7654, 8554, 9504, 10504, 11554, 12654, 13804, 15004, 16254, 17554, 18904, 20304, 21754, 23254, 24804, 26404, 28054, 29754, 31504, 33304, 35154, 37054, 39004, 41004, 43054

Offset: 0

N. J. A. Sloane

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. A000217, A001622, A016861, A016897, A177059, A200135, A200138.

Table[(5n+1)(5n+4),{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{4,54,154},60] (* Harvey P. Dale, Mar 17 2019 *)

a(n)=(5*n+1)*(5*n+4) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 50*A000217(n) + 4.
a(n) = 50*n + a(n-1) with a(0)=4. - Vincenzo Librandi, Jan 20 2011
From Amiram Eldar, Jan 23 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(1 + 2/sqrt(5))*Pi/15 = 0.2882687....
Sum_{n>=0} (-1)^n/a(n) = 2*log(phi)/(3*sqrt(5)) + 2*log(2)/15, where phi is the golden ratio (A001622).
Product_{n>=0} (1 - 1/a(n)) = sqrt(2 + 2/sqrt(5)) * cos(sqrt(13)*Pi/10).
Product_{n>=0} (1 + 1/a(n)) = sqrt(2 + 2/sqrt(5)) * cos(Pi/(2*sqrt(5))).
Product_{n>=0} (1 + 2/a(n)) = phi. (End)
G.f.: 2*(2+21*x+2*x^2)/(1-x)^3. - R. J. Mathar, May 30 2022
Sum_{n>=0} 1/a(n) = (Psi(4/5) - Psi(1/5))/15. See A200135, A200138. - R. J. Mathar, May 30 2022
From Elmo R. Oliveira, Oct 23 2024: (Start)
E.g.f.: exp(x)*(4 + 25*x*(2 + x)).
a(n) = A016861(n)*A016897(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

cp's OEIS Frontend

A005475 a(n) = n*(5*n+1)/2.

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A256779 Decimal expansion of the generalized Euler constant gamma(1,5).

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Magma

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A083912 Number of divisors of n that are congruent to 2 modulo 10.

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

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A200136 Decimal expansion of the negated value of the digamma function at 2/5.

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A262246 Decimal expansion of Sum_{k>=0} (-1)^k/(5k+2).

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A001545 a(n) = (5*n+1)*(5*n+4).

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A005475 a(n) = n(5n+1)/2.

A001545 a(n) = (5n+1)(5*n+4).