A087197 -id:A087197 - cp's OEIS frontend

A016814 a(n) = (4*n + 1)^2.

1, 25, 81, 169, 289, 441, 625, 841, 1089, 1369, 1681, 2025, 2401, 2809, 3249, 3721, 4225, 4761, 5329, 5929, 6561, 7225, 7921, 8649, 9409, 10201, 11025, 11881, 12769, 13689, 14641, 15625, 16641, 17689, 18769, 19881, 21025, 22201, 23409, 24649, 25921, 27225, 28561, 29929

Offset: 0

N. J. A. Sloane

A bisection of A016754. Sequence arises from reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..200 from Ivan Panchenko).
Leo Tavares, Illustration: Square trapeziums
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), this sequence (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).
Cf. A001539, A006752, A016742, A016802, A016813.
Cf. A016826, A016838, A017126, A068467, A087197.
Cf. A272399, A014105.

[(4*n+1)^2: n in [0..40]]; // G. C. Greubel, Dec 28 2022

A016814:=n->(4*n+1)^2; seq(A016814(k), k=0..100); # Wesley Ivan Hurt, Nov 02 2013

(4*Range[0,40] +1)^2 (* or *) LinearRecurrence[{3,-3,1}, {1,25,81}, 40] (* Harvey P. Dale, Nov 20 2012 *)
Accumulate[32Range[0, 47] - 8] + 9 (* Alonso del Arte, Aug 19 2017 *)

a(n)=(4*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

[(4*n+1)^2 for n in range(41)] # G. C. Greubel, Dec 28 2022

a(n) = a(n-1) + 32*n - 8, n > 0. - Vincenzo Librandi, Dec 15 2010
From George F. Johnson, Sep 28 2012: (Start)
G.f.: (1 + 22*x + 9*x^2)/(1 - x)^3.
a(n+1) = a(n) + 16 + 8*sqrt(a(n)).
a(n+1) = 2*a(n) - a(n-1) + 32 = 3*a(n) - 3*a(n-1) + a(n-2).
a(n-1)*a(n+1) = (a(n) - 16)^2 ; a(n+1) - a(n-1) = 16*sqrt(a(n)).
a(n) = A016754(2*n) = (A016813(n))^2. (End)
Sum_{n>=0} 1/a(n) = G/2 + Pi^2/16, where G is the Catalan constant (A006752). - Amiram Eldar, Jun 28 2020
Product_{n>=1} (1 - 1/a(n)) = 2*Gamma(5/4)^2/sqrt(Pi) = 2 * A068467^2 * A087197. - Amiram Eldar, Feb 01 2021
From G. C. Greubel, Dec 28 2022: (Start)
a(2*n) = A017078(n).
a(2*n+1) = A017126(n).
E.g.f.: (1 + 24*x + 16*x^2)*exp(x). (End)
a(n) = A272399(n+1) - A014105(n). - Leo Tavares, Dec 24 2023

A016838 a(n) = (4n + 3)^2.

Original entry on oeis.org

9, 49, 121, 225, 361, 529, 729, 961, 1225, 1521, 1849, 2209, 2601, 3025, 3481, 3969, 4489, 5041, 5625, 6241, 6889, 7569, 8281, 9025, 9801, 10609, 11449, 12321, 13225, 14161, 15129, 16129, 17161, 18225

Offset: 0

N. J. A. Sloane

If Y is a fixed 2-subset of a (4n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
A bisection of A016754. Sequence arises from reading the line from 9, in the direction 9, 49, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008
Using (n,n+1) to generate a Pythagorean triangle with sides of lengths xJ. M. Bergot, Jul 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..860
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A001539, A016742, A016754, A016802, A016814, A016826, A068465, A087197.

[(4*n+3)^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

A016838:=n->(4*n + 3)^2; seq(A016838(n), n=0..50); # Wesley Ivan Hurt, Feb 24 2014

Table[(4*n + 3)^2, {n, 0, 40}] (* Vaclav Kotesovec, Nov 14 2017 *)

a(n)=(4*n+3)^2 \\ Charles R Greathouse IV, Jun 17 2017

Denominators of first differences Zeta[2,(4n-1)/4]-Zeta[2,(4(n+1)-1)/4]. - Artur Jasinski, Mar 03 2010
From George F. Johnson, Oct 03 2012: (Start)
G.f.: (9+22*x+x^2)/(1-x)^3.
a(n+1) = a(n) + 16 + 8*sqrt(a(n)).
a(n+1) = 2*a(n) - a(n-1) + 32 = 3*a(n) - 3*a(n-1) + a(n-2).
a(n-1)*a(n+1) = (a(n)-16)^2; a(n+1) - a(n-1) = 16*sqrt(a(n)).
a(n) = A016754(2*n+1) = (A004767(n))^2.
(End)
Sum_{n>=0} 1/a(n) = Pi^2/16 - G/2, where G is the Catalan constant (A006752). - Amiram Eldar, Jun 28 2020
Product_{n>=0} (1 - 1/a(n)) = Gamma(3/4)^2/sqrt(Pi) = A068465^2 * A087197. - Amiram Eldar, Feb 01 2021

A243446 Decimal expansion of 3/(2*sqrt(Pi)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 05 2014

Expectation of the maximum of a size 3 sample from a normal (0,1) distribution.

			0.846284375321634430422119177341158878766...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A087197, A188340.

RealDigits[3/(2*Sqrt[Pi]), 10, 99] // First

3/(2*sqrt(Pi)) \\ G. C. Greubel, Jan 09 2017

A364394 G.f. satisfies A(x) = 1 + x/A(x)*(1 + 1/A(x)).

Original entry on oeis.org

1, 2, -6, 34, -238, 1858, -15510, 135490, -1223134, 11320066, -106830502, 1024144482, -9945711566, 97634828354, -967298498358, 9659274283650, -97119829841854, 982391779220482, -9990160542904134, 102074758837531810, -1047391288012377774, 10788532748880319298

Offset: 0

Seiichi Manyama, Jul 22 2023

sign

Cf. A112478, A364396, A364398.

Cf. A027307, A087197, A108424.

A364394 := proc(n)
    if n = 0 then
        1;
    else
    (-1)^(n-1)*add( binomial(n,k) * binomial(2*n+k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364394(n),n=0..80); # R. J. Mathar, Jul 25 2023

a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(2*n+k-2, n-1))/n);

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A027307.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n+k-2,n-1) = (-1)^(n-1) * A108424(n) for n > 0.
D-finite with recurrence n*(2*n-1)*a(n) +3*(6*n^2-10*n+3)*a(n-1) +(-46*n^2+227*n-279)*a(n-2) +2*(n-3)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Jul 25 2023
a(n) ~ c*(-1)^(n-1)*4^n*2F1([-n, 2*n-1], [n], -1)*n^(-3/2), with c = 1/(4*sqrt(Pi)) = A087197/4. - Stefano Spezia, Oct 21 2023

A087198 Decimal expansion of 1/(2*sqrt(Pi)).

Original entry on oeis.org

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

Offset: 0

Sven Simon, Aug 24 2003

Radius of a sphere with surface area 1.

According to Fouad (2004), to simulate the distance of a sound source under free field conditions, one can multiply "the waveform directly by a gain factor that is the square root of the intensity," which can be computed with the formula D = sqrt(1/(4 * Pi * d^2)) = 1/(3.55 * d), where d is the distance between the sound source and the listener and 3.55 is approximately 10(sqrt(Pi)/5) (A019707) (equation 15 in the chapter), though "in practice we usually drop the constant multiplier" (4 * Pi). If the distance is one unit, then D works out to this number. - Alonso del Arte, Jun 10 2012

			0.28209479177387814347...

Hesham Fouad, "Spatialization with Stereo Loudspeakers: Understanding Balance, Panning and Distance Attenuation" in Audio Anecdotes II: Tools, Tips, and Techniques for Digital Audio, K. Greenebaum & R. Barzel, eds. Wellesley, Massachusetts: A K Peters (2004): 150 - 153

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A086201, A087197, A087199.

RealDigits[Sqrt[1/(4Pi)], 10, 100][[1]] (* Alonso del Arte, Jun 10 2012 *)

sqrt(1/(4*Pi)) \\ G. C. Greubel, Jan 09 2017

1/(2 * sqrt(Pi)) = sqrt(1/(4 * Pi)).

A243447 Decimal expansion of 1-9/(4Pi)+sqrt(3)/(2Pi), an extreme value constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 05 2014

Variance of the maximum of a size 3 sample from a normal (0,1) distribution.

			0.55946720379736701379568631398017...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Index entries for transcendental numbers

Cf. A087197, A188340, A243446.

RealDigits[1 - 9/(4*Pi) + Sqrt[3]/(2*Pi), 10, 100] // First

1-9/4/Pi + sqrt(3)/2/Pi \\ Charles R Greathouse IV, Sep 28 2022

A076668 Decimal expansion of sqrt(2/Pi).

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Oct 25 2002

This is the limit of (n+1)!!/n!!/n^(1/2) at n_even->inf.

Expected value of |x - mu|/sigma for normal distribution with mean mu and standard deviation sigma (i.e., the normalized mean absolute deviation). - Stanislav Sykora, Jun 30 2017

			0.79788456080286535587989211986876373695171726232986931533...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Harmann König, Carsten Schütt, and Nicole Tomczak-Jaegermann, Projection constants of symmetric spaces and variants of Khintchine's inequality, J. reine angew. Math. 511 (1999), pp. 1-42.
Index entries for transcendental numbers

Cf. A004730, A004731, A019727, A060294 (Buffon's constant 2/Pi), A092678 (probable error).

pi:=Sqrt(2/Pi(RealField(110))); Reverse(Intseq(Floor(10^110*pi))); // Vincenzo Librandi, Jul 01 2017

RealDigits[Sqrt[2/Pi],10,120][[1]] (* Harvey P. Dale, Feb 05 2012 *)

sqrt(2/Pi) \\ G. C. Greubel, Sep 23 2017

Equals A087197*A002193. - R. J. Mathar Feb 05 2009

Equals integral_{-infinity..infinity} (1-erf(x)^2)/2 dx. - Jean-François Alcover, Feb 25 2015

More terms and better description from Benoit Cloitre and Michael Somos, Oct 29 2002

Leading zero removed, offset changed by R. J. Mathar, Feb 05 2009

A243448 Decimal expansion of 6*arcsec(sqrt(3))/Pi^(3/2), an extreme value constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2014

Expectation of the maximum of a size 4 sample from a normal (0,1) distribution.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Cf. A087197, A188340, A243446, A243447.

RealDigits[6*ArcSec[Sqrt[3]]/Pi^(3/2), 10, 99] // First

A243452 Decimal expansion of the variance of the maximum of a size 4 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 05 2014

			0.4917152368747417606817470099858870229...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

Cf. A087197, A188340, A243446, A243447, A243448.

RealDigits[1 + Sqrt[3]/Pi - 36*ArcSec[Sqrt[3]]^2/Pi^3, 10, 102] // First

1 + sqrt(3)/Pi - 36*arcsec(sqrt(3))^2/Pi^3.

A243453 Decimal expansion of the expectation of the maximum of a size 5 sample from a normal (0,1) distribution.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2014

			1.1629644736405196127722679885505014941...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.16 Extreme value constants, p. 365.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A087197, A188340, A243446, A243447, A243448, A243452.

RealDigits[5/Sqrt[Pi] - 15*ArcCsc[Sqrt[3]]/Pi^(3/2), 10, 102] // First

5/sqrt(Pi) - 15*asin(1/sqrt(3))/Pi^(3/2) \\ G. C. Greubel, Feb 01 2017

5/sqrt(Pi) - 15*arccsc(sqrt(3))/Pi^(3/2).

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A016814 a(n) = (4*n + 1)^2.

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A016838 a(n) = (4n + 3)^2.

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A243446 Decimal expansion of 3/(2*sqrt(Pi)).

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A364394 G.f. satisfies A(x) = 1 + x/A(x)*(1 + 1/A(x)).

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A087198 Decimal expansion of 1/(2*sqrt(Pi)).

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A243447 Decimal expansion of 1-9/(4*Pi)+sqrt(3)/(2*Pi), an extreme value constant.

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A076668 Decimal expansion of sqrt(2/Pi).

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A243447 Decimal expansion of 1-9/(4Pi)+sqrt(3)/(2Pi), an extreme value constant.