A069075 -id:A069075 - cp's OEIS frontend

A171522 Denominator of 1/n^2-1/(n+2)^2.

0, 9, 16, 225, 144, 1225, 576, 3969, 1600, 9801, 3600, 20449, 7056, 38025, 12544, 65025, 20736, 104329, 32400, 159201, 48400, 233289, 69696, 330625, 97344, 455625, 132496, 613089, 176400, 808201, 230400, 1046529, 295936, 1334025, 374544, 1677025, 467856

Offset: 0

Paul Curtz, Dec 11 2009

This is the third column in the table of denominators of the hydrogenic spectra (the main diagonal A147560):
0,   0,   0,   0,   0,   0,   0,   0... A000004
1,   4,   9,  16,  25,  36,  49,  64... A000290
1,  36,  16, 100,   9, 196,  64, 324... A061038
1, 144, 225,  12, 441, 576,  81, 900... A061040
1, 400, 144, 784,  64,1296, 400,1936... A061042
1, 900 1225,1600,2025, 100,3025,3600... A061044
1,1764, 576, 324, 225,4356,  48,6084... A061046
1,3136,3969,4900,5929,7056,8281, 196... A061048.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A105371. Bisections: A060300, A069075.

A171522 := proc(n) if n = 0 then 0 else lcm(n+2,n) ; %^2 ; end if ; end:
seq(A171522(n),n=0..70) ; # R. J. Mathar, Dec 15 2009

a[n_] := If[EvenQ[n], (n*(n+2))^2/4, (n*(n+2))^2]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jun 13 2017 *)

concat(0, Vec(x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5) + O(x^100))) \\ Colin Barker, Nov 05 2014

a(n) = (A066830(n+1))^2.
a(n) = -((-5+3*(-1)^n)*n^2*(2+n)^2)/8. - Colin Barker, Nov 05 2014
G.f.: x*(x^8+4*x^6+16*x^5+190*x^4+64*x^3+180*x^2+16*x+9) / ((x-1)^5*-(x+1)^5). - Colin Barker, Nov 05 2014

Edited and extended by R. J. Mathar, Dec 15 2009

A123092 Decimal expansion of Sum_{k>=1} 1/((2k-1)^2(2k+1)^2) = (Pi^2-8)/16.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Sep 27 2006

			0.116850275068084913677155687492259445957106212952549414150834336...

Erwin Kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley and Sons, Inc., NJ, 2006, page 506.

Index entries for transcendental numbers.

Cf. A000796, A002388, A069075, A111003.

Cf. A248895, A248896.

RealDigits[Sum[1/((2k - 1)^2(2k + 1)^2), {k, Infinity}], 10, 111][[1]]

(Pi^2-8)/16 \\ Charles R Greathouse IV, Sep 30 2022

Equals (A111003-1)/2. - Hugo Pfoertner, Aug 20 2024

Equals Sum_{k>=1} 1/(4*k^2-1)^2. - Sean A. Irvine, Mar 29 2025

A069076 a(n) = (4*n^2 - 1)^3.

Original entry on oeis.org

27, 3375, 42875, 250047, 970299, 2924207, 7414875, 16581375, 33698267, 63521199, 112678587, 190109375, 307546875, 480048687, 726572699, 1070599167, 1540798875, 2171747375, 3004685307, 4088324799, 5479701947, 7245075375

Offset: 1

Benoit Cloitre, Apr 05 2002

Konrad Knopp, Theory and application of infinite series, Dover, p. 269.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000466, A069075.

(4Range[30]^2-1)^3 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{27,3375,42875,250047,970299,2924207,7414875},30] (* Harvey P. Dale, Jan 20 2012 *)

Sum_{n>=1} 1/a(n) = (32 - 3*Pi^3)/64.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(1)=27, a(2)=3375, a(3)=42875, a(4)=250047, a(5)=970299, a(6)=2924207, a(7)=7414875. - Harvey P. Dale, Jan 20 2012
G.f: x*(x^6 - 34*x^5 - 3165*x^4 - 19852*x^3 - 19817*x^2 - 3186*x - 27)/(x-1)^7. - Harvey P. Dale, Jan 20 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^3/128 + 3*Pi/32 - 1/2. - Amiram Eldar, Feb 25 2022

A069073 a(n) = n*(4n^2 - 1)^2.

Original entry on oeis.org

0, 9, 450, 3675, 15876, 49005, 122694, 266175, 520200, 938961, 1592010, 2566179, 3967500, 5923125, 8583246, 12123015, 16744464, 22678425, 30186450, 39562731, 51136020, 65271549, 82372950, 102884175, 127291416, 156125025, 189961434, 229425075, 275190300, 327983301

Offset: 0

Benoit Cloitre, Apr 05 2002

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (104) on page 20.
Konrad Knopp, Theory and application of infinite series, Dover, p. 269.

Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")

Cf. A006752, A069075.

a[n_] := n*(4*n^2 - 1)^2; Array[a, 40, 0] (* Amiram Eldar, Mar 08 2022 *)

Sum_{n>=1} 1/a(n) = 3/2 -2*log(2) = 0.113705638880109...

Sum_{n>=1} (-1)^(n+1)/a(n) = G + log(2) - 3/2, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 08 2022

cp's OEIS Frontend

A171522 Denominator of 1/n^2-1/(n+2)^2.

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A123092 Decimal expansion of Sum_{k>=1} 1/((2k-1)^2(2k+1)^2) = (Pi^2-8)/16.

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A069076 a(n) = (4*n^2 - 1)^3.

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A069073 a(n) = n*(4n^2 - 1)^2.

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