A061039 -id:A061039 - cp's OEIS frontend

A061043 Numerator of 1/25 - 1/n^2.

0, 11, 24, 39, 56, 3, 96, 119, 144, 171, 8, 231, 264, 299, 336, 3, 416, 459, 504, 551, 24, 651, 704, 759, 816, 7, 936, 999, 1064, 1131, 48, 1271, 1344, 1419, 1496, 63, 1656, 1739, 1824, 1911, 16, 2091, 2184, 2279, 2376, 99, 2576, 2679, 2784

Offset: 5

N. J. A. Sloane, May 26 2001

From Pfund spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

a(n) = (n+5)^2-25 = n*(n+10) except a(5p) for p positive. Second (with m=5) of this kind after A061039, Paschen (m=3) and before A061047, Hansen-Strong (m=7). For the fourth, what is the value of m in 1/m^2-1/n^2? m=9? - Paul Curtz, Nov 01 2008

J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.

Reinhard Zumkeller, Table of n, a(n) for n = 5..10000
J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
Eric Weisstein's World of Physics, Balmer Formula

Cf. A061035-A061050.

import Data.Ratio ((%), numerator)
a061043 = numerator . (1 % 25 -) . recip . (^ 2) . fromIntegral
-- Reinhard Zumkeller, Jan 06 2014

Numerator[1/25-1/Range[5,60]^2] (* Harvey P. Dale, Jan 28 2013 *)

a(n)=numerator(1/25 - 1/n^2) \\ Charles R Greathouse IV, Feb 06 2017

from fractions import Fraction
def a(n): return (Fraction(1, 25) - Fraction(1, n*n)).numerator
print([a(n) for n in range(5, 54)]) # Michael S. Branicky, Nov 19 2021

A144437 Period 3: repeat [3, 3, 1].

Original entry on oeis.org

3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3

Offset: 1

Paul Curtz, Oct 05 2008

The sequence is generated from numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...
Conjecture: a(n) is the separatix. See A045944.
Also the decimal expansion of the constant 3310/999. - R. J. Mathar, May 21 2009
Continued fraction expansion of A171417.
Greatest common divisor of (n+1)^2-1 and (n+1)^2+2. - Bruno Berselli, Mar 08 2017

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000217, A045944, A109007, A164306, A167192, A169609, A256095.

Numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...

&cat [[3, 3, 1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

seq(op([3, 3, 1]), n=1..50); # Wesley Ivan Hurt, Jul 02 2016

A144437[n_]:=Denominator[n/3]; Array[A144437,100] (* Enrique Pérez Herrero, Oct 05 2011 *)
CoefficientList[Series[(3 + 3 x + x^2)/(1 - x^3), {x, 0, 120}], x] (* Michael De Vlieger, Jul 02 2016 *)
Table[Mod[2*n^2 + 1, 3,1], {n,1,50}] (* G. C. Greubel, Aug 24 2017 *)

a(n)=if(n%3,3,1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = (7-4*cos(2*Pi*n/3))/3. - Jaume Oliver Lafont, Nov 23 2008
G.f.: x*(3 + 3*x + x^2)/((1 - x)*(1 + x + x^2)). - R. J. Mathar, May 21 2009
a(n) = 3/gcd(n,3). - Reinhard Zumkeller, Oct 30 2009
a(n) = denominator(n^k/3), where k>0 is an integer. - Enrique Pérez Herrero, Oct 05 2011
a(n) = gcd(T(n+1), T(2)) = A256095(n+1, 2), with the triangular numbers T = A000217, for n >= 1. - Wolfdieter Lang, Mar 17 2015
a(n) = a(n-3) for n>3; a(n) = A169609(n) for n>0. - Wesley Ivan Hurt, Jul 02 2016
E.g.f.: (1/3)*(7*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2) - 3). - G. C. Greubel, Aug 24 2017
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 9/(a(n-2)*a(n-1)).
a(n) = 7 - a(n-2) - a(n-1). See also A052901 or A069705. (End)

Edited by R. J. Mathar, May 21 2009

A061040 Denominator of 1/9 - 1/n^2.

Original entry on oeis.org

1, 144, 225, 12, 441, 576, 81, 900, 1089, 48, 1521, 1764, 75, 2304, 2601, 324, 3249, 3600, 147, 4356, 4761, 64, 5625, 6084, 729, 7056, 7569, 100, 8649, 9216, 363, 10404, 11025, 1296, 12321, 12996, 507, 14400, 15129, 588, 16641, 17424

Offset: 3

N. J. A. Sloane, May 26 2001

See A061039 (numerators) for comments, references and links.

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
Friedrich Paschen, Zur Kenntnis ultraroter Linienspektra, Annalen der Physik 27, pp. 537-570 (1908).
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A061039.

import Data.Ratio ((%), denominator)
a061040 n = denominator $ 1%9 - 1%n^2 -- Reinhard Zumkeller, Jan 03 2012

Denominator[1/9-1/Range[3,50]^2] (* Harvey P. Dale, Jan 16 2012 *)

a(n)=denominator(1/9 - 1/n^2) \\ Charles R Greathouse IV, Feb 07 2017

from math import gcd
def A061040(n): return 9*n**2//gcd(n**2-9,9*n**2) # Chai Wah Wu, Apr 02 2021

[denominator(1/9 -1/n^2) for n in (3..50)] # G. C. Greubel, Mar 10 2022

a(n) = denominator(n^2 - 9)/(9*n^2), n >= 3.

a(n) = (n^2)/9 if n == 3 or 24 (mod 27), a(n) = (n^2)/3 if n == 6 or 12 or 15 or 21 (mod 27), a(n) = n^2 if n == 0 (mod 9) and 9*n^2 otherwise. From the period length 27 sequence gcd(n^2 - 9, 9*n^2). - Wolfdieter Lang, Mar 15 2018

A168668 a(n) = n(2 + 5n).

Original entry on oeis.org

0, 7, 24, 51, 88, 135, 192, 259, 336, 423, 520, 627, 744, 871, 1008, 1155, 1312, 1479, 1656, 1843, 2040, 2247, 2464, 2691, 2928, 3175, 3432, 3699, 3976, 4263, 4560, 4867, 5184, 5511, 5848, 6195, 6552, 6919, 7296, 7683, 8080, 8487, 8904, 9331, 9768, 10215, 10672

Offset: 0

Paul Curtz, Dec 02 2009

Appears on the main diagonal of the following table of terms of the Hydrogen series, A169603:
0,  3,  8, 15, 24,                        ... A005563
0,  7, 16,  1, 40,  55,                   ... A061039
0, 11, 24, 39, 56,   3,  96,              ... A061043
0, 15, 32, 51, 72,  95, 120,              ... A061047
0, 19, 40, 63, 88, 115, 144, 175, 208, 1, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature(3,-3,1).

Cf. A000217, A000384, A001622, A010692, A131242, A254963 (comment).

Cf. A005563, A061039, A061043, A061047, A169603.

[n*(2+5*n): n in [0..50] ]; // Vincenzo Librandi, Aug 06 2011

A168668:=n->n*(2+5*n): seq(A168668(n), n=0..50); # Wesley Ivan Hurt, Mar 28 2015

f[n_]:=n*(2+5*n);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
LinearRecurrence[{3,-3,1},{0,7,24},50] (* Harvey P. Dale, Sep 09 2021 *)

vector(50,n,n--;n*(2+5*n)) \\ Derek Orr, Jun 26 2015

G.f.: x*(7 + 3*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
First differences: a(n) - a(n-1) = 10*n-3.
Second differences: a(n) - 2*a(n-1) + a(n-2) = 10 = A010692(n).
a(n) = A131242(10n+6). - Philippe Deléham, Mar 27 2013
a(n) = A000384(n) + 6*A000217(n). - Luciano Ancora, Mar 28 2015
a(n) = A000217(n) + A000217(3*n). - Bruno Berselli, Jul 01 2016
E.g.f.: x*(7 + 5*x)*exp(x). - G. C. Greubel, Jul 29 2016
Sum_{n>=1} 1/a(n) = 5/4 - sqrt(1-2/sqrt(5))*Pi/4 + sqrt(5)*log(phi)/4 - 5*log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 17 2023

Edited and extended by R. J. Mathar, Dec 05 2009

A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

Original entry on oeis.org

1, 5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 125, 37, 173, 50, 229, 65, 293, 82, 365, 101, 445, 122, 533, 145, 629, 170, 733, 197, 845, 226, 965, 257, 1093, 290, 1229, 325, 1373, 362, 1525, 401, 1685, 442, 1853, 485, 2029, 530, 2213, 577, 2405, 626, 2605, 677

Offset: 0

Paul Curtz, Aug 15 2015

Using (n+sqrt(4+n^2))/2, after the integer 1 for n=0, the reduced metallic means are b(1) = (1+sqrt(5))/2, b(2) = 1+sqrt(2), b(3) = (3+sqrt(13))/2, b(4) = 2+sqrt(5), b(5) = (5+sqrt(29))/2, b(6) = 3+sqrt(10), b(7) = (7+sqrt(53))/2, b(8) = 4+sqrt(17), b(9) = (9+sqrt(85))/2, b(10) = 5+sqrt(26), b(11) = (11+sqrt(125))/2 = (11+5*sqrt(5))/2, ... . The last value yields the radicals in a(n) or A013946.
b(2) = 2.41, b(3) = 3.30, b(4) = 4.24, b(5) = 5.19 are "good" approximations of fractal dimensions corresponding to dimensions 3, 4, 5, 6: 2.48, 3.38, 4.33 and 5.45 based on models. See "Arbres DLA dans les espaces de dimension supérieure: la théorie des peaux entropiques" in Queiros-Condé et al. link. DLA: beginning of the title of the Witten et al. link.
Consider the symmetric array of the half extended Rydberg-Ritz spectrum of the hydrogen atom:
   0,    1/0,     1/0,     1/0,     1/0,      1/0,      1/0,     1/0, ...
-1/0,      0,     3/4,     8/9,   15/16,    24/25,    35/36,   48/49, ...
-1/0,   -3/4,       0,    5/36,    3/16,   21/100,      2/9,  45/196, ...
-1/0,   -8/9,   -5/36,       0,   7/144,   16/225,     1/12,  40/441, ...
-1/0, -15/16,   -3/16,  -7/144,       0,    9/400,    5/144,  33/784, ...
-1/0, -24/25, -21/100, -16/225,  -9/400,        0,   11/900, 24/1225, ...
-1/0, -35/36,    -2/9,   -1/12,  -5/144,  -11/900,        0, 13/1764, ...
-1/0, -48/49, -45/196, -40/441, -33/784, -24/1225, -13/1764,       0, ... .
The numerators are almost A165795(n).
Successive rows: A000007(n)/A057427(n), A005563(n-1)/A000290(n), A061037(n)/A061038(n), A061039(n)/A061040(n), A061041(n)/A061042(n), A061043(n)/A061044(n), A061045(n)/A061046(n), A061047(n)/A061048(n), A061049(n)/A061050(n).
A144433(n) or A195161(n+1) are the numerators of the second upper diagonal (denominators: A171522(n)).
c(n+1) = a(n) + a(n+1) = 6, 7, 15, 18, 34, 39, 63, 70, 102, 111, ... .
c(n+3) - c(n+1) = 9, 11, 19, 21, 29, 31, ... = A090771(n+2).
The final digit of a(n) is neither 4 nor 8. - Paul Curtz, Jan 30 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Diogo Queiros-Condé, Jean Chaline, and Jacques Dubois, Le monde des fractales La Nature trans-échelles, 478p., ellipses, Paris, 2015, page 220.
T. A. Witten, Jr. and L. M. Sander, Diffusion-Limited Aggregation, a Kinetic Critical Phenomenom, Phys. Rev. Lett., Vol. 47 (Nov 09 1981), pp. 1400-1403.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000007, A000290, A001622, A002522, A005563, A010685, A010698, A013946, A014176, A057427, A061035-A061050, A078370, A087475, A090771, A098316, A098317, A098318, A144433, A168077, A171621, A176398, A176439, A176458, A176522, A176537, A195161.

Cf. A069894.

[Numerator(1+n^2/4): n in [0..60]]; // Vincenzo Librandi, Aug 15 2015

A261327:=n->numer((4 + n^2)/4); seq(A261327(n), n=0..60); # Wesley Ivan Hurt, Aug 15 2015

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 5, 2, 13, 5, 29}, 60] (* Vincenzo Librandi, Aug 15 2015 *)
a[n_] := (n^2 + 4) / 4^Mod[n + 1, 2]; Table[a[n], {n, 0, 52}] (* Peter Luschny, Mar 18 2022 *)

vector(60, n, n--; numerator(1+n^2/4)) \\ Michel Marcus, Aug 15 2015

Vec((1+5*x-x^2-2*x^3+2*x^4+5*x^5)/(1-x^2)^3 + O(x^60)) \\ Colin Barker, Aug 15 2015

a(n)=if(n%2,n^2+4,(n/2)^2+1) \\ Charles R Greathouse IV, Oct 16 2015

[(n*n+4)//4**((n+1)%2) for n in range(60)] # Gennady Eremin, Mar 18 2022

[numerator(1+n^2/4) for n in (0..60)] # G. C. Greubel, Feb 09 2019

a(n) = numerator(1 + n^2/4). (Previous name.) See A010685 (denominators).
a(2*k) = 1 + k^2.
a(2*k+1) = 5 + 4*k*(k+1).
a(2*k+1) = 4*a(2*k) + 4*k + 1.
a(4*k+2) = A069894(k). - Paul Curtz, Jan 30 2019
a(-n) = a(n).
a(n+2) = a(n) + A144433(n) (or A195161(n+1)).
a(n) = A168077(n) + period 2: repeat 1, 4.
a(n) = A171621(n) + period 2: repeat 2, 8.
From Colin Barker, Aug 15 2015: (Start)
a(n) = (5 - 3*(-1)^n)*(4 + n^2)/8.
a(n) = n^2/4 + 1 for n even;
a(n) = n^2 + 4 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
G.f.: (1 + 5*x - x^2 - 2*x^3 + 2*x^4 + 5*x^5)/ (1 - x^2)^3. (End)
E.g.f.: (5/8)*(x^2 + x + 4)*exp(x) - (3/8)*(x^2 - x + 4)*exp(-x). - Robert Israel, Aug 18 2015
Sum_{n>=0} 1/a(n) = (4*coth(Pi)+tanh(Pi))*Pi/8 + 1/2. - Amiram Eldar, Mar 22 2022

New name by Peter Luschny, Mar 18 2022

A144433 Multiples of 8 interleaved with the sequence of odd numbers >= 3.

Original entry on oeis.org

8, 3, 16, 5, 24, 7, 32, 9, 40, 11, 48, 13, 56, 15, 64, 17, 72, 19, 80, 21, 88, 23, 96, 25, 104, 27, 112, 29, 120, 31, 128, 33, 136, 35, 144, 37, 152, 39, 160, 41, 168, 43, 176, 45, 184, 47, 192, 49, 200, 51, 208, 53, 216, 55, 224, 57, 232, 59, 240, 61, 248, 63, 256, 65, 264

Offset: 1

Paul Curtz, Oct 04 2008

For n >= 2, these are the numerators of 1/n^2 - 1/(n+1)^2: A061037(4), A061039(5), A061041(6), A061043(7), A061045(8), A061047(9), A061049(10), etc.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A120070.

[5+3/2*(-1)^(n-1)*(n-1)+3*(-1)^(n-1)+5/2*(n-1): n in [1..70]]; // Vincenzo Librandi, Jul 30 2011

A144433:=n->(n+1)*4^(n mod 2); seq(A144433(n), n=1..100); # Wesley Ivan Hurt, Nov 27 2013

Table[(n + 1)* 4^Mod[n, 2], {n, 100}] (* Wesley Ivan Hurt, Nov 27 2013 *)

x='x+O('x^50); Vec( x*(8+3*x-x^3)/((1-x)^2*(1+x)^2)) \\ G. C. Greubel, Sep 19 2018

a(2*n+1) = A008590(n+1), a(2*n) = A005408(n).
a(2*n+1) + a(2*n+2) = A017281(n+1).
From R. J. Mathar, Apr 01 2009: (Start)
a(n) = 2*a(n-2) - a(n-4).
G.f.: x*(8+3*x-x^3)/((1-x)^2*(1+x)^2). (End)
a(n) = (n + 1) * 4^(n mod 2). - Wesley Ivan Hurt, Nov 27 2013

Edited by R. J. Mathar, Apr 01 2009

A172157 Triangle T(n,m) = numerator of 1/n^2 - 1/m^2, read by rows, with T(n,0) = -1.

Original entry on oeis.org

-1, -1, -3, -1, -8, -5, -1, -15, -3, -7, -1, -24, -21, -16, -9, -1, -35, -2, -1, -5, -11, -1, -48, -45, -40, -33, -24, -13, -1, -63, -15, -55, -3, -39, -7, -15, -1, -80, -77, -8, -65, -56, -5, -32, -17, -1, -99, -6, -91, -21, -3, -4, -51, -9, -19, -1, -120, -117, -112

Offset: 1

Paul Curtz, Jan 27 2010

The triangle obtained by negating the values of the triangle A120072 and adding a row T(n,0) = -1.

			The full array of numerators starts in row n=1 with columns m>=0 as:
-1...0...3...8..15..24..35..48..63..80..99. A005563
-1..-3...0...5...3..21...2..45..15..77...6. A061037, A070262
-1..-8..-5...0...7..16...1..40..55...8..91. A061039
-1.-15..-3..-7...0...9...5..33...3..65..21. A061041
-1.-24.-21.-16..-9...0..11..24..39..56...3. A061043
-1.-35..-2..-1..-5.-11...0..13...7...5...4. A061045
-1.-48.-45.-40.-33.-24.-13...0..15..32..51. A061047
-1.-63.-15.-55..-3.-39..-7.-15...0..17...9. A061049
The triangle is the portion below the main diagonal, left from the zeros, 0<=m<n.

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A172370, A174233, A165795.

T[n_, 0] := -1; T[n_, k_] := 1/n^2 - 1/k^2; Table[Numerator[T[n, k]], {n, 1, 100}, {k, 0, n - 1}] // Flatten (* G. C. Greubel, Sep 19 2018 *)

A172370 Mirrored triangle A120072 read by rows.

Original entry on oeis.org

3, 5, 8, 7, 3, 15, 9, 16, 21, 24, 11, 5, 1, 2, 35, 13, 24, 33, 40, 45, 48, 15, 7, 39, 3, 55, 15, 63, 17, 32, 5, 56, 65, 8, 77, 80, 19, 9, 51, 4, 3, 21, 91, 6, 99, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 23, 11, 7, 5, 95, 1, 119, 1, 5, 35, 143, 25, 48, 69, 88, 105, 120, 133, 144

Offset: 2

Paul Curtz, Feb 01 2010

A table of numerators of 1/n^2 - 1/m^2 extended to negative m looks as follows, stacked such that values of common m are aligned
and the central column of -1 is defined for m=0:
.............................0..-1...0...3...8..15..24..35..48..63..80..99. A005563
.........................0..-3..-1..-3...0...5...3..21...2..45..15..77...6. A061037
.....................0..-5..-8..-1..-8..-5...0...7..16...1..40..55...8..91. A061039
.................0..-7..-3.-15..-1.-15..-3..-7...0...9...5..33...3..65..21. A061041
.............0..-9.-16.-21.-24..-1.-24.-21.-16..-9...0..11..24..39..56...3. A061043
.........0.-11..-5..-1..-2.-35..-1.-35..-2..-1..-5.-11...0..13...7...5...4. A061045
.....0.-13.-24.-33.-40.-45.-48..-1.-48.-45.-40.-33.-24.-13...0..15..32..51. A061047
.0.-15..-7.-39..-3.-55.-15.-63..-1.-63.-15.-55..-3.-39..-7.-15...0..17...9. A061049
The row-reversed variant of A120072 appears (negated) after the leftmost 0.
Equals A061035 with the first column removed. - Georg Fischer, Jul 26 2023

			The table starts
   3
   5   8
   7   3  15
   9  16  21  24
  11   5   1   2  35
  13  24  33  40  45  48
  15   7  39   3  55  15  63
  17  32   5  56  65   8  77  80
  19   9  51   4   3  21  91   6  99

G. C. Greubel, Rows n = 2..100 of triangle, flattened

Lower diagonal gives: A070262, A061037(n+2).

Cf. A061035, A172157, A165795.

[[Numerator(1/(n-k)^2 -1/n^2): k in [1..n-1]]: n in [2..20]]; // G. C. Greubel, Sep 20 2018

Table[Numerator[1/(n-k)^2 -1/n^2], {n, 2, 20}, {k, 1, n-1}]//Flatten (* G. C. Greubel, Sep 20 2018 *)

for(n=2,20, for(k=1,n-1, print1(numerator(1/(n-k)^2 -1/n^2), ", "))) \\ G. C. Greubel, Sep 20 2018

T(n,m) = numerator of 1/(n-m)^2 - 1/n^2, n >= 2, 1 <= m < n. - R. J. Mathar, Nov 23 2010

Comment rewritten and offset set to 2 by R. J. Mathar, Nov 23 2010

A143025 Period length 4: repeat [1, 8, 2, 8].

Original entry on oeis.org

1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8

Offset: 0

Paul Curtz, Oct 13 2008

Numerator of 1/n^2-1/(3n)^2 if n>0.
This can be generated from the transitions between principal quantum numbers n and 3n in the Hydrogen series: A005563(2), A061037(6), A061039(9), A061041(12), A061043(15), A061045(18), A061047(21), A061049(24),... (The mention of A005563(2) is somewhat a fluke to maintain the periodic pattern.)
Related to the continued fraction of (12*sqrt(55)-72)/19 = 0.89444115.. = 0+1/(1+1/(8+1/(2+...))). - R. J. Mathar, Jun 27 2011

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A045944, A144437.

&cat [[1, 8, 2, 8]^^30]; // Wesley Ivan Hurt, Jul 10 2016

seq(op([1, 8, 2, 8]), n=0..50); # Wesley Ivan Hurt, Jul 10 2016

PadRight[{}, 120, {1, 8, 2, 8}] (* Harvey P. Dale, Jul 01 2015 *)

a(n)=[1,8,2,8][n%4+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n+4) = a(n).
G.f.: (1+8*x+2*x^2+8*x^3)/(1-x^4).
From Wesley Ivan Hurt, Jul 10 2016: (Start)
a(n) = (19 - 13*I^(2*n) - I^(-n) - I^n)/4, where I = sqrt(-1).
a(n) = (19 - 2*cos(n*Pi/2) - 13*cos(n*Pi))/4. (End)

Partially edited by R. J. Mathar, Dec 10 2008

A165795 Array A(n, k) = numerator of 1/n^2 - 1/k^2 with A(0,k) = 1 and A(n,0) = -1, read by antidiagonals.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, -1, -3, 3, 1, -1, -8, 0, 8, 1, -1, -15, -5, 5, 15, 1, -1, -24, -3, 0, 3, 24, 1, -1, -35, -21, -7, 7, 21, 35, 1, -1, -48, -2, -16, 0, 16, 2, 48, 1, -1, -63, -45, -1, -9, 9, 1, 45, 63, 1, -1, -80, -15, -40, -5, 0, 5, 40, 15, 80, 1, -1, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, 1

Offset: 0

Paul Curtz, Sep 27 2009

A row of A(0,k)= 1 is added on top of the array shown in A172157, which is then read upwards by antidiagonals.

One may also interpret this as appending a 1 to each row of A173651 or adding a column of -1's and a diagonal of +1's to A165507.

			The array, A(n, k), of numerators starts in row n=0 with columns m>=0 as:
  .1...1...1...1...1...1...1...1...1...1...1.
  -1...0...3...8..15..24..35..48..63..80..99. A005563, A147998
  -1..-3...0...5...3..21...2..45..15..77...6. A061037, A070262
  -1..-8..-5...0...7..16...1..40..55...8..91. A061039
Antidiagonal triangle, T(n, k), begins as:
   1;
  -1,   1;
  -1,   0,   1;
  -1,  -3,   3,  1;
  -1,  -8,   0,  8,  1;
  -1, -15,  -5,  5, 15,  1;
  -1, -24,  -3,  0,  3, 24,  1;
  -1, -35, -21, -7,  7, 21, 35, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A158388, A165507, A172157, A173651.

Cf. A005408, A005563, A061037, A061039, A070262, A147998.

T[n_, k_]:= If[k==n, 1, If[k==0, -1, Numerator[1/(n-k)^2 - 1/k^2]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 10 2022 *)

def A165795(n,k):
    if (k==n): return 1
    elif (k==0): return -1
    else: return numerator(1/(n-k)^2 -1/k^2)
flatten([[A165795(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 10 2022

A(n, k) = numerator(1/n^2 - 1/k^2) with A(0,k) = 1 and A(n,0) = -1 (array).
A(n, 0) = -A158388(n).
A(n, k) = A172157(n,k), n>=1.
From G. C. Greubel, Mar 10 2022: (Start)
T(n, k) = numerator(1/(n-k)^2 -1/k^2), with T(n,n) = 1, T(n,0) = -1 (triangle).
A(n, n) = T(2*n, n) = 0^n.
Sum_{k=0..n} T(n, k) = 0^n.
T(n, n-k) = -T(n,k).
T(2*n+1, n) = -A005408(n). (End)

cp's OEIS Frontend

A061043 Numerator of 1/25 - 1/n^2.

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A144437 Period 3: repeat [3, 3, 1].

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A061040 Denominator of 1/9 - 1/n^2.

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A168668 a(n) = n*(2 + 5*n).

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A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

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A144433 Multiples of 8 interleaved with the sequence of odd numbers >= 3.

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A168668 a(n) = n(2 + 5n).