A061041 -id:A061041 - cp's OEIS frontend

A172370 Mirrored triangle A120072 read by rows.

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

A176126 Numerator of -A127276(n)/A001788(n).

Original entry on oeis.org

-1, -1, 1, 2, 4, 13, 19, 13, 17, 43, 53, 32, 38, 89, 103, 59, 67, 151, 169, 94, 104, 229, 251, 137, 149, 323, 349, 188, 202, 433, 463, 247, 263, 559, 593, 314, 332, 701, 739, 389, 409, 859, 901, 472, 494, 1033, 1079, 563, 587, 1223, 1273, 662, 688, 1429, 1483, 769, 797, 1651, 1709, 884, 914, 1889, 1951, 1007, 1039, 2143, 2209, 1138, 1172, 2413, 2483, 1277, 1313, 2699, 2773, 1424, 1462, 3001, 3079, 1579

Offset: 0

Paul Curtz, Dec 07 2010

The sequence of fractions starts -1/0, -1/1, 1/3, 2/3, 4/5, 13/15, 19/21, 13/14, 17/18, 43/45, 53/55, 32/33, 38/39, ...
The denominators are apparently A064038(n+1) = A061041(4+8*n) (i.e., specified as numerators in A061041).
The difference between denominator and numerator is A014695(n), n > 0.

Cf. A001788, A127276.

Cf. A061041, A064038, A014695.

A001788 := proc(n) n*(n+1)*2^(n-2) ; end proc:
A127276 := proc(n) 2^n-A001788(n) ; end proc:
A176126 := proc(n) if n = 0 then -1 else 2^n/A001788(n)-1 ; numer(-%) ; end if; end proc:
seq(A176126(n),n=0..40) ;

Conjecture: a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9) with g.f. (x^4-x^3+3*x^2-x+1)*(x^4-x^3-2*x^2-x+1) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Dec 12 2010

a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).

A225948 a(0) = -1; for n>0, a(n) = numerator(1/4 - 4/n^2).

Original entry on oeis.org

-1, -15, -3, -7, 0, 9, 5, 33, 3, 65, 21, 105, 2, 153, 45, 209, 15, 273, 77, 345, 6, 425, 117, 513, 35, 609, 165, 713, 12, 825, 221, 945, 63, 1073, 285, 1209, 20, 1353, 357, 1505, 99, 1665, 437, 1833, 30, 2009, 525, 2193, 143

Offset: 0

Paul Curtz, May 21 2013

Denominators are in A226008.
Fractions in lowest terms for n>0: -15/4, -3/4, -7/36, 0/1, 9/100, 5/36, 33/196, 3/16, 65/324, 21/100, 105/484, 2/9, 153/676, 45/196, 209/900, 15/64,...
If t(n) is the sequence with period 8: 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, ... (see A226044), then A226008(n) = 4*a(n) + t(n).

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

Cf. A000466, A028566, A061037, A061041, A078371, A106609, A142705, A145923, A226008.

[-1] cat [Numerator(1/4-4/n^2): n in [1..50]]; // Bruno Berselli, May 22 2013

Join[{-1}, Table[Numerator[1/4 - 4/n^2], {n, 50}]] (* Bruno Berselli, May 24 2013 *)

concat([-1], vector(100, n, numerator(1/4 - 4/n^2))) \\ G. C. Greubel, Sep 19 2018

a(n) = 3*a(n-8) -3*a(n-16) +a(n-24).
a(2n) = A061037(n), a(2n+1) = A145923(n-2) for A145923(-2)=-15, A145923(-1)=-7.
a(4n)   = A142705(n) for A142705(0)=-1, a(8n) = A000466(n);
a(4n+1) = A028566(4n-3) for A028566(-3)=-15;
a(4n+2) = A078371(n-1) for A078371(-1)=-3;
a(4n+3) = A028566(4n-1) for A028566(-1)=-7.
a(n+4)  = A106609(n) * A106609(n+8).
G.f.: -(1 +15*x +3*x^2 +7*x^3 -9*x^5 -5*x^6 -33*x^7 -6*x^8 -110*x^9 -30*x^10 -126*x^11 -2*x^12 -126*x^13 -30*x^14 -110*x^15 -3*x^16 -33*x^17 -5*x^18 -9*x^19 +7*x^21 +3*x^22 +15*x^23)/(1-x^8)^3. - Bruno Berselli, May 22 2013
a(n) = (n^2-16)*(6*cos(Pi*n/4)-54*cos(Pi*n/2)+6*cos(3*Pi*n/4)-219*(-1)^n+293)/512. - Bruno Berselli, May 22 2013
a(n+10) = a(n+2)*(n+14)/(n-2) for n=0,1 and n>2. - Bruno Berselli, May 22 2013

Edited by Bruno Berselli, May 22 2013

A061044 Denominator of 1/25 - 1/n^2.

Original entry on oeis.org

1, 900, 1225, 1600, 2025, 100, 3025, 3600, 4225, 4900, 225, 6400, 7225, 8100, 9025, 80, 11025, 12100, 13225, 14400, 625, 16900, 18225, 19600, 21025, 180, 24025, 25600, 27225, 28900, 1225, 32400, 34225, 36100, 38025, 1600, 42025

Offset: 5

N. J. A. Sloane, May 26 2001

Reinhard Zumkeller, Table of n, a(n) for n = 5..10000
A. H. Pfund, The emission of nitrogen and hydrogen in the infrared, Journal of the Optical Society of America, Vol. 9, Issue 3, (1924) pp. 193-196.

See A061041 for comments, references, links.

Cf. A061043 (numerator).

import Data.Ratio ((%), denominator)
a061044 = denominator . (1 % 25 -) . recip . (^ 2) . fromIntegral
-- Reinhard Zumkeller, Jan 06 2014

Table[Denominator[1/5^2 - 1/n^2], {n, 7, 50}] (* G. C. Greubel, Jul 07 2017 *)

a(n) = denominator(1/25 - 1/n^2); \\ Michel Marcus, Aug 15 2013

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

A165207 Period 4: repeat [2, 2, 4, 4].

Original entry on oeis.org

2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2

Offset: 0

Paul Curtz, Sep 07 2009

Continued fraction expansion of (21+5*sqrt(26))/19 = A177153. - Klaus Brockhaus, May 03 2010

A045572(n)^a(n) == 1 (mod 10). For n>1, a(n) is the smallest positive exponent with this property. - Christina Steffan, Sep 08 2015

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

Cf. A002378, A045572, A061037, A061041, A064038, A130658, A177153.

&cat[[2,2,4,4]^^30]; // Vincenzo Librandi, Feb 08 2016

seq(op([2, 2, 4, 4]), n=0..50); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 120, {2,2,4,4}] (* Harvey P. Dale, Oct 08 2014 *)

a(n)=n\2*2%4 + 2 \\ Charles R Greathouse IV, Jul 17 2016

a(n) = 2*A130658(n).
a(n) = A002378(n+1)/A064038(n+2) = A061037(4n+6)/A064038(n+2) = A061037(4n+6)/A061041(8n+12).
From R. J. Mathar, Sep 11 2009: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n>2.
G.f.: 2*(1+2*x^2)/((1-x)*(1+x^2)). (End)
a(n) = 3-cos(Pi*n/2)-sin(Pi*n/2). - R. J. Mathar, Oct 08 2011
a(n) = 2 + (2*floor(n/2) mod 4). - Wesley Ivan Hurt, Apr 20 2015
a(n) = a(n-4) for n>3. - Wesley Ivan Hurt, Jul 09 2016

Edited, offset set to 0, by R. J. Mathar, Sep 11 2009

A174680 Numerator of 1/16 - 1/n^2, using -1 at the pole where n=0.

Original entry on oeis.org

-1, -15, -3, -7, 0, 9, 5, 33, 3, 65, 21, 105, 1, 153, 45, 209, 15, 273, 77, 345, 3, 425, 117, 513, 35, 609, 165, 713, 3, 825, 221, 945, 63, 1073, 285, 1209, 5, 1353, 357, 1505, 99, 1665, 437, 1833, 15, 2009, 525, 2193, 143, 2385, 621, 2585, 21, 2793, 725, 3009, 195, 3233, 837, 3465, 14, 3705

Offset: 0

Paul Curtz, Nov 30 2010

Extends the Brackett spectrum to negative principal quantum numbers in the spirit of A144477 and A171709.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A174233, A067998.

Table[(n^2 - 16)/(GCD[n^2 - 16, 16*n^2]), {n, 0, 100}] (* G. C. Greubel, Sep 16 2018 *)

{a(n) = (n^2 - 16) / gcd(n^2 - 16, 16 * n^2)}; /* Michael Somos, Jan 06 2011 */

a(n) = A061041(n), n >= 4.
a(n) = A172157(4,n), n >= 1.
a(n) = a(-n) for all n in Z.

removed a(-4)-a(-1) since a(-n)=a(n) by Michael Somos, Jan 06 2011

A171372 a(n) = Numerator of 1/(2n)^2 - 1/(3n)^2 for n > 0, a(0) = 1.

Original entry on oeis.org

1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5

Offset: 0

Paul Curtz, Dec 07 2009

The diagonal of a table of numerators of the Rydberg-Ritz spectrum of hydrogen:
  1,  1,  1,   1,  1,   1,  1,   1,  1,   1,   1, ... A000012
  0,  5,  3,  21,  2,  45, 15,  77,  6, 117,  35, ... A061037
  0,  9,  5,  33,  3,  65, 21, 105,  1, 153,  45, ... A061041
  0, 13,  7,   5,  4,  85,  1, 133, 10,   7,  55, ... A061045
  0, 17,  9,  57,  5, 105, 33, 161,  3, 225,  65, ... A061049
  0, 21, 11,  69,  6,   1, 39, 189, 14, 261,   3, ...
  0, 25, 13,   1,  7, 145,  5, 217,  1,  11,  85, ...
  0, 29, 15,  93,  8, 165, 51,   5, 18, 333,  95, ...
  0, 33, 17, 105,  9, 185, 57, 273,  5, 369, 105, ...
  0, 37, 19,  13, 10, 205,  7, 301, 22,   5, 115, ...
  0, 41, 21, 129, 11,   9, 69, 329,  3, 441,   1, ...
In that respect, constructed similar to A144437.

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

Cf. A171373 (binomial transform), A171408, A105371.

[1] cat [Numerator(5/(6*n)^2): n in [1..100]]; // G. C. Greubel, Sep 20 2018

Table[If[n==0,1,Numerator[5/(6*n)^2]], {n,0,100}] (* G. C. Greubel, Sep 20 2018 *)

concat([1], vector(100, n, numerator(5/(6*n)^2))) \\ G. C. Greubel, Sep 20 2018

a(n) = numerator of 5/(6*n)^2 .
Period 5: repeat [1,5,5,5,5].
G.f.: (1 + 5*x + 5*x^2 + 5*x^3 + 5*x^4)/((1-x)*(1 + x + x^2 + x^3 + x^4)).
a(n) = 1 + 4*sign(n mod 5). - Wesley Ivan Hurt, Sep 26 2018
a(n) = (21-8*cos(2*n*Pi/5)-8*cos(4*n*Pi/5))/5. - Wesley Ivan Hurt, Sep 27 2018

Edited by R. J. Mathar, Dec 15 2009

A178395 Triangle T(n,m) read by rows: the numerator of the coefficient [x^m] of the inverse Euler polynomial E^{-1}(n,x), 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 15, 10, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 4, 14, 28, 35, 28, 14, 4, 1, 1, 9, 18, 42, 63, 63, 42, 18, 9, 1, 1, 5, 45, 60, 105, 126, 105, 60, 45, 5, 1, 1, 11, 55, 165, 165, 231, 231, 165, 165, 55, 11, 1, 1, 6, 33, 110, 495, 396, 462, 396, 495, 110, 33, 6, 1

Offset: 0

Paul Curtz, May 27 2010

The triangle of fractions A060096(n,m)/A060097(n,m) contains the coefficients of the Euler Polynomial E(n,x) in row n. The matrix inverse of this triangle is
   1;
  1/2,  1;
  1/2,  1,   1;
  1/2, 3/2, 3/2,  1;
  1/2,  2,   3,   2,   1;
  1/2, 5/2,  5,   5,  5/2,  1;
and defines inverse Euler polynomials E^{-1}(n,x) assuming that row n and column m contain the coefficient [x^m] E^{-1}(n,x). The column m=0 is 1 if n=0, otherwise 1/2.
The current triangle T(n,m) shows the numerator of [x^m] E^{-1}(n,x).
Numerators of exponential Riordan array [(1+exp(x))/2,x]. Central coefficients T(2n,n) are A088218. - Paul Barry, Sep 07 2010

			From _Paul Barry_, Sep 07 2010: (Start)
Triangle begins
  1;
  1,   1;
  1,   1,   1;
  1,   3,   3,   1;
  1,   2,   3,   2,   1;
  1,   5,   5,   5,   5,   1;
  1,   3,  15,  10,  15,   3,   1;
  1,   7,  21,  35,  35,  21,   7,   1;
  1,   4,  14,  28,  35,  28,  14,   4,   1;
  1,   9,  18,  42,  63,  63,  42,  18,   9,   1;
  1,   5,  45,  60, 105, 126, 105,  60,  45,   5,   1; (End)

T.-X. He, L. C. Hsu, P. J.-S. Shiue, The Sheffer group and the Riordan group, Discr. Appl. Math. 155 (2007) 1895-1909.

Cf. A178474 (denominators).
Cf. A060096, A060097.
Cf. A088218, A026741, A061041, A064038.

nm := 15 : eM := Matrix(nm,nm) :
for n from 0 to nm-1 do for m from 0 to n do eM[n+1,m+1] := coeff(euler(n,x),x,m) ; end do: for m from n+1 to nm-1 do eM[n+1,m+1] := 0 ; end do: end do:
eM := LinearAlgebra[MatrixInverse](eM) :
for n from 1 to nm do for m from 1 to n do printf("%d,", numer(eM[n,m])) ; end do: end do: # R. J. Mathar, Dec 21 2010

(* The function RiordanArray is defined in A256893. *)
rows = 13;
R = RiordanArray[(1 + E^#)/2&, #&, rows, True];
R // Flatten // Numerator (* Jean-François Alcover, Jul 20 2019 *)

T(n,k)=numerator((binomial(n,k)+binomial(0,n-k))/2);
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print());

T(n,0) = 1.
T(n,m) = T(n,n-m).
T(n,1) = A026741(n).
T(n,2) = A064038(n) (numerators related to A061041).
Number triangle T(n,k) = [k<=n]*numerator((C(n,k) + C(0,n-k))/2). - Paul Barry, Sep 07 2010

A165983 Period 16: repeat 1,1,1,2,1,1,1,2,1,1,1,4,1,1,1,4.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 0

Paul Curtz, Oct 03 2009

The numerator of the reduced fraction A061037(n+3)/A061041(2n+6).

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

Cf. A064038.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(( -1-x-x^2-2*x^3-x^8-x^9-x^10-4*x^11 )/((x-1)*(1+x)*(1+x^2)*(x^8+1)))); // G. C. Greubel, Sep 20 2018

LinearRecurrence[{0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1}, {1, 1, 1, 2,  1, 1, 1, 2, 1, 1, 1, 4}, 50] (* G. C. Greubel, Apr 20 2016 *)

x='x+O('x^50); Vec(( -1-x-x^2-2*x^3-x^8-x^9-x^10-4*x^11 )/((x-1)*(1+x)*(1+x^2)*(x^8+1))) \\ G. C. Greubel, Sep 20 2018

a(n) = a(n-4) - a(n-8) + a(n-12). - R. J. Mathar, Dec 17 2010
G.f.: ( -1 - x - x^2 - 2*x^3 - x^8 - x^9 - x^10 - 4*x^11 ) / ( (x-1)*(1+x)*(1+x^2)*(x^8+1) ). - R. J. Mathar, Dec 17 2010
a(4n) = a(4n+1) = a(4n+2) = 1. a(4n+3) = A165207(n).

cp's OEIS Frontend

A172370 Mirrored triangle A120072 read by rows.

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A176126 Numerator of -A127276(n)/A001788(n).

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A225948 a(0) = -1; for n>0, a(n) = numerator(1/4 - 4/n^2).

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A061044 Denominator of 1/25 - 1/n^2.

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

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A165207 Period 4: repeat [2, 2, 4, 4].

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A174680 Numerator of 1/16 - 1/n^2, using -1 at the pole where n=0.

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A171372 a(n) = Numerator of 1/(2n)^2 - 1/(3n)^2 for n > 0, a(0) = 1.