A061038 -id:A061038 - cp's OEIS frontend

A154615 a(n) = A022998(n)^2.

0, 1, 16, 9, 64, 25, 144, 49, 256, 81, 400, 121, 576, 169, 784, 225, 1024, 289, 1296, 361, 1600, 441, 1936, 529, 2304, 625, 2704, 729, 3136, 841, 3600, 961, 4096, 1089, 4624, 1225, 5184, 1369, 5776, 1521, 6400, 1681, 7056, 1849, 7744, 2025, 8464, 2209, 9216

Offset: 0

Paul Curtz, Jan 13 2009

Multiplicative because A022998 is. - Andrew Howroyd, Jul 25 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A016754, A022998, A061038.

Join[{0}, Denominator[Table[(1/4)*(1 - 1/n^2), {n, 1, 50}]]] (* or *) Table[(1/2)*(5 + 3*(-1)^n)*n^2 {n,0,50}] (* G. C. Greubel, Jul 20 2017 *)

for(n=0, 50, print1((1/2)*(5 + 3*(-1)^n)*n^2, ", ")) \\ G. C. Greubel, Jul 20 2017

Denominators of 1/4 - 1/(2n)^2, if n>0.
a(2n+1) = A016754(n). a(2n) = 16*A000290(n).
a(n) = A061038(2*n) (bisection).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
G.f.: x*(1+16*x+6*x^2+16*x^3+x^4)/((1-x)^3*(1+x)^3).
From G. C. Greubel, Jul 20 2017: (Start)
a(n) = (1/2)*(5 + 3*(-1)^n)*n^2.
E.g.f.: x*( (4*x +1)*cosh(x) + (x+4)*sinh(x) ). (End)
Sum_{n>=1} 1/a(n) = 13*Pi^2/96. - Amiram Eldar, Aug 13 2022

Edited, offset set to 1, and extended by R. J. Mathar, Sep 07 2009

a(0) added Oct 21 2009

A177427 Numerators of the Inverse Akiyama-Tanigawa transform of the aerated even-indexed Bernoulli numbers 1, 0, 1/6, 0, -1/30, 0, 1/42, ...

Original entry on oeis.org

1, 1, 13, 7, 149, 157, 383, 199, 7409, 7633, 86231, 88331, 1173713, 1197473, 1219781, 620401, 42862943, 43503583, 279379879, 283055551, 57313183, 19328341, 449489867, 1362695813, 34409471059, 34738962067, 315510823603, 45467560829, 9307359944587, 9382319148907, 293103346860157, 147643434162641, 594812856101039, 54448301591149

Offset: 0

Paul Curtz, May 07 2010

These are the numerators of the first row of a Table T(n,k) which contains the even-indexed Bernoulli numbers in the first column: T(2n,0) = A000367(n)/A002445(n), T(2n+1,0)=0, and which generates rows with the Akiyama-Tanigawa transform. (Because the first column is given, the algorithm is an inverse Akiyama-Tanigawa transform.)

These are the absolute values of the numerators of the Taylor expansion of sinh(log(x+1))*log(x+1)at x=0. - Gary Detlefs, Aug 31 2011

			The table T(n,k) of fractions generated by the Akiyama-Tanigawa transform, with the column T(n,0) equal to Bernoulli(n) for even n and equal to 0 for odd n, starts in row n=0 as:
1, 1, 13/12, 7/6, 149/120, 157/120, 383/280, 199/140, ...
0, -1/6, -1/4, -3/10, -1/3, -5/14, -3/8, -7/18, -2/5, -9/22, ...
1/6, 1/6, 3/20, 2/15, 5/42, 3/28, 7/72, 4/45, 9/110, 5/66, ...
0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495, 3/55, 15/286, ...
-1/30, -1/30, -3/140, -1/105, 0, 1/140, 49/3960, 8/495, ...
0, -1/42, -1/28, -4/105, -1/28, -29/924, -7/264, -28/1287, -87/5005, ...
1/42, 1/42, 1/140, -1/105, -5/231, -9/308, -343/10296, -1576/45045, ...

L. A. Medina, V. H. Moll, E. S. Rowland, Iterated primitives of logarithmic powers, arXiv:0911.1325, arXiv:0911.1325 [math.NT], 2009-2010.
D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A177690 (denominators).

t[n_, 0] := BernoulliB[n]; t[1, 0]=0; t[n_, k_] := t[n, k] = (t[n, k-1] + (k-1)*t[n, k-1] - t[n+1, k-1])/k; Table[t[0, k], {k, 0, 33}] // Numerator (* Jean-François Alcover, Aug 09 2012 *)

From Groux Roland, Jan 07 2011: (Start)
T(0,k) = H(k)/2 + 1/(k+1) with H(k) harmonic number of order k.
T(0,k)= -(1/2)*(k+1)*Integral_{x=0..1} x^n*log(x*(1-x)) dx.
G.f.: Sum_{k>=0} T(0,k) x^k = (x-2)*(log(1-x))/(2*x*(1-x)). (End)
T(1,n) = -A191567(n)/A061038(n+2) = -A060819(n)/A145979(n). - Paul Curtz, Jul 19 2011
(T(1,n))^2 = A181318(n)/A061038(n+2). - Paul Curtz, Jul 19 2011, index corrected by R. J. Mathar, Sep 09 2011

A226008 a(0) = 0; for n>0, a(n) = denominator(1/4 - 4/n^2).

Original entry on oeis.org

0, 4, 4, 36, 1, 100, 36, 196, 16, 324, 100, 484, 9, 676, 196, 900, 64, 1156, 324, 1444, 25, 1764, 484, 2116, 144, 2500, 676, 2916, 49, 3364, 900, 3844, 256, 4356, 1156, 4900, 81, 5476, 1444, 6084, 400, 6724, 1764, 7396, 121, 8100

Offset: 0

Paul Curtz, May 22 2013

Numerators are in A225948.

Repeated terms of A016826 are in the positions 1, 2, 3, 6, 5, 10, ... (A043547).

			a(0) = (-1+1)^2 = 0, a(1) = (-3+5)^2 = 4, a(2) = (-1+3)^2 = 4.

Cf. A016754, A016802, A016826, A017090, A017138, A154615, A225948 (numerators).

Cf. A225975 (associated square roots).

[0] cat [Denominator(1/4-4/n^2): n in [1..50]]; // Bruno Berselli, May 23 2013

Join[{0},Table[Denominator[1/4 - 4/n^2], {n, 49}]] (* Alonso del Arte, May 22 2013 *)

a(n)    = 3*a(n-8) -3*a(n-16) +a(n-24).
a(8n)   = A016802(n), a(8n+4) = A016754(n).
a(4n)   = A154615(n).
a(4n+1) = A017090(n).
a(4n+2) = a(2n+1) = A016826(n); a(2n) = A061038(n).
a(4n+3) = A017138(n).
From Bruno Berselli, May 23 2013: (Start)
G.f.: x*(4 +4*x +36*x^2 +x^3 +100*x^4 +36*x^5 +196*x^6 +16*x^7 +312*x^8 +88*x^9 +376*x^10 +6*x^11 +376*x^12 +88*x^13 +312*x^14 +16*x^15 +196*x^16 +36*x^17 +100*x^18 +x^19 +36*x^20 +4*x^21 +4*x^22)/(1-x^8)^3.
a(n) = n^2*(6*cos(3*Pi*n/4)+6*cos(Pi*n/4)-54*cos(Pi*n/2)-219*(-1)^n+293)/128.
a(n+9) = a(n+1)*((n+9)/(n+1))^2. (End)
Sum_{n>=1} 1/a(n) = 19*Pi^2/96. - Amiram Eldar, Aug 14 2022

Edited by Bruno Berselli, May 23 2013

A176672 a(2n) = 1 + 6n, a(2*n+1) = A165367(n).

Original entry on oeis.org

1, 1, 7, 5, 13, 4, 19, 11, 25, 7, 31, 17, 37, 10, 43, 23, 49, 13, 55, 29, 61, 16, 67, 35, 73, 19, 79, 41, 85, 22, 91, 47, 97, 25, 103, 53, 109, 28, 115, 59, 121, 31, 127, 65, 133, 34, 139, 71, 145, 37, 151, 77, 157, 40, 163, 83, 169, 43, 175, 89, 181, 46, 187, 95, 193

Offset: 0

Paul Curtz, Apr 23 2010

Motivation: Start an array from a left column of fractions 0, 1/6, 0, -1/30, 0, ... = A176327(.)/A176592(.), which is zero followed by the Bernoulli numbers from B_2 onwards.
Construct more columns of the array by iteration of the Akiyama-Tanigawa algorithm working backwards through the rows of the table. In our case, the array starts with column indices k>=0:
0, -1/6, -1/4, -3/10, -1/3, -5/14, -3/8, -7/18, ...
1/6, 1/6, 3/20, 2/15, 5/42, 3/28, 7/72, 4/45, 9/110, ...
0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495, ...
-1/30, -1/30, -3/140, -1/105, 0, 1/140, 49/3960, ...
0, -1/42, -1/28, -4/105, -1/28, -29/924, ...
1/42, 1/42, 1/140, -1/105, -5/231, -9/308, -343/10296, ...
The fractions of the top row are -A060819(n)/A145979(n). The current sequence contains essentially the difference between numerator and denominator of each fraction, a(2)=6+1, a(3)=4+1, a(4)=10+3, ... The sum of numerator and denominator is essentially A060819.
Also, numerators of (3*n + 1)/12. - Bruno Berselli, Apr 13 2018
Also, numerators of (3*n + 1)/4. - Altug Alkan, Apr 17 2018

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A061038, A145979.

From R. J. Mathar, Jan 06 2011: (Start)
a(n) = 2*a(n-4) - a(n-8).
G.f.: (1 + x + 7*x^2 + 5*x^3 + 11*x^4 + 2*x^5 + 5*x^6 + x^7) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2). (End)
a(n) = (2*(3*n + 1)*(11 + 5*(-1)^n) + (6*n + 5 + 3*(-1)^n)*(1 - (-1)^n)*(-1)^((2*n + 3 + (-1)^n)/4))/32. - Luce ETIENNE, Jan 28 2015

A165441 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2.

Original entry on oeis.org

1, 4, 4, 9, 1, 9, 16, 36, 36, 16, 25, 16, 1, 16, 25, 36, 100, 144, 144, 100, 36, 49, 9, 225, 1, 225, 9, 49, 64, 196, 12, 400, 400, 12, 196, 64, 81, 64, 441, 144, 1, 144, 441, 64, 81, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 121, 25, 81, 64, 1225, 1, 1225, 64, 81, 25, 121

Offset: 1

Paul Curtz, Sep 19 2009

A synopsis of the denominators of the transitions in the Rydberg-Ritz spectrum of hydrogenic atoms.

			.1,   4,   9,   16,   25,   36,   49,   64,   81, ... A000290
.4,   1,  36,   16,  100,    9,  196,   64,  324, ... A061038
.9,  36,   1,  144,  225,   12,  441,  576,   81, ... A061040
16,  16, 144,    1,  400,  144,  784,   64, 1296, ... A061042
25, 100, 225,  400,    1,  900, 1225, 1600, 2025, ... A061044
36,   9,  12,  144,  900,    1, 1764,  576,  324, ... A061046
49, 196, 441,  784, 1225, 1764,    1, 3136, 3969, ... A061048
64,  64, 576,   64, 1600,  576, 3136,    1, 5184, ... A061050
81, 324,  81, 1296, 2025,  324, 3969, 5184,    1, ...

Alois P. Heinz, Table of n, a(n) for n = 1..1035

T:= (k,n)-> denom(1/min (n,k)^2 -1/max (n, k)^2):
seq(seq(T(k, d-k), k=1..d-1), d=2..12);

T[n_, k_] := Denominator[1/Min[n, k]^2 - 1/Max[n, k]^2];
Table[T[n-k, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 04 2020 *)

T(n,k) = A165727(n,k).

Edited by R. J. Mathar, Feb 27 2010, Mar 03 2010

A191567 Four interlaced 2nd order polynomials: a(4k) = k(1+2k); a(1+2k) = 2(1+2k)(3+2k); a(2+4k) = 4(1+k)(1+2k).

Original entry on oeis.org

0, 6, 4, 30, 3, 70, 24, 126, 10, 198, 60, 286, 21, 390, 112, 510, 36, 646, 180, 798, 55, 966, 264, 1150, 78, 1350, 364, 1566, 105, 1798, 480, 2046, 136, 2310, 612, 2590, 171, 2886, 760, 3198, 210, 3526, 924, 3870, 253, 4230, 1104, 4606, 300, 4998, 1300, 5406, 351

Offset: 0

Paul Curtz, Jun 12 2011

a(n) = T(0,n) and differences T(n,k) = T(n-1,k+1) - T(n-1,k) define the array
0,   6,  4,  30,    3,  70,   24,  126,   10,  198,   60,  286,   21,  390,  ..
6,  -2, 26, -27,   67, -46,  102, -116,  188, -138,  226, -265,  369, -278, ..
-8, 28 -53,  94, -113, 148, -218,  304, -326,  364, -491,  634, -647,  676, ...
T(3,n) mod 9 is the sequence 1, 1, 1, 4, 4, 4, 7, 7, 7, 4, 4, 4 (and periodically repeated with period 12).
A064680(2+n) divides a(n), where b(n) = a(n)/A064680(2+n) = 0, 1, 2, 3, 1, 5, 6, 7, 2,... for n>=0, obeys b(4*k) = k and has recurrence b(n) = 2*b(n-4) - b(n-8).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A014105, A000466, A000384, A177427.

a:=[0,6,4,30,3,70,24,126,10,198,60,286];; for n in [13..60] do a[n]:= 3*a[n-4]-3*a[n-8]+a[n-12]; od; a; # G. C. Greubel, Feb 26 2019

I:=[0,6,4,30,3,70,24,126,10,198,60,286]; [n le 12 select I[n] else 3*Self(n-4)-3*Self(n-8)+Self(n-12): n in [1..60]]; // Vincenzo Librandi, Apr 23 2017

Table[Which[OddQ@ n, 2 (1 + 2 #) (3 + 2 #) &[(n - 1)/2], Mod[n, 4] == 0, # (1 + 2 #) &[n/4], True, 4 (1 + #) (1 + 2 #) &[(n - 2)/4]], {n, 0, 60}] (* or *)
CoefficientList[Series[x(6 +4x +30x^2 +3x^3 +52x^4 +12x^5 +36x^6 +x^7 +6x^8 -2x^10)/((1-x)^3*(1+x)^3*(1+x^2)^3), {x, 0, 60}], x] (* Michael De Vlieger, Apr 22 2017 *)
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1}, {0,6,4,30,3,70,24,126,10,198,60, 286}, 80] (* Vincenzo Librandi, Apr 23 2017 *)

m=60; v=concat([0,6,4,30,3,70,24,126,10,198,60,286], vector(m-12)); for(n=13, m, v[n]=3*v[n-4]-3*v[n-8]+v[n-12]); v \\ G. C. Greubel, Feb 26 2019

(x*(6+4*x+30*x^2+3*x^3+52*x^4+12*x^5+36*x^6+x^7+6*x^8-2*x^10)/((1-x)^3 *(1+x)^3*(1+x^2)^3 )).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n) = A061037(n+2) + A181318(n). - Paul Curtz, Jul 19 2011
a(n) = A060819(n) * A145979(n). - Paul Curtz, Sep 06 2011
G.f.: x*(6+4*x+30*x^2+3*x^3+52*x^4+12*x^5+36*x^6+x^7+6*x^8-2*x^10) /( (1-x)^3 *(1+x)^3 *(1+x^2)^3 ). - R. J. Mathar, Jun 17 2011
Let BEB(n) = a(n)/A061038(n+2) = A060819(n)/A145979(n). Then (BEB(n))^2 = A181318(n)/A061038(n+2) = BEB(n) - A061037(n+2)/A061038(n+2). - Paul Curtz, Jul 19 2011, index corrected by R. J. Mathar, Sep 09 2011
From Luce ETIENNE, Apr 18 2017: (Start)
a(n) = n*(n + 2)*(37 - 27*(-1)^n - 3*((-1)^((2*n + 1 - (-1)^n)/4) + (-1)^((2*n - 1 + (-1)^n)/4)))/32.
a(n) = n*(n+2)*(37-27*cos(n*Pi) - 6*cos(n*Pi/2))/32.
a(n) = n*(n + 2)*(37 - 27*(-1)^n - 3*(i^n + (-i)^n))/32, where i=sqrt(-1). (End)

A165727 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2 with T(0,n) = T(k,0) = 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 4, 0, 0, 9, 1, 9, 0, 0, 16, 36, 36, 16, 0, 0, 25, 16, 1, 16, 25, 0, 0, 36, 100, 144, 144, 100, 36, 0, 0, 49, 9, 225, 1, 225, 9, 49, 0, 0, 64, 196, 12, 400, 400, 12, 196, 64, 0, 0, 81, 64, 441, 144, 1, 144, 441, 64, 81, 0, 0, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 0

Offset: 0

Paul Curtz, Sep 25 2009

A synopsis of the denominators of the transitions in the Rydberg-Ritz spectrum of hydrogenic atoms.

			0,  0,   0,   0,    0,    0,    0,    0,    0,    0, ... A000004
0,  1,   4,   9,   16,   25,   36,   49,   64,   81, ... A000290
0,  4,   1,  36,   16,  100,    9,  196,   64,  324, ... A061038
0,  9,  36,   1,  144,  225,   12,  441,  576,   81, ... A061040
0, 16,  16, 144,    1,  400,  144,  784,   64, 1296, ... A061042
0, 25, 100, 225,  400,    1,  900, 1225, 1600, 2025, ... A061044
0, 36,   9,  12,  144,  900,    1, 1764,  576,  324, ... A061046
0, 49, 196, 441,  784, 1225, 1764,    1, 3136, 3969, ... A061048
0, 64,  64, 576,   64, 1600,  576, 3136,    1, 5184, ... A061050
0, 81, 324,  81, 1296, 2025,  324, 3969, 5184,    1, ...

Alois P. Heinz, Table of n, a(n) for n = 0..1034

Cf. A165441 (top row and left column removed)

T:= (k,n)-> `if` (n=0 or k=0, 0, denom (1/min (n,k)^2 -1/max (n, k)^2)):
seq (seq (T (k, d-k), k=0..d), d=0..11);

Edited by R. J. Mathar, Feb 27 2010, Mar 03 2010

A168068 Array T(n,k) read by antidiagonals: T(n,2k+1) = 2k+1. T(n,2k) = 2^n*k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 2, 0, 1, 8, 3, 4, 5, 0, 1, 16, 3, 8, 5, 3, 0, 1, 32, 3, 16, 5, 6, 7, 0, 1, 64, 3, 32, 5, 12, 7, 4, 0, 1, 128, 3, 64, 5, 24, 7, 8, 9, 0, 1, 256, 3, 128, 5, 48, 7, 16, 9, 5, 0, 1, 512, 3, 256, 5, 96, 7, 32, 9, 10, 11, 0, 1, 1024, 3, 512, 5, 192, 7, 64, 9, 20, 11, 6, 0, 1, 2048, 3, 1024, 5

Offset: 0

Paul Curtz, Nov 18 2009

The array is constructed multiplying the even-indexed A026741(k) by 2^n, and keeping the odd-indexed A026471(k) as they are.

Connections to the hydrogen spectrum: The squares of the second row are T(1,k)^2 = A001477(k)^2 = A000290(k) which are the denominators of the Lyman lines (see A171522). The squares of the row T(2,k) are in A154615, denominators of the Balmer series. Row T(3,k) is related to A106833 and A061038.

			The array starts in row n=0 with columns k>=0 as:
0,1,1,3,2,5,3,7,4, A026741
0,1,2,3,4,5,6,7,8, A001477
0,1,4,3,8,5,12,7,16, A022998
0,1,8,3,16,5,24,7,32, A144433
0,1,16,3,32,5,48,7,64,
0,1,32,3,64,5,96,7,128,

A168068 := proc(n,k) if type(k,'odd') then k; else 2^(n-1)*k ; end if; end proc: # R. J. Mathar, Jan 22 2011

A174683 Denominator of 1/16 - 1/n^2.

Original entry on oeis.org

0, 16, 16, 144, 16, 400, 144, 784, 64, 1296, 400, 1936, 18, 2704, 784, 3600, 256, 4624, 1296, 5776, 50, 7056, 1936, 8464, 576, 10000, 2704, 11664, 49, 13456, 3600, 15376, 1024, 17424, 4624, 19600, 81, 21904, 5776, 24336, 1600, 26896, 7056, 29584, 242, 32400, 8464, 35344, 2304, 38416

Offset: 0

Paul Curtz, Nov 30 2010

The value of a(n)=0 is substituted at the pole n=0.

Extends the Bracket spectrum to negative quantum numbers in the fashion of A061038 (1/4-1/n^2) and A181759 (1/9-1/n^2).

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

Cf A174680 (numerators).

Table[If[n == 0, 0, If[n == 4, 16, Denominator[(n^2 - 16)/(4*n)^2]]], {n, 0, 100}] (* G. C. Greubel, Sep 16 2018 *)
Table[Which[n==0,0,n==4,16,True,Denominator[(n^2-16)/(16n^2)]],{n,0,100}] (* Harvey P. Dale, Dec 13 2024 *)

for(n=0,100, print1(if(n==0,0, if(n==4,16, denominator((n^2 - 16)/(4*n)^2))), ", ")) \\ G. C. Greubel, Sep 16 2018

a(n) = A061042(n), n>=4.

a(n) = LCM[n^2 - 16, 16*n^2]/(n^2 - 16), for n>=5. - G. C. Greubel, Sep 16 2018

Removed a(-4)-a(-1) since a(-n)=a(n) by G. C. Greubel, Sep 16 2018

A226044 Period of length 8: 1, 64, 16, 64, 4, 64, 16, 64.

Original entry on oeis.org

1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64, 1, 64, 16, 64, 4, 64, 16, 64

Offset: 0

Paul Curtz, May 24 2013

A002378(n)/A016754(n) gives 0/1, 2/9, 6/25, 12/49, 20/81, 30/121, 42/169, 56/225,..., where A016754(n) = 4*A002378(n) + 1;
A142705(n)/A154615(n+1) gives 0/1, 3/16,  2/9,  15/64,  6/25,  35/144, 12/49,  63/256,..., where A142705(n) = 4*A154615(n+1) + A010685(n);
A061037(n)/A061038(n) gives 0/1, 5/36,  3/16, 21/100, 2/9,   45/196, 15/64,  77/324,..., where A061038(n) = 4*A061037(n) + A177499(n);
A225948(n)/A226008(n) gives 0/1, 9/100, 5/36, 33/196, 3/16,  65/324, 21/100, 105/484,..., where A226008(n) = 4*A225948(n) + a(n).
See also the triangle in Example lines.

			Triangle in which the terms of each line are repeated:
A000012: 1,   ...
A010685: 1,   4,  ...
A177499: 1,  16,  4,  16,  ...
A226044: 1,  64, 16,  64,  4,  64, 16,  64, ...
         1, 256, 64, 256, 16, 256, 64, 256, 4, 256, 64, 256, 16, 256, 64, 256, ...

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

Cf. A010685, A177499, A205383, A225948.

Table[{1, 64, 16, 64, 4, 64, 16, 64}, {7}] // Flatten (* Jean-François Alcover, May 24 2013 *)

a(n) = A205383(n+7)^2.

G.f.: (1+64*x+16*x^2+64*x^3+4*x^4+64*x^5+16*x^6+64*x^7)/((1-x)*(1+x)*(1+x^2)*(1+x^4)). [Bruno Berselli, May 25 2013]

cp's OEIS Frontend

A154615 a(n) = A022998(n)^2.

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A177427 Numerators of the Inverse Akiyama-Tanigawa transform of the aerated even-indexed Bernoulli numbers 1, 0, 1/6, 0, -1/30, 0, 1/42, ...

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A226008 a(0) = 0; for n>0, a(n) = denominator(1/4 - 4/n^2).

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A176672 a(2*n) = 1 + 6*n, a(2*n+1) = A165367(n).

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A165441 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2.

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A191567 Four interlaced 2nd order polynomials: a(4*k) = k*(1+2*k); a(1+2*k) = 2*(1+2*k)*(3+2*k); a(2+4*k) = 4*(1+k)*(1+2*k).

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A165727 Table T(k,n) read by antidiagonals: denominator of 1/min(n,k)^2 -1/max(n,k)^2 with T(0,n) = T(k,0) = 0.

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A176672 a(2n) = 1 + 6n, a(2*n+1) = A165367(n).

A191567 Four interlaced 2nd order polynomials: a(4k) = k(1+2k); a(1+2k) = 2(1+2k)(3+2k); a(2+4k) = 4(1+k)(1+2k).