A001859 -id:A001859 - cp's OEIS frontend

A359860 Number of regions among all distinct circles that can be constructed from a 2 X n square grid of points using only a compass.

3, 45, 231, 865, 2081, 4489, 8211, 14401, 22857, 35445, 51741, 74397, 102271, 138801, 182739, 238181, 303175, 383097, 474995, 586021, 712003, 860829, 1028225, 1223773, 1440593, 1689993, 1965525, 2279509, 2622993, 3011405, 3433615, 3907241, 4419261, 4988781, 5603271

Offset: 1

Scott R. Shannon, Jan 16 2023

nonn

See A359859 for further details. No formula for a(n) is known.

Lucas A. Brown, Python program.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 8.

Cf. A359859 (vertices), A359861 (edges), A359862 (k-gons), A001859, A359253.

a(n) = A359861(n) - A359859(n) + 1 by Euler's formula.

a(19)-a(35) from Lucas A. Brown, Oct 11 2024

A359862 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from a 2 x n square grid of points using only a compass.

Original entry on oeis.org

3, 0, 16, 29, 0, 102, 117, 10, 2, 4, 368, 402, 64, 26, 1, 12, 860, 903, 252, 52, 0, 2, 12, 1812, 2028, 520, 110, 4, 3, 24, 3168, 3841, 960, 204, 8, 6, 32, 5420, 6804, 1748, 362, 24, 11, 44, 8388, 10987, 2826, 552, 46, 14, 56, 12808, 17122, 4448, 922, 72, 17, 64, 18348, 25257, 6594, 1370, 82, 26

Offset: 1

Scott R. Shannon, Jan 16 2023

See A359859 and A359860 for further details and images of the circles.

			The table begins:
   3;
   0,    16,    29;
   0,   102,   117,   10,    2;
   4,   368,   402,   64,   26,  1;
  12,   860,   903,  252,   52,  0,  2;
  12,  1812,  2028,  520,  110,  4,  3;
  24,  3168,  3841,  960,  204,  8,  6;
  32,  5420,  6804, 1748,  362, 24, 11;
  44,  8388, 10987, 2826,  552, 46, 14;
  56, 12808, 17122, 4448,  922, 72, 17;
  64, 18348, 25257, 6594, 1370, 82, 26;
  ...

Scott R. Shannon, Image for n = 7.

Cf. A359859 (vertices), A359860 (regions), A359861 (edges), A001859, A359258.

Sum of row n = A359860(n).

A003984 Table of max(x,y), where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 13, 12, 11, 10, 9, 8, 7, 7, 8

Offset: 0

Marc LeBrun

			Top left corner of array:
0 1 2 3
1 1 2 3
2 2 2 3
3 3 3 3

Cf. A051125, A003056, A004197.

Antidiagonal sums are in A001859.

From  Franklin T. Adams-Watters, Feb 06 2006: (Start)
G.f.: (x+y-3xy+x^2 y^2)/((1-x)^2 (1-y)^2 (1-xy)).
T(n,m) = n + m - min(n,m); a(n) = A003056(n) - A004197(n). (End)
a(n) = (1/2)*(t - 1 + abs(t^2 - 2*n - 1)), where t = floor(sqrt(2*n+1)+1/2). - Ridouane Oudra, May 03 2019

A102214 Expansion of (1 + 4x + 4x^2)/((1+x)*(1-x)^3).

Original entry on oeis.org

1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900

Offset: 0

Creighton Dement, Feb 17 2005

A floretion-generated sequence.
a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
Number of different partitions of numbers x + y = z such that {x,y,z} are integers {1,2,3,...,3n} and z > y > x. - Artur Jasinski, Feb 09 2010
Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011
a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020
One odd followed by three evens.
From Paul Curtz, Mar 06 2020: (Start)
b(n) = 0, 1, 6, 16,  30, 49, ... = 0, a(n).
(  25, 12,  4, 0,  1,  6,  16, 30, ...
  -13, -8, -4  1,  5, 10,  14, 19, ...
    5,  4,  5, 4,  5,  4,   5,  4, ... .)
b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
A154589(n) are in the main diagonal of b(n) and b(-n). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.

Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.

[(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011

aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (* Artur Jasinski, Feb 09 2010 *)

a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ Charles R Greathouse IV, Apr 16 2020

G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
a(2n) = A016778(n) = (3n+1)^2.
a(n) + a(n+1) = A038764(n+1).
a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - Bruno Berselli, Oct 25 2011
a(n) = A198392(n) + A198392(n-1). - Bruno Berselli, Nov 06 2011
From Paul Curtz, Mar 04 2020: (Start)
a(n) = A006578(n) + A001859(n) + A077043(n+1).
a(n) = A274221(2+2*n).
a(20+n) - a(n) = 30*(32+3*n).
a(1+2*n) = 3*(1+n)*(2+3*n).
a(n) = A047237(n) * A047251(n).
a(n) = A001651(n+1) * A032766(n).(End)
E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

Definition rewritten by Bruno Berselli, Oct 25 2011

A115006 Row 2 of array in A114999.

Original entry on oeis.org

0, 3, 8, 16, 26, 39, 54, 72, 92, 115, 140, 168, 198, 231, 266, 304, 344, 387, 432, 480, 530, 583, 638, 696, 756, 819, 884, 952, 1022, 1095, 1170, 1248, 1328, 1411, 1496, 1584, 1674, 1767, 1862, 1960, 2060, 2163, 2268, 2376, 2486, 2599, 2714, 2832, 2952, 3075, 3200

Offset: 0

N. J. A. Sloane, Feb 23 2006

Number of lattice points (x,y) in the region of the coordinate plane bounded by y < 3x+1, y > x/2 and x <= n. - Wesley Ivan Hurt, Oct 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A114999, A000217 (triangular numbers), A002620 (quarter-squares), A001859 (triangular numbers plus quarter-squares), A017305 (10n+3), A147874 (zero followed by partial sums of A017305).

Partial Sums of A047218.

[ n*(n+1) + (n+1)^2 div 4: n in [0..50] ];

A115006:=n->(10*n^2 + 12*n + 1 - (-1)^n)/8: seq(A115006(n), n=0..50); # Wesley Ivan Hurt, Oct 27 2014

Table[(10*n^2 + 12*n + 1 - (-1)^n)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 27 2014 *)
LinearRecurrence[{2,0,-2,1},{0,3,8,16},60] (* Harvey P. Dale, Jan 13 2015 *)

{for(n=0, 50, print1(n*(n+1)+floor((n+1)^2/4), ","))}

a(n) = floor((n+1)^2/4)+n*(n+1).
G.f.: x*(2*x+3)/((1-x)^3*(1+x)).
From Wesley Ivan Hurt, Oct 27 2014: (Start)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = (10*n^2 + 12*n + 1 - (-1)^n)/8.
a(n) = Sum_{i=1..n+1} (10*i + (-1)^i - 9)/4. (End)
E.g.f.: (x*(11 + 5*x)*cosh(x) + (1 + 11*x + 5*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Edited by Klaus Brockhaus, Nov 18 2008

A359861 Number of edges among all distinct circles that can be constructed from a 2 X n square grid of points using only a compass.

Original entry on oeis.org

4, 84, 420, 1604, 3904, 8444, 15524, 27356, 43540, 67720, 99088, 142912, 196820, 267580, 352844, 460432, 586592, 741852, 920528, 1136888, 1382360, 1672384, 1998964, 2380940, 2804292, 3291376, 3830048, 4444176, 5116128, 5876580, 6703220, 7631352, 8634796, 9751064, 10956320

Offset: 1

Scott R. Shannon, Jan 16 2023

nonn

See A359859 and A359860 for further details and images of the circles. No formula for a(n) is known.

			a(n) = A359859(n) + A359860(n) - 1 by Euler's formula.

Lucas A. Brown, Python program.

Cf. A359859 (vertices), A359860 (regions), A359862 (k-gons), A001859, A359254.

a(19)-a(35) from Lucas A. Brown, Oct 11 2024

A185212 a(n) = 12n^2 - 8n + 1.

Original entry on oeis.org

1, 5, 33, 85, 161, 261, 385, 533, 705, 901, 1121, 1365, 1633, 1925, 2241, 2581, 2945, 3333, 3745, 4181, 4641, 5125, 5633, 6165, 6721, 7301, 7905, 8533, 9185, 9861, 10561, 11285, 12033, 12805, 13601, 14421, 15265, 16133, 17025, 17941, 18881, 19845, 20833

Offset: 0

Reinhard Zumkeller, Dec 20 2012

Sequence found by reading the line from 1, in the direction 1, 5, and the same line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, May 08 2018

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Square Block Rays
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

For n > 0: odd terms in A001859.
Cf. A001082.
Cf. A016754, A000384.

Haskell
```
a185212 = (+ 1) . (* 4) . a000567
```

Table[12n^2-8n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,5,33},50] (* Harvey P. Dale, Jul 07 2015 *)

a(n)=12*n^2-8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*A000567(n) + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0)=1, a(1)=5, a(2)=33. - Harvey P. Dale, Jul 07 2015
G.f.: (-1 - 2*x - 21*x^2)/(-1+x)^3. - Harvey P. Dale, Jul 07 2015
E.g.f.: (12*x^2 + 4*x + 1)*exp(x). - G. C. Greubel, Jun 25 2017
a(n) = A016754(n-1) + 4*A000384(n). - Leo Tavares, May 21 2022
From Amiram Eldar, May 28 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/8 - 3*log(3)/8 + 1.
Sum_{n>=0} (-1)^n/a(n) = Pi/8 - sqrt(3)*arccoth(sqrt(3))/2 + 1. (End)

A254708 Expansion of (1 + 2x^2) / (1 - 2x^2 - x^3 + 2x^5 + 2x^6 - x^8 - 2*x^9 + x^11) in powers of x.

Original entry on oeis.org

2, 0, 5, 2, 10, 5, 18, 10, 29, 18, 43, 29, 62, 43, 85, 62, 113, 85, 147, 113, 187, 147, 233, 187, 287, 233, 348, 287, 417, 348, 495, 417, 582, 495, 678, 582, 785, 678, 902, 785, 1030, 902, 1170, 1030, 1322, 1170, 1486, 1322, 1664, 1486, 1855, 1664, 2060, 1855

Offset: 0

Michael Somos, Feb 06 2015

The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+7 = x+u, (u+v < x+w and x+u+v+w is even) or (u+v > x+w and x+u+v+w is odd).

			G.f. = 2 + 5*x^2 + 2*x^3 + 10*x^4 + 5*x^5 + 18*x^6 + 10*x^7 + 29*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1).

Cf. A001859, A254707, A254594.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2 + x^2)/(1-2*x^2-x^3+2*x^5+2*x^6-x^8-2*x^9+x^11))); // G. C. Greubel, Aug 03 2018

a[ n_] := Quotient[ n^3 + If[ OddQ[n], 10 n^2 + 21 n + 12, 19 n^2 + 108 n + 192], 96];
a[ n_] := Module[{m = n}, SeriesCoefficient[ If[ n < 0, m = -9 - n; -1 - 2 x^2, 2 + x^2]/ ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 7, (u + v < x + w && x + u + v + w == 2 k) || (u + v > x + w && x + u + v + w == 2 k + 1)}, {x, u, v, w, k}, Integers, 10^9];
CoefficientList[Series[(2 + x^2)/(1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11), {x,0,50}], x] (* G. C. Greubel, Apr 14 2017 *)
LinearRecurrence[{0,2,1,0,-2,-2,0,1,2,0,-1},{2,0,5,2,10,5,18,10,29,18,43},60] (* Harvey P. Dale, Mar 13 2023 *)

{a(n) = (n^3 + if( n%2, 10*n^2 + 21*n + 12, 19*n^2 + 108*n + 192)) \ 96};

{a(n) = polcoeff( if( n<0, n = -9-n; -1 - 2*x^2, 2 + x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

G.f.: (2 + x^2) / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
0 = a(n) + a(n+1) - a(n+2) - 2*a(n+3) - 2*a(n+4) + 2*a(n+6) + 2*a(n+7) + a(n+8) - a(n+9) - a(n+10) + 3 for all n in Z.
a(n+3) - a(n) = 0 if n even else A001859((n+5)/2) for all n in Z.
a(n) = A254594(n-2) + 2*A254594(n) for all n in Z.
a(n) = -A254707(-9 - n) for all n in Z.

A330707 a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4.

Original entry on oeis.org

0, 1, 3, 7, 13, 20, 28, 38, 50, 63, 77, 93, 111, 130, 150, 172, 196, 221, 247, 275, 305, 336, 368, 402, 438, 475, 513, 553, 595, 638, 682, 728, 776, 825, 875, 927, 981, 1036, 1092, 1150, 1210, 1271, 1333, 1397, 1463

Offset: 0

Paul Curtz, Dec 27 2019

Essentially four odds followed by four evens.
Last digit is neither 4 nor 9.
Essentially twice or twin sequences in the hexagonal spiral from A002265.
                  21  21  21  22  22  22  22
                21  14  14  14  14  15  15  23
              20  13   8   8   8   9   9  15  23
            20  13   8   4   4   4   4   9  15  23
          20  13   7   3   1   1   1   5   9  16  23
        20  13   7   3   1   0   0   2   5  10  16  24
          19  12   7   3   0   0   2   5  10  16  24
            19  12   7   3   2   2   5  10  16  24
              19  12   6   6   6   6  10  17  24
                19  12  11  11  11  11  17  25
                  18  18  18  18  17  17  25
.
There are 12 twin sequences. 6 of them (A001859, A006578, A077043, A231559, A024219, A281026) are in the OEIS. a(n) is the seventh.
  0, 1, 3, 7, 13, 20, 28, 38, 50, ...
  1, 2, 4, 6,  7,  8, 10, 12, 13, ...
  1, 2, 2, 1,  1,  2,  2,  1,  1, ... period 4. See A014695.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A001859, A002265, A006578, A014695, A016921, A024219, A033579, A077043, A231559, A281026.

[(3*n^2+n-1+ (-1)^Floor(n/2))/4: n in [0..60]]; // G. C. Greubel, Dec 30 2019

seq((3*n^2+n-1+sqrt(2)*sin((2*n+1)*Pi/4))/4, n = 0..60); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{3,-4,4,-3,1}, {0,1,3,7,13}, 60] (* Amiram Eldar, Dec 27 2019 *)

concat(0, Vec(x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^60))) \\ Colin Barker, Dec 27 2019

[(3*n^2+n-1+(-1)^floor(n/2))/4 for n in (0..60)] # G. C. Greubel, Dec 30 2019

a(n) = A231559(-n).
a(1+2*n) + a(2+2*n) = A033579(n+1).
a(40+n) - a(n) = 1210, 1270, 1330, 1390, 1450, ... . See 10*A016921(n).
From Colin Barker, Dec 27 2019: (Start)
G.f.: x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.
(End)
E.g.f.: (cos(x) + sin(x) + (-1 + 4*x + 3*x^2)*exp(x))/4. - Stefano Spezia, Dec 27 2019
a(n) = ( 3*n^2 + n - 1 + sqrt(2)*sin((2*n+1)*Pi/4) )/4 = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4. - G. C. Greubel, Dec 30 2019

A210797 Triangle of coefficients of polynomials u(n,x) jointly generated with A210798; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 4, 5, 3, 1, 6, 10, 10, 5, 1, 7, 16, 22, 18, 8, 1, 9, 24, 42, 47, 33, 13, 1, 10, 33, 69, 98, 95, 59, 21, 1, 12, 44, 108, 182, 220, 188, 105, 34, 1, 13, 56, 156, 308, 444, 472, 363, 185, 55, 1, 15, 70, 220, 490, 818, 1034, 985, 690, 324, 89, 1

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
Column 2: A032766
Column 3: A001859
Row sums: A099232
Alternating row sums: A008346
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...1
1...3...2
1...4...5....3
1...6...10...10...5
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + 2x^2.

Cf. A210798, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210797 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210798 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A099232 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A006130 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A008346 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A039834 *)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

cp's OEIS Frontend

A359860 Number of regions among all distinct circles that can be constructed from a 2 X n square grid of points using only a compass.

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A359862 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from a 2 x n square grid of points using only a compass.

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A003984 Table of max(x,y), where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

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A102214 Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).

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Magma

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A115006 Row 2 of array in A114999.

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A359861 Number of edges among all distinct circles that can be constructed from a 2 X n square grid of points using only a compass.

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A185212 a(n) = 12*n^2 - 8*n + 1.

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A254708 Expansion of (1 + 2*x^2) / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11) in powers of x.

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Examples

A102214 Expansion of (1 + 4x + 4x^2)/((1+x)*(1-x)^3).

A185212 a(n) = 12n^2 - 8n + 1.

A254708 Expansion of (1 + 2x^2) / (1 - 2x^2 - x^3 + 2x^5 + 2x^6 - x^8 - 2*x^9 + x^11) in powers of x.