A064038 -id:A064038 - cp's OEIS frontend

A165207 Period 4: repeat [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, 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

A181407 a(n) = (n-4)(n-3)2^(n-2).

Original entry on oeis.org

3, 3, 2, 0, 0, 16, 96, 384, 1280, 3840, 10752, 28672, 73728, 184320, 450560, 1081344, 2555904, 5963776, 13762560, 31457280, 71303168, 160432128, 358612992, 796917760, 1761607680, 3875536896, 8489271296, 18522046464, 40265318400, 87241523200, 188441690112

Offset: 0

Paul Curtz, Jan 28 2011

Binomial transform of (3, 0, -1, followed by A005563).
The sequence and its successive differences are:
   3,  3,  2,   0,   0,   16,   96,  384,      a(n),
   0, -1, -2,   0,  16,   80,  288,  896,      A178987,
  -1, -1,  2,  16,  64,  208,  608, 2688,     -A127276,
   0,  3, 14,  48, 144,  400, 1056, 2688,      A176027,
   3, 11, 34,  96, 256,  656, 1632, 3968,      A084266(n+1)
   8, 23, 62, 160, 400,  976, 2336, 5504,
  15, 39, 98, 240, 576, 1360, 3168, 7296.
Division of the k-th column by abs(A174882(k)) gives
   3,  3,  1,  0,  0,  1,  3,  3,  5,  15,  21, 14,   A064038(n-3),
   0, -1, -1,  0,  1,  5,  9,  7, 10,  27,  35, 22,   A160050(n-3),
  -1, -1,  1,  2,  4, 13, 19, 13, 17,  43,  53, 32,   A176126,
   0,  3,  7,  6,  9, 25, 33, 21, 26,  63,  75, 44,   A178242,
   3, 11, 17, 12, 16, 41, 51, 31, 37,  87, 101, 58,
   8  23, 31, 20, 25, 61, 73, 43, 50, 115, 131, 74,
  15, 39, 49, 30, 36, 85, 99, 57, 65, 147, 165, 92.
Columns are (or from) A005563, A142463, A056220, A002378, A000290, A001844, A058331, A002061, A002522, A097080, A093328, A014206.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A176027, A181318.

List([0..40], n-> (n-4)*(n-3)*2^(n-2)); # G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2): n in [0..40] ]; // Vincenzo Librandi, Feb 01 2011

Table[(n-4)*(n-3)*2^(n-2), {n,0,40}] (* G. C. Greubel, Feb 21 2019 *)

vector(40, n, n--; (n-4)*(n-3)*2^(n-2)) \\ G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2) for n in (0..40)] # G. C. Greubel, Feb 21 2019

a(n) = 16*A001788(n-4).
a(n+1) - a(n) = A178987(n).
G.f.: (3 - 15*x + 20*x^2) / (1-2*x)^3. - R. J. Mathar, Jan 30 2011
E.g.f.: (x^2 - 3*x + 3)*exp(2*x). - G. C. Greubel, Feb 21 2019

A182868 a(n) = -1 + n + 4*n^2.

Original entry on oeis.org

-1, 4, 17, 38, 67, 104, 149, 202, 263, 332, 409, 494, 587, 688, 797, 914, 1039, 1172, 1313, 1462, 1619, 1784, 1957, 2138, 2327, 2524, 2729, 2942, 3163, 3392, 3629, 3874, 4127, 4388, 4657, 4934, 5219, 5512, 5813, 6122, 6439, 6764, 7097, 7438, 7787, 8144, 8509, 8882, 9263, 9652

Offset: 0

Paul Curtz, Feb 01 2011

First quadrisection of A176126(n). Take clockwise (square) spiral from A023443(n)=n-1: a(n) is on the negative x-axis. Fourth quadrisection (-1-n+4*n^2) is on the negative y-axis.
Conjecture: the 4 quadrisections of (the family) A064038, A160050, A176126, A178242 (see A181407) come from square spiral.
a(n) mod 9 has period 9: 8,4,8,2,4,5,5,4,2. a(n) mod 10 has period 10: 9,4,7,8,7,4,9,2,3,2. Each polynomial modulo some constant c has a period of length c (and perhaps shorter ones). - Paul Curtz and Bruno Berselli, Feb 05 2011

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[-1+n+4*n^2: n in [0..700] ] // Vincenzo Librandi, Feb 01 2011

f[n_]:=-1+n+4*n^2;f[Range[0,100]] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n)=-1+n+4*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A176126(4*n).
a(n) = 4*n^2 + n - 1.
a(n) = a(n-1) - 3 + 8*n.
a(n) = 2*a(n) - a(n-2) + 8.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(1 - 7*x - 2*x^2)/(1-x)^3. - Bruno Berselli, Feb 05 2011

A227316 a(n) = n(n+1) if n == 0 or 1 (mod 4), otherwise a(n) = n(n+1)/2.

Original entry on oeis.org

0, 2, 3, 6, 20, 30, 21, 28, 72, 90, 55, 66, 156, 182, 105, 120, 272, 306, 171, 190, 420, 462, 253, 276, 600, 650, 351, 378, 812, 870, 465, 496, 1056, 1122, 595, 630, 1332, 1406, 741, 780, 1640, 1722, 903, 946, 1980, 2070, 1081, 1128

Offset: 0

Paul Curtz, Jul 06 2013

			a(0) = 2*0 = 0, a(1) = 2*1 = 2, a(2) = 1*3 = 3, a(3) = 1*6 = 6, a(4) = 2*10 = 20.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A000217, A002378, A130658, A169642 (first bisection), A176743, A109043, A227380.

[(3+(-1)^Floor(n/2))*n*(n+1)/4: n in [0..50]]; // Bruno Berselli, Jul 10 2013

a[n_] := n*(n+1)/4*GCD[n-1, 4]*GCD[n, 4]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 10 2013 *)
Table[If[Mod[n,4]<2,n(n+1),(n(n+1))/2],{n,0,50}] (* or *) LinearRecurrence[ {3,-6,10,-12,12,-10,6,-3,1},{0,2,3,6,20,30,21,28,72},50] (* Harvey P. Dale, Aug 26 2016 *)

a(n) = A130658(n+2)*A000217(n), a(-n-1) = A130658(n)*A000217(n).
a(2n) = A169642(n), a(2n+1) = 2*(2*n+1)*A026741(n+1).
a(n) = A176743(n-2)*A176743(n-1).
a(n) = A177002(n+2)*A064038(n+1).
a(n) = 3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*(n-6) +6*(n-7) -3*a(n-8) +a(n-9) = 3*a(n-4) -3*a(n-8) +a(n-12).
G.f.: x*(2-3*x+9*x^2+3*x^5+x^6)/((1-x)^3*(1+x^2)^3). - Bruno Berselli, Jul 10 2013
a(n) = (3+(-1)^floor(n/2))*n*(n+1)/4. - Bruno Berselli, Jul 10 2013
Sum_{n>=1} 1/a(n) = 1 + log(2)/2. - Amiram Eldar, Aug 12 2022

A086314 Total number of edges in the distinct simple graphs on n nodes.

Original entry on oeis.org

0, 1, 6, 33, 170, 1170, 10962, 172844, 4944024, 270116280, 28022441260, 5448008695536, 1969579223350128, 1321964082404214704, 1649890513414726210320, 3840060942271653473695680, 16723638762440239422492944768, 136749695973639295091912681599872

Offset: 1

Eric W. Weisstein, Jul 15 2003

nonn

a(n) = A000088(n)*n(n-1)/4.

a(n) = A000088(n)*A064038(n)/A014695(n).

Eric Weisstein's World of Mathematics, Simple Graph
Eric Weisstein's World of Mathematics, Graph Edge

Cf. A000088, A014695, A064038.

<< Combinatorica`; Table[D[GraphPolynomial[n, x], x] /. x -> 1, {n, 18}]  (* Geoffrey Critzer, Sep 29 2012 *)
<< Combinatorica`; Table[Binomial[n, 2] NumberOfGraphs[n]/2, {n, 18}] (* Eric W. Weisstein, May 17 2017 *)

A185138 a(4n) = n(4n-1); a(2n+1) = n(n+1)/2; a(4n+2) = (2n+1)(4*n+1).

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 15, 6, 14, 10, 45, 15, 33, 21, 91, 28, 60, 36, 153, 45, 95, 55, 231, 66, 138, 78, 325, 91, 189, 105, 435, 120, 248, 136, 561, 153, 315, 171, 703, 190, 390, 210, 861, 231, 473, 253, 1035, 276, 564, 300, 1225, 325

Offset: 0

Paul Curtz, Mar 12 2012

a(n) is divisible by the n-th term of the sequence 3, 3, 1, 1, 3, 3 (periodically repeated with period 6).
a(n) is divisible by b(floor((n-1)/3)), where b(n) = 1, 3, 2, 3, 7, 3, 5, 3, 13, 3, 8, 3, 19, 3,... , n>=0, is defined by inserting a 3 after each entry of A165355.
(n+1)*(n+2)*(n+3)/2=3*A000292(n+1) is divisible by a(n+2), so there is an integer sequence c(n)= 3*A000292(n+1)/a(n+2) = 3, 12, 10, 20, 7, 28, 18,... with c(2*n)=A123167(n+1) and c(n)/A109613(n+2)=A176895(n).
The sequence of denominators of a(n+2)/n has period length 8: 1, 2, 1, 4, 1, 1, 1, 4.
A table T(k,c) = a(1+c*(1+2k)) of (2*k+1)-sections starts as follows:
0  1  1   3   3   15...
0  3  6  45  21   60...
0 15 15  60  55  325...
0 14 28 231 105  315...
0 45 45 189 171 1035...
The table of T'(k,c) = T(k,c)/(2k+1), columns c>=0, looks as follows, construction similar to A165943:
0 1 1  3  3  15  6  14   k=0
0 1 2 15  7  20 15  77   k=1
0 3 3 12 11  65 24  63   k=2
0 2 4 33 15  45 33 175   k=3
0 5 5 21 19 115 42 112   k=4
0 3 6 51 23  70 51 273   k=5
The entries T'(k,c) are divisible by A060819(c).
Differences are T'(2,c)-T'(0,c) = T'(4,c)-T'(2,c) = 0, 2, 2, 9, 8, 50, 18, 49, 32, ... which is A168077(c) multiplied by the c-th term of the period-4 sequence 2, 2, 2, 1.
Differences are T'(3,c)- T'(1,c) = T'(5,c)-T'(3,c) = 0, 1, 2, 18, 8, 25, 18, 98, 32,... which is  A168077(c) multiplied by the period-4 sequence 2, 1, 2, 2.
The reduced fractions T'(0,c)/T'(1,c) = 1, 1/2, 1/5, 3/7, 3/4, 2/5, 2/11, 5/13, 5/7, 3/8, 3/17, 7/19, .., c>=1, have a numerator sequence A026741(floor(c/2)+1). The denominator sequence is f(c) = 1, 2, 5, 7, 4, 5,.. = A001651(c+1)/A130658(c+1), with f(2*c+1) +f(2*c+2) = 3, 12, 9, 24 .. =3*A022998(c).

G. C. Greubel, 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. A000217, A014634, A026741, A033991, A064038, A060819, A165355, A208950.

A185138 := proc(n)
        if n mod 4 = 0 then
                return n/4*(n-1) ;
        elif n mod 2 = 1 then
                return (n-1)*(n+1)/8 ;
        else
                return (n-1)*n/2 ;
        end if;
end proc: # R. J. Mathar, Apr 05 2012

Clear[b];b[1] = 0; b[2] = 0; b[3] = 1; b[4] = 1; b[5] = 3; b[6] = 3; b[7] = 15;b[8] = 6;b[n_Integer] := b[n] = ((-2 + n) (-4 (-4 + n) (-3 + n) (-2 + n) (8 + n (-9 + 2 n)) b[-3 + n] + (-5 + n) ((-3 +n) ((-4 + n) (211 + 2 n (-215 + n (147 + n (-41 + 4 n)))) - 4 (-1 + n) (19 + n (-13 + 2 n)) b[-2 + n]) - 4 (-4 + n)^2 (8 + n (-9 + 2 n)) b[-1 + n])))/(4 (-5 + n) (-4 + n) (-3 + n)^2 (19 + n (-13 + 2 n)))
a = Table[b[n], {n, 1, 52}] (* Roger L. Bagula, Mar 14 2012 *)
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{0,0,1,1,3,3,15,6,14,10,45,15},60] (* Harvey P. Dale, Nov 23 2015 *)

x='x+O('x^50); concat([0,0], Vec(-x^2*(3*x^8+x^7+5*x^6+3*x^5+12*x^4+3*x^3+3*x^2+x+1)/ ((x-1)^3*(x+1)^3*(x^2+1)^3))) \\ G. C. Greubel, Jun 23 2017

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(2*n) = A064038(2*n), a(2*n+1) = A000217(n).
a(n) = 3*A208950(n)/A109613(n).
a(n+1) = A060819(n) * A026741(n+2)(floor(n/2)).
G.f.: -x^2*(3*x^8+x^7+5*x^6+3*x^5+12*x^4+3*x^3+3*x^2+x+1)/ ((x-1)^3*(x+1)^3*(x^2+1)^3). - R. J. Mathar, Mar 22 2012
a(n) = (4*n^2-3*n-1+(2*n^2-3*n+1)*(-1)^n + n*(n-1)*(1+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4))/16. - Luce ETIENNE, May 13 2016
Sum_{n>=2} 1/a(n) = 2 - Pi/4 + 7*log(2)/2. - Amiram Eldar, Aug 12 2022

A177049 Numerator of (3n+1)*(3n+2)/4.

Original entry on oeis.org

1, 5, 14, 55, 91, 68, 95, 253, 325, 203, 248, 595, 703, 410, 473, 1081, 1225, 689, 770, 1711, 1891, 1040, 1139, 2485, 2701, 1463, 1580, 3403, 3655, 1958, 2093, 4465, 4753, 2525, 2678, 5671, 5995, 3164, 3335, 7021, 7381, 3875, 4064, 8515, 8911

Offset: 0

Paul Curtz, Dec 09 2010

A trisection of A064038.

Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A064038, A127922.

Table[Numerator[(3 n + 1) (3 n + 2)/4], {n, 0, 50}] (* Wesley Ivan Hurt, Jun 14 2014 *)
LinearRecurrence[{3,-6,10,-12,12,-10,6,-3,1},{1,5,14,55,91,68,95,253,325},50] (* Harvey P. Dale, Jan 18 2020 *)

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^2+4*x+1)*(x^6-2*x^5+12*x^4-13*x^3+12*x^2-2*x+1) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Dec 12 2010
The conjecture is correct. - Charles R Greathouse IV, Feb 08 2012
a(n) ~ -27/8*n^2 - 27/8*n. - Ralf Stephan, Jun 16 2014
Sum_{n>=0} 1/a(n) = (4/(3*sqrt(3)) - 1/3)*Pi. - Amiram Eldar, Aug 13 2022

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

A215189 Array t(n,k) of the family ((n+k)/gcd(n+k,4))*(n/gcd(n,4)), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 9, 3, 3, 0, 1, 3, 1, 1, 0, 25, 5, 15, 5, 5, 0, 9, 15, 3, 9, 3, 3, 0, 49, 21, 35, 7, 21, 7, 7, 0, 4, 14, 6, 10, 2, 6, 2, 2, 0, 81, 18, 63, 27, 45, 9, 27, 9, 9, 0, 25, 45, 10, 35, 15, 25, 5, 15, 5, 5, 0, 121, 55, 99, 22, 77, 33, 55, 11, 33, 11, 11, 0, 9, 33, 15, 27, 6, 21, 9, 15, 3, 9, 3, 3, 0

Offset: 0

Jean-François Alcover, Jun 12 2013

Identification of rows and columns:
Row 2, n=1: A060819,
row 3, n=2: A060819 (shifted),
row 4, n=3: A068219,
row 5, n=4: A060819 (shifted),
row 6, n=5: A060819 (shifted and multiplied by 5),
row 7, n=6: A068219 (shifted),
row 8, n=7: A060819 (shifted and multiplied by 7);
column 1, k=0: A181318,
column 2, k=1: A064038,
column 3, k=2: A198148,
column 4, k=3: A160050,
column 5, k=4: A061037,
column 6, k=5: A178242,
column 7, k=6: A217366,
column 8, k=7: A217367.
This array is the transposition of the array given by Paul Curtz in the comments in A181318.

			Array begins:
   0,  0,  0,  0,  0,  0,  0, ...
   1,  1,  3,  1,  5,  3,  7, ...
   1,  3,  1,  5,  3,  7,  2, ...
   9,  3, 15,  9, 21,  6, 27, ...
   1,  5,  3,  7,  2,  9,  5, ...
  25, 15, 35, 10, 45, 25, 55, ...
   9, 21,  6, 27, 15, 33,  9, ...
  49, 14, 63, 35, 77, 21, 91, ...
  ...
Triangle begins:
    0;
    1,  0;
    1,  1,  0;
    9,  3,  3,  0;
    1,  3,  1,  1,  0;
   25,  5, 15,  5,  5,  0;
    9, 15,  3,  9,  3,  3,  0;
   49, 21, 35,  7, 21,  7,  7,  0;
    4, 14,  6, 10,  2,  6,  2,  2,  0;
   81, 18, 63, 27, 45,  9, 27,  9,  9,  0;
   25, 45, 10, 35, 15, 25,  5, 15,  5,  5,  0;
  121, 55, 99, 22, 77, 33, 55, 11, 33, 11, 11,  0;
    9, 33, 15, 27,  6, 21,  9, 15,  3,  9,  3,  3,  0;
  ...

G. C. Greubel, Antidiagonals n=0..100 of triangle, flattened

Cf. A060819, A181318, A064038, A198148, A160050, A061037, A178242, A217366, A217367.

/* As triangle: */ [[(n-k)/GCD(n-k, 4)*n/GCD(n, 4): k in [0..n]]: n in [0..12]]; // Bruno Berselli, Jun 13 2013

t[n_, k_] := (n+k)/GCD[n+k, 4]*n/GCD[n, 4];  Table[t[n-k, k], {n, 0, 12}, {k, 0, n}] // Flatten

t(n,k) = ((n+k)/gcd(n+k,4))*(n/gcd(n,4)).

A227168 a(n) = gcd(2n, n(n+1)/2)^2.

Original entry on oeis.org

1, 1, 36, 4, 25, 9, 196, 16, 81, 25, 484, 36, 169, 49, 900, 64, 289, 81, 1444, 100, 441, 121, 2116, 144, 625, 169, 2916, 196, 841, 225, 3844, 256, 1089, 289, 4900, 324, 1369, 361, 6084, 400

Offset: 1

Paul Curtz, Jul 03 2013

a(n) is defined as A062828(n)^2 for n >= 1. If we extend the sequence to n=0 and negative n by use of the recurrence that relates a(n) to a(n+12), a(n+8) and a(n+4), we obtain a(0)=0, a(-1)=4 and a(-n) = A176743(n-2)^2 for n >= 2.
Define c(n) = a(n+2) - a(n-2) for c >= 0. Because a(n) is a shuffle of three interleaved 2nd-order polynomials, c(n) is a shuffle of three interleaved 1st-order polynomials: c(n) = 4* A062828(n)*(periodically repeated 1, 8, 1, 1).
The sequence a(n) is case p=0 of the family A062828(n)*A062828(n+p):
0, 1, 1, 36,  4, 25,  9, 196, ... = a(n).
0, 1, 6, 12, 10, 15, 42,  56, ... = A130658(n)*A000217(n) = A177002(n-1)*A064038(n+1).
0, 6, 2, 30,  6, 70, 12, 126, ... = 2*A198148(n)
0, 2, 5, 18, 28, 20, 27,  70, ... = A177002(n+2)*A160050(n+1) = A014695(n+2)*A000096(n).

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

Cf. A062828, A016814, A017138, A005565, A006752, A061037, A061038.

[GCD(2*n, n*(n+1)/2)^2: n in [1..50]]; // G. C. Greubel, Sep 20 2018

A227168 := proc(n)
    A062828(n)^2 ;
end proc: # R. J. Mathar, Jul 25 2013

a[n_] := GCD[2*n, n*(n + 1)/2]^2; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 03 2013 *)

a(n)=if(n%2, n*if(n%4>2, 2, 1), n/2)^2 \\ Charles R Greathouse IV, Jul 07 2013

a(n) = A062828(n)^2.
a(4n) = (4*n+1)^2; a(2n+1) = (n+1)^2; a(4n+2) = 4*(4*n+3)^2.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n) * (period 4: repeat 4, 1, 1, 4) = A061038(n).
A005565(n-3) = a(n+1) * A061037(n). - Corrected by R. J. Mathar, Jul 25 2013
a(n) = A130658(n-1)^2 * A181318(n). - Corrected by R. J. Mathar, Aug 01 2013
G.f.: -x*(1 + x + 36*x^2 + 4*x^3 + 22*x^4 + 6*x^5 + 88*x^6 + 4*x^7 + 9*x^8 + x^9 + 4*x^10) / ( (x-1)^3*(1+x)^3*(x^2+1)^3 ). - R. J. Mathar, Jul 20 2013
Sum_{n>=1} 1/a(n) = 47*Pi^2/192 + 3*G/8, where G is Catalan's constant (A006752). - Amiram Eldar, Aug 21 2022

cp's OEIS Frontend

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

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A181407 a(n) = (n-4)*(n-3)*2^(n-2).

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A182868 a(n) = -1 + n + 4*n^2.

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A227316 a(n) = n(n+1) if n == 0 or 1 (mod 4), otherwise a(n) = n(n+1)/2.

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A086314 Total number of edges in the distinct simple graphs on n nodes.

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A185138 a(4*n) = n*(4*n-1); a(2*n+1) = n*(n+1)/2; a(4*n+2) = (2*n+1)*(4*n+1).

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A177049 Numerator of (3n+1)*(3n+2)/4.

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A181407 a(n) = (n-4)(n-3)2^(n-2).

A185138 a(4n) = n(4n-1); a(2n+1) = n(n+1)/2; a(4n+2) = (2n+1)(4*n+1).

A227168 a(n) = gcd(2n, n(n+1)/2)^2.