A227380 -id:A227380 - cp's OEIS frontend

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

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

A266491 a(n) = n*A130658(n).

Original entry on oeis.org

0, 1, 4, 6, 4, 5, 12, 14, 8, 9, 20, 22, 12, 13, 28, 30, 16, 17, 36, 38, 20, 21, 44, 46, 24, 25, 52, 54, 28, 29, 60, 62, 32, 33, 68, 70, 36, 37, 76, 78, 40, 41, 84, 86, 44, 45, 92, 94, 48, 49, 100, 102, 52, 53, 108, 110, 56, 57, 116, 118, 60, 61, 124, 126, 64

Offset: 0

Paul Curtz, Dec 30 2015

Successive differences:
r(0):    0,   1,    4,    6,   4,    5,   12,   14, ...
r(1):    1,   3,    2,   -2,   1,    7,    2,   -6, ...
r(2):    2,  -1,   -4,    3,   6,   -5,   -8,    7, ... (see A103889)
r(3):   -3,  -3,    7,    3, -11,   -3,   15,    3, ...
r(4):    0,  10,   -4,  -14,   8,   18,  -12,  -22, ...
r(5):   10, -14,  -10,   22,  10,  -30,  -10,   38, ...
r(6):  -24,   4,   32,  -12, -40,   20,   48,  -28, ...
r(7):   28,  28,  -44,  -28,  60,   28,  -76,  -28, ...
r(8):    0, -72,   16,   88, -32, -104,   48,  120, ...
r(9):  -72,  88,   72, -120, -72,  152,   72, -184, ...
r(10): 160, -16, -192,   48, 224,  -80, -256,  112, ...
etc.
Let b(n) = 1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 16, 16, ..., with n>=0, which is formed from the terms of A011782 repeated twice.
Conjecture: all terms of the row r(i) are divisible by b(i).
Conjecture: the terms of the first column divided by b(n) provide 0, 1, 2, -3, 0, 5, -6, 7, 0, -9, 10, -11, ..., the absolute values of which are listed in A190621.

Cf. A005408, A011782, A042963, A058962, A103889, A128135, A130658, A190621, A227380.

[n*(3-(-1)^((n-1)*n div 2))/2: n in [0..70]]; // Vincenzo Librandi, Jan 08 2016

Table[n (3 - (-1)^((n - 1) n/2))/2, {n, 0, 55}]
Table[n (Boole@ OddQ@ Floor[n/2] + 1), {n, 0, 55}] (* or *) Table[SeriesCoefficient[x (3/(1 - x)^2 + 2 x/(1 + x^2)^2 - (1 - x^2)/(1 + x^2)^2)/2, {x, 0, n}], {n, 0, 55}] (* Michael De Vlieger, Jan 04 2016 *)

vector(60, n, n--; n*(3-(-1)^((n-1)*n/2))/2) \\ Altug Alkan, Jan 04 2016

a(n) = n*(3 - (-1)^((n-1)*n/2))/2.
a(n) = a(n-4) + 4*A130658(n) for n>3.
a(n) = 2*a(n-1) -3*a(n-2) +4*a(n-3) -3*(n-4) +2*a(n-5) -a(n-6) for n>5.
G.f.: x*(3/(1 - x)^2 + 2*x/(1 + x^2)^2 - (1 - x^2)/(1 + x^2)^2)/2. - Michael De Vlieger, Jan 04 2016

Edited by Bruno Berselli, Jan 07 2016

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula