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

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

Author

Paul Curtz, Dec 30 2015

Keywords

• nonn
• easy

Comments

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.

Crossrefs

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

Programs

Magma

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

Mathematica

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 *)

PARI

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

Formula

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

Extensions

Edited by Bruno Berselli, Jan 07 2016