A057029 Central column of arrays in A057027 and A057028.

1, 6, 12, 27, 39, 64, 82, 117, 141, 186, 216, 271, 307, 372, 414, 489, 537, 622, 676, 771, 831, 936, 1002, 1117, 1189, 1314, 1392, 1527, 1611, 1756, 1846, 2001, 2097, 2262, 2364, 2539, 2647, 2832, 2946, 3141, 3261, 3466, 3592, 3807, 3939, 4164, 4302, 4537

Offset: 1

Author

Clark Kimberling, Jul 28 2000

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A057027, A057028.

Programs

Magma

[(5-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4 : n in [1..80]]; // Wesley Ivan Hurt, Jul 03 2016

Maple

A057029:=n->(5-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: seq(A057029(n), n=1..80); # Wesley Ivan Hurt, Jul 03 2016

Mathematica

Table[(5 - (-1)^n + 2 (-4 + (-1)^n) n + 8 n^2)/4, {n, 49}] (* or *)
Table[If[OddQ@ n, Binomial[2 n - 1, 2] + (n + 1)/2 , Binomial[2 n, 2] - (n - 2)/2], {n, 49}] (* or *)
Rest@ CoefficientList[Series[x (1 + 5 x + 4 x^2 + 5 x^3 + x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 49}], x] (* Michael De Vlieger, Jul 03 2016 *)

PARI

Vec(x*(1+5*x+4*x^2+5*x^3+x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jul 02 2016

Formula

a(n) = C(2n-1, 2)+(n+1)/2 if n is odd, else a(n) = C(2n, 2)-(n-2)/2.

From Colin Barker, Jul 02 2016: (Start)

a(n) = (5-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4.

a(n) = (4*n^2-3*n+2)/2 for n even, a(n) = (4*n^2-5*n+3)/2 for n odd.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.

G.f.: x*(1+5*x+4*x^2+5*x^3+x^4) / ((1-x)^3*(1+x)^2). (End)

E.g.f.: ((2 - x + 4*x^2)*cosh(x) + (3 + x + 4*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Sep 10 2024