A329583 Numerators of 1 + n^2/4 + period 3: repeat [-1, 1, 1].

0, 6, 3, 12, 6, 30, 9, 54, 18, 84, 27, 126, 36, 174, 51, 228, 66, 294, 81, 366, 102, 444, 123, 534, 144, 630, 171, 732, 198, 846, 225, 966, 258, 1092, 291, 1230, 324, 1374, 363, 1524, 402, 1686, 441, 1854, 486, 2028, 531, 2214, 576, 2406, 627

Offset: 0

Author

Paul Curtz, Nov 17 2019

Keywords

• nonn
• easy

Comments

First bisection is 3*A008810.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,2,3,0,-3,-2,1,1).

Crossrefs

Cf. A008810, A042965, A047395, A130823, A131561, A261327.

Programs

Mathematica

MapIndexed[#1 - 2 Boole[Mod[First@ #2, 3] == 1] + 1 &, CoefficientList[Series[(1 + 5 x - x^2 - 2 x^3 + 2 x^4 + 5 x^5)/(1 - x^2)^3, {x, 0, 44}], x]] (* Michael De Vlieger, Nov 18 2019 *)

PARI

concat(0, Vec(3*x*(2 + 3*x + x^2 - 2*x^3 + x^4 + 3*x^5 + 2*x^6) / ((1 - x)^3*(1 + x)^3*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Nov 24 2019

Formula

a(n) = A261327(n) + A131561(n+2) = (n^2 + 4)*(5 - 3*(-1)^n)/8 + (-1)^((n+1) mod 3).

From Colin Barker, Nov 24 2019: (Start)

G.f.: 3*x*(2 + 3*x + x^2 - 2*x^3 + x^4 + 3*x^5 + 2*x^6) / ((1 - x)^3*(1 + x)^3*(1 + x + x^2)).

a(n) = -a(n-1) + 2*a(n-2) + 3*a(n-3) - 3*a(n-5) - 2*a(n-6) + a(n-7) + a(n-8) for n>8. (End)

Extensions

Incorrect 129 replaced with 123 by Colin Barker, Nov 24 2019