A288994 a(n) = n*(n+3) when n is congruent to 0 or 3 (mod 4), and n*(n+3)/2 otherwise.

0, 2, 5, 18, 28, 20, 27, 70, 88, 54, 65, 154, 180, 104, 119, 270, 304, 170, 189, 418, 460, 252, 275, 598, 648, 350, 377, 810, 868, 464, 495, 1054, 1120, 594, 629, 1330, 1404, 740, 779, 1638, 1720, 902, 945, 1978, 2068, 1080, 1127, 2350, 2448, 1274, 1325

Offset: 0

Author

Jean-François Alcover and Paul Curtz, Jun 21 2017

Keywords

• nonn
• easy

Links

• Colin Barker, 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).

Crossrefs

Cf. A000096, A014695, A028552, A060819, A130658, A145979, A227316.

Programs

Mathematica

a[n_] := n (n+3) Switch[Mod[n, 4], 0|3, 1, _, 1/2]; Table[a[n], {n, 0, 50}]
Table[If[MemberQ[{0,3},Mod[n,4]],n(n+3),(n(n+3))/2],{n,0,50}] (* or *) LinearRecurrence[{3,-6,10,-12,12,-10,6,-3,1},{0,2,5,18,28,20,27,70,88},60] (* Harvey P. Dale, Jun 05 2021 *)

PARI

concat(0, Vec(x*(2 - x + 15*x^2 - 16*x^3 + 18*x^4 - 9*x^5 + 5*x^6 - 2*x^7) / ((1 - x)^3*(1 + x^2)^3) + O(x^60))) \\ Colin Barker, Jun 21 2017

PARI

i=I; a(n) = (1/8 + i/8)*(((3 - 3*i) - i*(-i)^n + i^n)*n*(3 + n)) \\ Colin Barker, Jun 21 2017

Formula

a(n) = n*(n+3)/2 * (2 - floor((n+1)/2) mod 2), where n*(n+3)/2 is A000096(n).

a(n) = A060819(n+3)*A145979(n-2).

a(n) = (2*n*(n+3))/(GCD(4, n+2)*GCD(4, n+3)).

a(n) = A227316(n+1) - (period 4 repeat 2,1,1,2).

From Colin Barker, Jun 21 2017: (Start)

G.f.: x*(2 - x + 15*x^2 - 16*x^3 + 18*x^4 - 9*x^5 + 5*x^6 - 2*x^7) / ((1 - x)^3*(1 + x^2)^3).

a(n) = (1/8 + i/8)*(((3 - 3*i) - i*(-i)^n + i^n)*n*(3 + n)), where i=sqrt(-1). (End)

Sum_{n>=1} 1/a(n) = 17/18 + log(2)/6. - Amiram Eldar, Aug 12 2022