OFFSET
1,2
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).
FORMULA
a(n) = n*(1 + floor((n-2)/3) - floor(n/3)) + 3*floor(n/3)^2*(1 + floor(n/3)) + floor((n+2)/3)*(3*floor((n+2)/3) - 1)/2.
From Colin Barker, Sep 16 2018: (Start)
G.f.: x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.
(End)
EXAMPLE
a(1) = 1;
a(2) = 1 + 2 = 3;
a(3) = 1 + 2*3 = 7;
a(4) = 1 + 2*3 + 4 = 11;
a(5) = 1 + 2*3 + 4 + 5 = 16;
a(6) = 1 + 2*3 + 4 + 5*6 = 41;
a(7) = 1 + 2*3 + 4 + 5*6 + 7 = 48;
a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8 = 56;
a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;
a(10) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;
a(11) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11 = 141;
a(12) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 = 262; etc.
MATHEMATICA
Table[n (1 + Floor[(n - 2)/3] - Floor[n/3]) + 3 Floor[n/3]^2 (1 + Floor[n/3]) + Floor[(n + 2)/3] (3 Floor[(n + 2)/3] - 1)/2, {n, 50}]
PROG
(PARI) Vec(x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Sep 16 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Sep 16 2018
STATUS
approved