OFFSET
0,3
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of the initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).
FORMULA
O.g.f.: x*(1 + x + x^2)/((1 + x^2)*(1 - x)^3).
E.g.f.: -(1 - 6*x - 3*x^2)*exp(x)/4 - (1 + i)*(i - exp(2*i*x))*exp(-i*x)/8, where i=sqrt(-1).
a(n) = a(-n-1) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) = a(n-4) + 6*n - 9.
a(n) = 3*n*(n+1)/4 + (i^(n*(n+1)) - 1)/4. Therefore:
a(4*k+r) = 12*k^2 + 3*(2*r+1)*k + r^2, where 0 <= r <= 3.
a(n) = n^2 - floor((n-1)*(n-2)/4).
a(n) = A011865(3*n+2).
MAPLE
MATHEMATICA
Table[Floor[3 n (n + 1)/4], {n, 0, 60}]
LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 4, 9, 15}, 60] (* Harvey P. Dale, Jun 04 2023 *)
PROG
(PARI) vector(60, n, n--; floor(3*n*(n+1)/4))
(Python) [int(3*n*(n+1)/4) for n in range(60)]
(Sage) [floor(3*n*(n+1)/4) for n in range(60)]
(Maxima) makelist(floor(3*n*(n+1)/4), n, 0, 60);
(Magma) [3*n*(n+1) div 4: n in [0..60]];
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Jan 13 2017
STATUS
approved