|
|
A178946
|
|
a(n) = n*(n+1)*(2*n+1)/6 - n*floor(n/2).
|
|
1
|
|
|
1, 3, 11, 22, 45, 73, 119, 172, 249, 335, 451, 578, 741, 917, 1135, 1368, 1649, 1947, 2299, 2670, 3101, 3553, 4071, 4612, 5225, 5863, 6579, 7322, 8149, 9005, 9951, 10928, 12001, 13107, 14315, 15558, 16909, 18297, 19799, 21340, 23001
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Previous name was: A modified variant of A005900.
Let S(x) = (1, 3, 5, 7,...); then A178946 = (1/2) * ((S(x)^2 + S(x^2)).
If n is even, a(n) is the sum of the first n squares minus n^2/2. If n is odd, a(n) is the sum of the first n squares minus n(n-1)/2. - Wesley Ivan Hurt, Sep 17 2013
|
|
LINKS
|
|
|
FORMULA
|
a(n) = +2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6). G.f.: x*(1+x+4*x^2+x^4+x^3) / ( (1+x)^2*(x-1)^4 ). - R. J. Mathar, Jan 03 2011
|
|
EXAMPLE
|
(1/2) *((1, 6, 19, 44, 85, 146, 231,...) + (1, 0, 3, 0, 5, 0, 7, 0, 9,...)) =
(1, 3, 11, 22, 45, 73, 119,...).
|
|
MAPLE
|
A005900 := proc(n) n*(2*n^2+1)/3 ; end proc:
seq(k*(k+1)*(2*k+1)/6 - k*floor(k/2), k=1..100); # Wesley Ivan Hurt, Sep 17 2013
|
|
MATHEMATICA
|
Table[n(n+1)(2n+1)/6-n*Floor[n/2], {n, 100}] (* Wesley Ivan Hurt, Sep 17 2013 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 3, 11, 22, 45, 73}, 50] (* Harvey P. Dale, Mar 20 2018 *)
|
|
PROG
|
(Magma) [n*(n+1)*(2*n+1)/6 - n*Floor(n/2): n in [1..50]]; // Vincenzo Librandi, Sep 17 2013
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|