OFFSET
0,3
COMMENTS
Difference table for 0 followed by a(n):
0, 0, 1, 2, 7, 11, 24, 33,...
0, 1, 1, 5, 4, 13, 9, 25,... =A147685(n)
-1, 4, -5, 10, -13, 20, -25, 34,...
5, -9, 15, -23, 33, -45, 59, -75,... =(-1)^n*A027688(n+2).
a(-n) = -a(n-1).
From the second row, signature (0,3,0,-3,0,1).
Consider a(n+2k+1)+a(2k-n):
1, 2, 6, 9, 17, 22, 34,...
9, 12, 24, 33, 57, 72, 108,...
35, 40, 60, 75, 115, 140, 200,...
91, 98, 126, 147, 203, 238, 322,...
189, 198, 234, 261, 333, 378, 486,... .
The first column is A005898(n).
The rows are successively divisible by 2*k+1. Hence
1, 2, 6, 9, 17, 22, 34,...
3, 4, 8, 11, 19, 24, 36,...
7, 8, 12, 15, 23, 28, 40,...
13, 14, 18, 21, 29, 34, 46,...
21, 22, 26, 29, 37, 42, 54,...
The first column is A002061(n+1).
The main diagonal is A212965(n).
The first difference of every row is A022998(n+1).
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).
FORMULA
a(n) = 4*a(n-2) -6*a(n-4) +4*a(n-6) -a(n-8), n>7.
a(2n) + a(2n+1) = A005898(n).
a(2n-1) + a(2n) = A061317(n).
a(n) = (-1)*((-1+(-1)^n-2*n)*(2+n+n^2))/16. a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). G.f.: x*(x^2+1)*(x^2+x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Jan 20 2014
EXAMPLE
a(1)=1, a(2)=2, a(3)=3+4=7, a(4)=5+6=11, a(5)=7+8+9=24, a(6)=10+11+12=33.
MATHEMATICA
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {0, 1, 2, 7, 11, 24, 33}, 50] (* Harvey P. Dale, Nov 22 2014 *)
PROG
(PARI) Vec(x*(x^2+1)*(x^2+x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Jan 20 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Jan 07 2014
EXTENSIONS
More terms from Colin Barker, Jan 20 2014
STATUS
approved