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A235355
0 followed by the sum of (1),(2), (3,4),(5,6), (7,8,9),(10,11,12) from the natural numbers.
3
0, 1, 2, 7, 11, 24, 33, 58, 74, 115, 140, 201, 237, 322, 371, 484, 548, 693, 774, 955, 1055, 1276, 1397, 1662, 1806, 2119, 2288, 2653, 2849, 3270, 3495, 3976, 4232, 4777, 5066, 5679, 6003, 6688, 7049, 7810, 8210, 9051, 9492, 10417, 10901, 11914, 12443, 13548
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, 0, 4, -1, 9, -4, 16, -9,... =interleave A000290(n+1),-A000290(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).
Compare to the (2k+1)-sections of A061037 in A165943.
FORMULA
a(n) = 4*a(n-2) -6*a(n-4) +4*a(n-6) -a(n-8), n>7.
a(2n) = 0 followed by A085786(n). a(2n+1) = A081436(n).
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
Sequence in context: A228434 A031873 A075356 * A103184 A093039 A201630
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Jan 07 2014
EXTENSIONS
More terms from Colin Barker, Jan 20 2014
STATUS
approved