A235355 0 followed by the sum of (1),(2), (3,4),(5,6), (7,8,9),(10,11,12) from the natural numbers.

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

Author

Paul Curtz, Jan 07 2014

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

Cf. A075356, A234305.

Programs

Mathematica

LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,2,7,11,24,33},50] (* Harvey P. Dale, Nov 22 2014 *)

PARI

Vec(x*(x^2+1)*(x^2+x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Jan 20 2014

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

Extensions

More terms from Colin Barker, Jan 20 2014