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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 (list; graph; refs; listen; history; text; internal format)
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.

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) = 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

Cf. A075356, A234305.

Sequence in context: A228434 A031873 A075356 * A103184 A093039 A201630

Adjacent sequences:  A235352 A235353 A235354 * A235356 A235357 A235358

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Jan 07 2014

EXTENSIONS

More terms from Colin Barker, Jan 20 2014

STATUS

approved

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Last modified April 3 14:34 EDT 2020. Contains 333197 sequences. (Running on oeis4.)