A173154 a(n) = n^3/6 + 3*n^2/4 + 7*n/3 + 7/8 + (-1)^n/8.

1, 4, 10, 19, 33, 52, 78, 111, 153, 204, 266, 339, 425, 524, 638, 767, 913, 1076, 1258, 1459, 1681, 1924, 2190, 2479, 2793, 3132, 3498, 3891, 4313, 4764, 5246, 5759, 6305, 6884, 7498, 8147, 8833, 9556, 10318, 11119, 11961, 12844, 13770, 14739, 15753, 16812, 17918, 19071, 20273, 21524

Offset: 0

Author

Paul Curtz, Feb 11 2010

Keywords

• nonn
• easy

Comments

Generated by reading the table shown in A172002 down the diagonal starting at 1.

The inverse binomial transform yields 1, 3, 3, 0, 2, -4, 8, -16, 32, -64, 128, -256, 512, -1024, ... with a pattern of powers of 2.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Programs

Magma

[n^3/6 + 3*n^2/4 + 7*n/3 + 7/8 + (-1)^n/8: n in [0..50]]; // Vincenzo Librandi, Aug 05 2011

Mathematica

Table[n^3/6+(3n^2)/4+(7n)/3+7/8+(-1)^n/8,{n,0,50}] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,4,10,19,33},50] (* Harvey P. Dale, Jan 04 2012 *)

Formula

G.f.: ( 1 + x - x^3 + x^4 ) / ( (1+x)*(x-1)^4 ).

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).

a(n+4) - a(n) = 4*A152948(n+5) = 4*A089071(n+5).

First differences: a(n+1) - a(n) = A061925(n+2).

Second differences: a(n+2) - 2*a(n+1) + a(n) = n + 5/2 + (-1)^n/2 = 3, 3, 5, 5, 7, 7, 9, 9, ... , duplicated A144396.