A196514 Partial sums of A100381.

0, 4, 28, 124, 444, 1404, 4092, 11260, 29692, 75772, 188412, 458748, 1097724, 2588668, 6029308, 13893628, 31719420, 71827452, 161480700, 360710140, 801112060, 1769996284, 3892314108, 8522825724, 18589155324, 40399536124, 87509958652

Offset: 0

Author

R. J. Mathar, Oct 03 2011

Keywords

• nonn
• easy
• less

Comments

Like any sequence with a linear recurrence, this has a Pisano period length modulo any k >= 1. The period lengths for this sequence are (modulo k >= 1) 1, 1, 6, 1, 20, 6, 21, 1, 18, 20, 110, 6, 156, 21, 60, 1, 136, 18, 342, 20, ....

References

• Jolley, Summation of Series, Dover (1961), eq (53) page 10.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Programs

Magma

[(n^2-n+2)*2^(n+1)-4 : n in [0..30]]; // Vincenzo Librandi, Oct 05 2011

Mathematica

Table[2^n*Binomial[n, 2], {n, 1, 27}] // Accumulate (* Jean-François Alcover, Jun 24 2013 *)
LinearRecurrence[{7,-18,20,-8},{0,4,28,124},30] (* Harvey P. Dale, Jan 12 2016 *)

PARI

a(n)=(n^2-n+2)<<(n+1)-4 \\ Charles R Greathouse IV, Oct 05 2011

Formula

G.f.: 4*x / ( (x-1)*(2*x-1)^3 ).

a(n) = (n^2 - n + 2)*2^(n+1) - 4 = 4*A055580(n-1).

a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4); a(0)=0, a(1)=4, a(2)=28, a(3)=124. - Harvey P. Dale, Jan 12 2016