A131090 First differences of A131666.

0, 1, 0, 1, 1, 4, 7, 15, 28, 57, 113, 228, 455, 911, 1820, 3641, 7281, 14564, 29127, 58255, 116508, 233017, 466033, 932068, 1864135, 3728271, 7456540, 14913081, 29826161, 59652324, 119304647, 238609295, 477218588, 954437177, 1908874353

Offset: 0

Author

Paul Curtz, Sep 24 2007

Keywords

• easy
• nonn

Comments

The first differences b(n)=a(n+1)-a(n) obey the recurrence b(n+1)-2b(n) = (-3,3,-2,3,-3,2), continued with period 6.

The 2nd differences c(n)=b(n+1)-b(n) obey the recurrence c(n+1)-2c(n) = (6,-5,5,-6,5,-5), periodically continued with period 6.

The hexaperiodic coefficients in these recurrences for A113405, A131666 and their higher order differences define a table,

0, 0, 1, 0, 0, -1 <- A113405

0, 1, -1, 0, -1, 1 <- A131666

1, -2, 1, -1, 2, -1 <- a(n)

-3, 3, -2, 3, -3, 2 <- b(n)

6, -5, 5, -6, 5, -5 <- c(n)

-11,10,-11, 11,-10, 11

21,-21,22,-21, 21,-22

...

in which the first three columns are A024495, A131708 and A024493, multiplied by a checkerboard pattern of signs.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Programs

Mathematica

LinearRecurrence[{2,0,-1,2},{0,1,0,1},40] (* Harvey P. Dale, Jan 15 2016 *)

Formula

a(n) = A131666(n+1)-A131666(n).

a(n+1)-2a(n) = A131556(n), a sequence with period length 6.

G.f.: -(x-1)^2*x / ((x+1)*(2*x-1)*(x^2-x+1)). - Colin Barker, Mar 04 2013

Extensions

Edited by R. J. Mathar, Jun 28 2008