Jacobsthal Number

Author

Eric W. Weisstein
April 12, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/JacobstahlNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/JacobstahlNumber.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Code

JacobsthalJ[n_]:=(1/3) (-(((-1))^n)+(2^n))
Jacobsthalj[n_]:=(((-1))^n)+(2^n)

Recurrence

V6

RSolve[{J[n]==J[n-1]+2J[n-2],J[0]==0,J[1]==1},J[n],n]

RSolve[{j[n]==j[n-1]+2j[n-2],j[0]==2,j[1]==1},j[n],n]

V7

RSolve[{J[n]==J[n-1]+2J[n-2],J[0]==0,J[1]==1},J[n],n]

RSolve[{j[n]==j[n-1]+2j[n-2],j[0]==2,j[1]==1},j[n],n]

Sums

(∑_(r=0)^(⌊(n-1)/2⌋))(2^r) (Binomial[n-r-1, r])==JacobsthalJ(n)

V5

Simplify[Sum[Binomial[n-1-r,r](2^r),{r,0,Floor[(n-1)/2]}],n∈Integers]

V7

Sum[Binomial[n-1-r,r](2^r),{r,0,Floor[(n-1)/2]}]

(∑_(r=0)^(⌊n/2⌋))((n (2^r) (Binomial[n-r, r]))/(n-r))==Jacobsthalj(n)

Sum[(n/(n-r))Binomial[n-r,r](2^r),{r,0,Floor[n/2]}]

Sum Identities

(∑_n^∞)JacobsthalJ(n) (x^(n-1))==(1/(-2 (x^2)-x+1))

V5

Sum[JacobsthalJ[n](x^(n-1)),{n,∞}]//Simplify

V7

Sum[JacobsthalJ[n](x^(n-1)),{n,∞}]

FullSimplify[%]

(∑_n^∞)Jacobsthalj(n) (x^(n-1))==((-4 x-1)/(2 (x^2)+x-1))

V5

Sum[Jacobsthalj[n](x^(n-1)),{n,∞}]//Simplify

V7

Sum[Jacobsthalj[n](x^(n-1)),{n,∞}]

FullSimplify[%]

(∑_(n=2)^n)JacobsthalJ(n)==(1/2) (JacobsthalJ(n+2)-3)

Sum[JacobsthalJ[n],{n,2,n}]

(1/2)(JacobsthalJ[n+2]-3)//Simplify

Subtract@@{Sum[JacobsthalJ[n],{n,2,n}],(1/2)(JacobsthalJ[n+2]-3)}//Simplify

(∑_n^n)Jacobsthalj(n)==(1/2) (Jacobsthalj(n+2)-5)

Sum[Jacobsthalj[n],{n,n}]

(1/2)(Jacobsthalj[n+2]-5)//Simplify

Subtract@@{Sum[Jacobsthalj[n],{n,n}],(1/2)(Jacobsthalj[n+2]-5)}//Simplify

Identities

JacobsthalJ(n+1) JacobsthalJ(n-1)==((JacobsthalJ(n))^2)

JacobsthalJ[n+1]JacobsthalJ[n-1]-((JacobsthalJ[n])^2)//Simplify

Jacobsthalj(n+1) Jacobsthalj(n-1)==((Jacobsthalj(n))^2)

Jacobsthalj[n+1]Jacobsthalj[n-1]-((Jacobsthalj[n])^2)//Simplify

Jacobsthalj(n) JacobsthalJ(n)==JacobsthalJ(2 n)

Jacobsthalj[n]JacobsthalJ[n]-JacobsthalJ[2n]//Simplify