[seqfan] Symmetric and Antisymmetric Linear Recurrences?

[Ron Hardin]

Is there any general significance to linear recurrences that are always symmetric or antisymmetric?

For instance the rows (not columns) of the following have symmetric or antisymmetric recurrences


/tmp/ejo
T(n,k)=Number of length n+3 0..k arrays with no disjoint pairs in any consecutive four terms having the same sum

Table starts
.8....48.....168......440.......960.......1848........3248.........5328
.8....90.....456.....1592......4344......10098.......20816........39264
.8...172....1248.....5796.....19744......55372......133780.......290004
.8...334....3424....21152.....89836.....303924......860360......2143214
.8...656....9392....77236....408644....1668072.....5532212.....15837692
.8..1300...25822...282384...1859736....9157806....35577396....117045466
.8..2584...71060..1032952...8465936...50284864...228817500....865051288
.8..5148..195536..3779018..38539276..276119316..1471661464...6393427268
.8.10272..537880.13825712.175434372.1516191100..9465023576..47252411120
.8.20520.1480026.50587924.798617096.8325624724.60874728614.349232818280

Empirical for column k:
k=1: a(n)=a(n-1)
k=2: a(n)=2*a(n-1)+2*a(n-4)-4*a(n-5)
k=3: [order 27]
k=4: [order 45]
k=5: [order 76]
Empirical for row n:
n=1: a(n) = n^4 + 2*n^3 + 3*n^2 + 2*n
n=2: a(n)=4*a(n-1)-4*a(n-2)-4*a(n-3)+10*a(n-4)-4*a(n-5)-4*a(n-6)+4*a(n-7)-a(n-8)
n=3: a(n)=2*a(n-1)+a(n-2)-2*a(n-3)-2*a(n-4)-2*a(n-5)+5*a(n-6)+2*a(n-7)-2*a(n-9)-5*a(n-10)+2*a(n-11)+2*a(n-12)+2*a(n-13)-a(n-14)-2*a(n-15)+a(n-16)

n=4: a(n)=a(n-2)+2*a(n-3)+2*a(n-4)-a(n-5)-2*a(n-6)-4*a(n-7)-3*a(n-8)-a(n-9)+a(n-10)+4*a(n-11)+6*a(n-12)+5*a(n-13)+2*a(n-14)-2*a(n-15)-6*a(n-16)-6*a(n-17)-6*a(n-18)-2*a(n-19)+2*a(n-20)+5*a(n-21)+6*a(n-22)+4*a(n-23)+a(n-24)-a(n-25)-3*a(n-26)-4*a(n-27)-2*a(n-28)-a(n-29)+2*a(n-30)+2*a(n-31)+a(n-32)-a(n-34)

n=5: a(n)=-3*a(n-1)-6*a(n-2)-9*a(n-3)-10*a(n-4)-7*a(n-5)+2*a(n-6)+17*a(n-7)+36*a(n-8)+54*a(n-9)+64*a(n-10)+59*a(n-11)+35*a(n-12)-9*a(n-13)-67*a(n-14)-128*a(n-15)-177*a(n-16)-199*a(n-17)-182*a(n-18)-122*a(n-19)-24*a(n-20)+98*a(n-21)+222*a(n-22)+323*a(n-23)+378*a(n-24)+372*a(n-25)+299*a(n-26)+167*a(n-27)-5*a(n-28)-189*a(n-29)-354*a(n-30)-470*a(n-31)-516*a(n-32)-482*a(n-33)-372*a(n-34)-202*a(n-35)+202*a(n-37)+372*a(n-38)+482*a(n-39)+516*a(n-40)+470*a(n-41)+354*a(n-42)+189*a(n-43)+5*a(n-44)-167*a(n-45)-299*a(n-46)-372*a(n-47)-378*a(n-48)-323*a(n-49)-222*a(n-50)-98*a(n-51)+24*a(n-52)+122*a(n-53)+182*a(n-54)+199*a(n-55)+177*a(n-56)+128*a(n-57)+67*a(n-58)+9*a(n-59)-35*a(n-60)-59*a(n-61)-64*a(n-62)-54*a(n-63)-36*a(n-64)-17*a(n-65)-2*a(n-66)+7*a(n-67)+10*a(n-68)+9*a(n-69)+6*a(n-70)+3*a(n-71)+a(n-72)


 
rhhardin at mindspring.com
rhhardin at att.net (either)