A241615 Number of length n+2 0..9 arrays with no consecutive three elements summing to more than 9

220, 1210, 6655, 34243, 180829, 963886, 5093737, 26932543, 142701909, 755538278, 3999038946, 21172904049, 112098384491, 593455432350, 3141868198978, 16633824615067, 88062718713584, 466221475528171, 2468274573927916

Offset

1,1

Comments

Column 9 of A241619

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Robert Israel, Maple-assisted proof of empirical formula

Formula

Empirical: a(n) = 4*a(n-1) +2*a(n-2) +44*a(n-3) -69*a(n-4) -79*a(n-5) -507*a(n-6) +572*a(n-7) +514*a(n-8) +2973*a(n-9) -3097*a(n-10) -1820*a(n-11) -10364*a(n-12) +10800*a(n-13) +4269*a(n-14) +25019*a(n-15) -25821*a(n-16) -6914*a(n-17) -44207*a(n-18) +44275*a(n-19) +8829*a(n-20) +59359*a(n-21) -57787*a(n-22) -9308*a(n-23) -62456*a(n-24) +58989*a(n-25) +8291*a(n-26) +52174*a(n-27) -48385*a(n-28) -5846*a(n-29) -35493*a(n-30) +32403*a(n-31) +3452*a(n-32) +19719*a(n-33) -17810*a(n-34) -1563*a(n-35) -9053*a(n-36) +8178*a(n-37) +608*a(n-38) +3390*a(n-39) -3025*a(n-40) -167*a(n-41) -1072*a(n-42) +973*a(n-43) +42*a(n-44) +259*a(n-45) -227*a(n-46) -6*a(n-47) -56*a(n-48) +52*a(n-49) +a(n-50) +7*a(n-51) -6*a(n-52) -a(n-54) +a(n-55).

Empirical formula verified: see link. - Robert Israel, Sep 03 2019

Example

Some solutions for n=5
..0....1....0....1....1....7....5....5....1....0....0....1....1....1....1....0
..5....2....1....2....1....0....1....1....3....1....1....2....2....0....7....1
..2....1....3....1....2....0....0....3....1....2....2....4....1....0....1....7
..0....3....1....5....0....4....1....2....2....5....3....3....1....0....0....0
..3....1....0....0....0....0....3....0....4....0....2....0....5....1....4....1
..2....4....8....1....1....3....0....4....0....1....2....3....1....8....1....3
..3....1....1....1....2....3....6....5....4....2....4....4....0....0....2....4

Maple

r:= [seq(seq([i, j], j=0..9-i), i=0..9)]:
T:= Matrix(55, 55, proc(i, j) if r[i][1]=r[j][2] and r[i][1]+r[i][2]+r[j][1]<=9 then 1 else 0 fi end proc):
U[0]:= Vector(55, 1):
for n from 1 to 50 do U[n]:= T . U[n-1] od:
seq(U[0]^%T . U[j], j=1..50); # Robert Israel, Sep 03 2019

Crossrefs

Sequence in context: A260086 A339678 A339682 * A180219 A233626 A251322

Adjacent sequences: A241612 A241613 A241614 * A241616 A241617 A241618

Keyword

nonn

Author

R. H. Hardin, Apr 26 2014

Status

approved