A269606 -id:A269606 - cp's OEIS frontend

A269600 Number of length-n 0..n arrays with no repeated value differing from the previous repeated value by one or less.

2, 9, 60, 572, 7030, 105892, 1886252, 38768412, 902743646, 23482784400, 674804472216, 21227322458468, 725465523386294, 26765145874993234, 1060182321437102136, 44874027105660248520, 2021234030434596074938

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

Diagonal of A269606.

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

R. H. Hardin, Table of n, a(n) for n = 1..193

Cf. A269606.

A269601 Number of length-n 0..2 arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

3, 9, 24, 62, 154, 376, 902, 2142, 5040, 11786, 27418, 63544, 146822, 338478, 778944, 1790282, 4110730, 9432424, 21633542, 49603134, 113717328, 260691722, 597649402, 1370287384, 3142264070, 7206988878, 16533117792, 37935980618, 87065571946

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

			Some solutions for n=7:
..2. .2. .2. .2. .2. .0. .1. .0. .2. .2. .2. .1. .0. .1. .2. .2
..1. .1. .1. .1. .0. .2. .2. .2. .0. .0. .1. .2. .2. .2. .0. .1
..2. .1. .0. .2. .2. .1. .1. .2. .0. .1. .0. .2. .1. .1. .1. .0
..0. .2. .2. .1. .1. .0. .1. .1. .2. .1. .1. .0. .0. .0. .0. .1
..1. .1. .0. .0. .0. .2. .2. .0. .0. .0. .0. .1. .2. .1. .1. .2
..1. .0. .1. .2. .2. .0. .0. .2. .1. .1. .0. .2. .1. .0. .0. .1
..2. .1. .0. .0. .1. .0. .1. .0. .2. .2. .2. .0. .1. .2. .1. .0

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

Column 2 of A269606.

Empirical: a(n) = 5*a(n-1) - 5*a(n-2) - 8*a(n-3) + 12*a(n-4).

Empirical g.f.: x*(3 - 6*x - 6*x^2 + 11*x^3) / ((1 - 2*x)^2*(1 - x - 3*x^2)). - Colin Barker, Jan 24 2019

A269602 Number of length-n 0..4 arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

5, 25, 120, 572, 2692, 12570, 58280, 268704, 1233046, 5636046, 25675580, 116635916, 528551090, 2390183046, 10789164304, 48625122028, 218845692934, 983773248134, 4417701453060, 19819733378212, 88847987191058, 398004177820814

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

			Some solutions for n=7:
..0. .1. .1. .4. .3. .2. .1. .2. .1. .2. .4. .4. .1. .3. .4. .1
..1. .0. .4. .1. .1. .1. .4. .4. .1. .3. .2. .1. .4. .4. .1. .1
..4. .0. .4. .0. .0. .0. .1. .1. .2. .0. .2. .4. .0. .3. .0. .2
..4. .4. .2. .1. .0. .1. .1. .1. .3. .3. .4. .4. .1. .2. .4. .1
..0. .4. .4. .3. .2. .0. .3. .2. .2. .1. .2. .2. .1. .2. .2. .3
..1. .2. .3. .1. .3. .3. .4. .1. .0. .0. .4. .3. .2. .0. .1. .1
..3. .4. .1. .0. .1. .4. .4. .3. .4. .3. .0. .2. .4. .3. .0. .4

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

Column 4 of A269606.

Empirical: a(n) = 13*a(n-1) - 49*a(n-2) - 5*a(n-3) + 292*a(n-4) - 70*a(n-5) - 658*a(n-6) - 344*a(n-7).

Empirical g.f.: x*(5 - 40*x + 40*x^2 + 262*x^3 - 199*x^4 - 748*x^5 - 362*x^6) / ((1 - 4*x)*(1 - 9*x + 13*x^2 + 57*x^3 - 64*x^4 - 186*x^5 - 86*x^6)). - Colin Barker, Jan 24 2019

A269603 Number of length-n 0..5 arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

6, 36, 210, 1220, 7030, 40288, 229754, 1304934, 7385898, 41679780, 234601902, 1317558578, 7385249086, 41325945826, 230904832646, 1288466651340, 7181415415962, 39985405920156, 222432351559566, 1236355637246456

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof for empirical recursion

Column 5 of A269606.

T:= Matrix(42,42):
for x from 0 to 5 do
  for v from 0 to 6 do
    i:= 1 + x + 6*v;
    for y in {$0..5} minus {x} do
      T[i,1+y+6*v]:= 1;
    od:
    if abs(x-v) > 1 or v=6 then T[i,1+x+6*x]:= 1 fi
od od:
u:= Vector([0$36,1$6]): v:= Vector(42,1):
Tv[1]:= v:
for n from 2 to 50 do Tv[n]:= T . Tv[n-1] od:
seq(u^%T . Tv[n], n=1..50); # Robert Israel, Jan 24 2019

Empirical: a(n) = 17*a(n-1) - 91*a(n-2) + 83*a(n-3) + 542*a(n-4) - 550*a(n-5) - 1651*a(n-6) - 745*a(n-7).
Empirical g.f.: 2*x*(3 - 33*x + 72*x^2 + 214*x^3 - 420*x^4 - 922*x^5 - 393*x^6) / ((1 - 5*x)*(1 - 12*x + 31*x^2 + 72*x^3 - 182*x^4 - 360*x^5 - 149*x^6)). - Colin Barker, Jan 24 2019
Empirical recursion verified - see link. - Robert Israel, Jan 24 2019

A269604 Number of length-n 0..6 arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

7, 49, 336, 2298, 15630, 105892, 714874, 4811578, 32300252, 216337084, 1446056046, 9648789758, 64281141440, 427655897226, 2841661493142, 18861464959350, 125070420653458, 828618463551536, 5485481885293294, 36288577806336542

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of empirical recursion

Column 6 of A269606.

with(LinearAlgebra):
T:= Matrix(56,56):
for x from 0 to 6 do
  for v from 0 to 7 do
    i:= 1 + x + 7*v;
    for y in {$0..6} minus {x} do
      T[i,1+y+7*v]:= 1;
    od:
    if abs(x-v) > 1 or v=7 then T[i,1+x+7*x]:= 1 fi
od od:
u:= Vector([0$49,1$7]): v:= Vector(56,1):
Tv[1]:= v:
for n from 2 to 50 do Tv[n]:= T . Tv[n-1] od:
seq(u^%T . Tv[n], n=1..50); # Robert Israel, Jan 24 2019

Empirical: a(n) = 26*a(n-1) - 243*a(n-2) + 833*a(n-3) + 567*a(n-4) - 7567*a(n-5) - 1006*a(n-6) + 27361*a(n-7) + 31306*a(n-8) + 9984*a(n-9).
Empirical g.f.: x*(7 - 133*x + 763*x^2 - 362*x^3 - 7256*x^4 + 3224*x^5 + 32851*x^6 + 34133*x^7 + 10511*x^8) / ((1 - 6*x)*(1 - 20*x + 123*x^2 - 95*x^3 - 1137*x^4 + 745*x^5 + 5476*x^6 + 5495*x^7 + 1664*x^8)). - Colin Barker, Jan 24 2019
Empirical recursion verified: see link. - Robert Israel, Jan 24 2019

A269605 Number of length-n 0..7 arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

8, 64, 504, 3962, 31024, 242226, 1886252, 14654952, 113629480, 879470154, 6796127732, 52443005888, 404170590152, 3111359345068, 23927329547328, 183840499514208, 1411335451447128, 10826702362761906, 82998453154738884

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

Column 7 of A269606.

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of empirical recursion

Cf. A269606.

with(LinearAlgebra):
T:= Matrix(72,72):
for x from 0 to 7 do
  for v from 0 to 8 do
    i:= 1 + x + 8*v;
    for y in {$0..7} minus {x} do
      T[i,1+y+8*v]:= 1;
    od:
    if abs(x-v) > 1 or v=8 then T[i,1+x+8*x]:= 1 fi
od od:
u:= Vector([0$64,1$8]): v:= Vector(72,1):
Tv[1]:= v:
for n from 2 to 50 do Tv[n]:= T . Tv[n-1] od:
seq(u^%T . Tv[n], n=1..50); # Robert Israel, Jan 24 2019

Empirical: a(n) = 31*a(n-1) -353*a(n-2) +1601*a(n-3) -435*a(n-4) -14505*a(n-5) +7118*a(n-6) +65542*a(n-7) +66279*a(n-8) +19971*a(n-9)

Empirical formula verified: see link. - Robert Israel, Jan 24 2019

A269607 Number of length-4 0..n arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

8, 62, 222, 572, 1220, 2298, 3962, 6392, 9792, 14390, 20438, 28212, 38012, 50162, 65010, 82928, 104312, 129582, 159182, 193580, 233268, 278762, 330602, 389352, 455600, 529958, 613062, 705572, 808172, 921570, 1046498, 1183712, 1333992

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

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

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

Row 4 of A269606.

Empirical: a(n) = n^4 + 4*n^3 + 4*n^2 - n.
Conjectures from Colin Barker, Jan 25 2019: (Start)
G.f.: 2*x*(4 + 11*x - 4*x^2 + x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A269608 Number of length-5 0..n arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

10, 154, 804, 2692, 7030, 15630, 31024, 56584, 96642, 156610, 243100, 364044, 528814, 748342, 1035240, 1403920, 1870714, 2453994, 3174292, 4054420, 5119590, 6397534, 7918624, 9715992, 11825650, 14286610, 17141004, 20434204, 24214942

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

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

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

Row 5 of A269606.

Empirical: a(n) = n^5 + 5*n^4 + 7*n^3 - 4*n^2 + n.
Conjectures from Colin Barker, Jan 25 2019: (Start)
G.f.: 2*x*(5 + 47*x + 15*x^2 - 11*x^3 + 4*x^4) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A269609 Number of length-6 0..n arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

12, 376, 2878, 12570, 40288, 105892, 242226, 499798, 952180, 1702128, 2888422, 4693426, 7351368, 11157340, 16477018, 23757102, 33536476, 46458088, 63281550, 84896458, 112336432, 146793876, 189635458, 242418310, 306906948, 385090912

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

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

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

Row 6 of A269606.

Empirical: a(n) = n^6 + 6*n^5 + 11*n^4 - 8*n^3 + n^2 + 3*n - 2.
Conjectures from Colin Barker, Jan 25 2019: (Start)
G.f.: 2*x*(6 + 146*x + 249*x^2 - 50*x^3 - 2*x^4 + 12*x^5 - x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A269610 Number of length-7 0..n arrays with no repeated value differing from the previous repeated value by one or less.

Original entry on oeis.org

14, 902, 10192, 58280, 229754, 714874, 1886252, 4405772, 9366790, 18476654, 34284584, 60459952, 102126002, 166254050, 262123204, 401850644, 600997502, 879255382, 1261218560, 1777246904, 2464424554, 3367619402, 4540648412

Offset: 1

R. H. Hardin, Mar 01 2016

nonn

			Some solutions for n=4:
..1. .4. .2. .0. .2. .1. .4. .3. .1. .3. .4. .2. .4. .3. .3. .2
..4. .1. .0. .3. .4. .3. .2. .0. .3. .4. .1. .1. .1. .1. .4. .3
..0. .3. .3. .1. .1. .0. .4. .3. .1. .4. .4. .4. .0. .1. .2. .0
..1. .2. .2. .2. .1. .4. .0. .0. .0. .3. .2. .1. .3. .0. .4. .2
..1. .0. .0. .3. .3. .2. .1. .2. .2. .1. .2. .3. .4. .3. .3. .3
..4. .1. .3. .2. .3. .3. .4. .0. .1. .1. .4. .3. .4. .0. .2. .1
..4. .3. .1. .1. .2. .2. .0. .3. .1. .3. .1. .0. .3. .3. .1. .4

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

Row 7 of A269606.

Empirical: a(n) = n^7 + 7*n^6 + 16*n^5 - 12*n^4 - 5*n^3 + 18*n^2 - 15*n + 4.
Conjectures from Colin Barker, Jan 25 2019: (Start)
G.f.: 2*x*(7 + 395*x + 1684*x^2 + 608*x^3 - 321*x^4 + 143*x^5 + 6*x^6 - 2*x^7) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

cp's OEIS Frontend

A269600 Number of length-n 0..n arrays with no repeated value differing from the previous repeated value by one or less.

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A269601 Number of length-n 0..2 arrays with no repeated value differing from the previous repeated value by one or less.

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A269602 Number of length-n 0..4 arrays with no repeated value differing from the previous repeated value by one or less.

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A269603 Number of length-n 0..5 arrays with no repeated value differing from the previous repeated value by one or less.

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Maple

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A269604 Number of length-n 0..6 arrays with no repeated value differing from the previous repeated value by one or less.

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Maple

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A269605 Number of length-n 0..7 arrays with no repeated value differing from the previous repeated value by one or less.

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Maple

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A269607 Number of length-4 0..n arrays with no repeated value differing from the previous repeated value by one or less.

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A269608 Number of length-5 0..n arrays with no repeated value differing from the previous repeated value by one or less.

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A269609 Number of length-6 0..n arrays with no repeated value differing from the previous repeated value by one or less.

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