A269656 -id:A269656 - cp's OEIS frontend

A269657 Number of length-4 0..n arrays with no adjacent pair x,x+1 repeated.

1, 15, 79, 253, 621, 1291, 2395, 4089, 6553, 9991, 14631, 20725, 28549, 38403, 50611, 65521, 83505, 104959, 130303, 159981, 194461, 234235, 279819, 331753, 390601, 456951, 531415, 614629, 707253, 809971, 923491, 1048545, 1185889, 1336303

Offset: 0

R. H. Hardin, Mar 02 2016

nonn

I.e., a(n) = # {x in {0..n}^4 | x[1] != x[0]+1 or x[2] != x[0] or x[3] != x[1]}. The only possibility to have an adjacent x,x+1 pair repeated in a length-4 array is to have the array (x,x+1,x,x+1), with 0 <= x <= n-1 given the restriction on the domain of coefficients. This implies a(n) = (n+1)^4 - n and previously conjectured formulas. - M. F. Hasler, Feb 29 2020

			From _M. F. Hasler_, Feb 29 2020: (Start)
For n=0, the only length-4 0..0 array is (0,0,0,0) and it satisfies the restriction, so a(0) = 1.
For n=1, there is only one 4-tuple with coefficients in 0..1 which has a repeated pair (x,x+1), namely (0,1,0,1). Thus, a(1) = 2^4 - 1 = 15.
For n=2, there are two 4-tuples with coefficients in 0..2 which have a repeated pair (x,x+1), namely (0,1,0,1) and (1,2,1,2). Thus, a(1) = 3^4 - 2 = 79.
(End)
Some solutions for n=3 (length-4 arrays shown as columns):
  1  1  0  2  0  2  2  3  0  3  2  1  0  3  1  1
  1  0  0  1  3  2  0  1  2  3  2  1  2  0  0  2
  1  1  2  2  1  0  0  2  2  0  2  0  0  0  0  1
  3  3  0  1  0  0  1  2  2  1  3  3  2  2  0  3

R. H. Hardin, Table of n, a(n) for n = 0..210 (a(0) = 1 inserted by _M. F. Hasler_, Feb 29 2020).
John Elias, Illustration of initial terms: chain-linked cubes

Row 4 of A269656.

Denominator/@Flatten[Table[x/.Solve[m-Sqrt[-1/(1/(1/(1-x)-(m-1))-(m+1))]==0],{m,2,34}]] (* Ed Pegg Jr, Jan 14 2020 *)

apply( {A269657(n)=(n+1)^4-n}, [0..44]) \\ M. F. Hasler, Feb 29 2020

Empirical: a(n) = n^4 + 4*n^3 + 6*n^2 + 3*n + 1.
Conjectures from Colin Barker, Jan 25 2019: (Start)
G.f.: (1 + 10*x + 14*x^2 - 2*x^3 + x^4) / (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)
a(n) = (n+1)^4 - n, cf. comment, confirming the above conjectured formulas. - M. F. Hasler, Feb 29 2020

Extended to a(0) = 1 by M. F. Hasler, Feb 29 2020

A269658 Number of length-5 0..n arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

1, 26, 225, 988, 3065, 7686, 16681, 32600, 58833, 99730, 160721, 248436, 370825, 537278, 758745, 1047856, 1419041, 1888650, 2475073, 3198860, 4082841, 5152246, 6434825, 7960968, 9763825, 11879426, 14346801, 17208100, 20508713, 24297390

Offset: 0

R. H. Hardin, Mar 02 2016

nonn

The repeated pair is of the form (x,x+1) with 0 <= x <= n-1. Together with its repetition it occupies all but one position of the length-5 array. There are three choices for this position (beginning, middle, end; cf. example for n=1) and n+1 choices for the element in this position. This makes n*3*(n+1) forbidden arrays out of the (n+1)^5 possible ones. - M. F. Hasler, Feb 29 2020

			From _M. F. Hasler_, Feb 29 2020: (Start)
For n=0, there is only one array of length 5 with coefficients in 0..0, (0,0,0,0,0), and it satisfies the requirement, so a(0) = 1.
For n=1, the six arrays of length 5 with coefficients in 0..1 which do not satisfy the requirement are {(0,1,0,1,x), (0,1,x,0,1), (x,0,1,0,1); 0 <= x <= 1}, so a(1) = 2^5 - 6 = 26.
(End)
Some solutions for n=3:
  2  3  1  0  0  3  3  0  0  0  0  3  2  3  2  2
  2  0  3  3  2  3  3  1  2  3  0  3  3  1  0  3
  1  0  2  1  1  1  3  2  3  2  3  2  0  1  2  0
  0  2  0  1  0  2  2  2  1  3  3  2  2  1  3  3
  2  3  0  0  0  1  1  3  1  0  1  0  1  1  1  1

R. H. Hardin, Table of n, a(n) for n = 0..210 (a(0) = 1 inserted by _M. F. Hasler_, Feb 29 2020).

Row 5 of A269656.

apply( {A269658(n)=(n+1)^5-3*n*(n+1)}, [0..44]) \\ M. F. Hasler, Feb 29 2020

Empirical: a(n) = n^5 + 5*n^4 + 10*n^3 + 7*n^2 + 2*n + 1.
Conjectures from Colin Barker, Jan 25 2019: (Start)
G.f.: (1 + 20*x + 84*x^2 + 8*x^3 + 7*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)
a(n) = (n+1)^5 - 3*n*(n+1) = A000584(n+1) - A028896(n), cf. comment, which confirms the above conjectured formulas. - M. F. Hasler, Feb 29 2020

Extended to a(0) = 1 by M. F. Hasler, Feb 29 2020

A269659 Number of length-6 0..n arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

42, 626, 3816, 15036, 45590, 115902, 259476, 527576, 994626, 1764330, 2976512, 4814676, 7514286, 11371766, 16754220, 24109872, 33979226, 47006946, 63954456, 85713260, 113318982, 147966126, 191023556, 244050696, 308814450, 387306842

Offset: 1

R. H. Hardin, Mar 02 2016

nonn

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

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

Row 6 of A269656.

Empirical: a(n) = n^6 + 6*n^5 + 15*n^4 + 14*n^3 + 3*n^2 + 3*n.
Conjectures from Colin Barker, Jan 25 2019: (Start)
G.f.: 2*x*(21 + 166*x + 158*x^2 + 17*x^4 - 2*x^5) / (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)

A269660 Number of length-7 0..n arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

64, 1710, 14596, 73348, 269472, 803434, 2061940, 4725456, 9911008, 19355302, 35643204, 62486620, 105058816, 170389218, 267823732, 409555624, 611232000, 892640926, 1278484228, 1799241012, 2492126944, 3402154330, 4583298036

Offset: 1

R. H. Hardin, Mar 02 2016

nonn

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

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

Row 7 of A269656.

Empirical: a(n) = n^7 + 7*n^6 + 21*n^5 + 25*n^4 + 5*n^3 + 2*n^2 + 11*n - 8.
Conjectures from Colin Barker, Jan 26 2019: (Start)
G.f.: 2*x*(32 + 599*x + 1354*x^2 + 438*x^3 + 48*x^4 + 71*x^5 - 26*x^6 + 4*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)

A269649 Number of length-n 0..n arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

2, 9, 64, 621, 7686, 115902, 2061940, 42277329, 981592964, 25455321375, 729233710320, 22870710513322, 779384544768952, 28676004488510379, 1132947222095403840, 47837636065465105365, 2149808857973618974752

Offset: 1

R. H. Hardin, Mar 02 2016

nonn

Diagonal of A269656.

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

R. H. Hardin, Table of n, a(n) for n = 1..20

Cf. A269656.

A269650 Number of length-n 0..2 arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

3, 9, 27, 79, 225, 626, 1710, 4605, 12259, 32320, 84504, 219356, 565816, 1451349, 3704271, 9412153, 23818707, 60055275, 150913073, 378064818, 944442242, 2353140149, 5848794543, 14504575980, 35894673012, 88654500384, 218560230944

Offset: 1

R. H. Hardin, Mar 02 2016

nonn

			Some solutions for n=9:
..1. .2. .2. .0. .0. .2. .1. .2. .2. .2. .0. .0. .2. .1. .1. .0
..1. .1. .0. .0. .0. .0. .2. .1. .1. .2. .2. .2. .0. .1. .2. .0
..1. .2. .2. .0. .0. .1. .0. .0. .0. .1. .2. .2. .1. .1. .2. .0
..1. .1. .1. .0. .2. .0. .2. .1. .2. .2. .0. .2. .2. .1. .1. .1
..0. .1. .0. .2. .1. .0. .2. .1. .2. .0. .2. .2. .1. .2. .0. .1
..0. .0. .1. .0. .1. .2. .2. .2. .2. .2. .2. .0. .1. .2. .0. .1
..2. .2. .2. .2. .2. .0. .2. .2. .1. .2. .2. .2. .0. .0. .2. .0
..0. .2. .0. .0. .2. .0. .1. .2. .2. .0. .2. .0. .0. .0. .2. .2
..2. .2. .0. .1. .0. .2. .1. .0. .2. .1. .2. .2. .0. .1. .2. .1

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

Column 2 of A269656.

T:= Matrix(12,12):
for i from 1 to 12 do T[i,i]:= 1 od:
T[1,6]:= 1: T[3,8]:= 1:
T[5,11]:= 1: T[6,12]:= 1:
for i from 1 to 4 do T[i,i+8]:= 1; T[i+4,i]:= 1; T[i+8,i]:= 1; T[i+8,i+4]:= 1 od:
u:= <1,0,0,0,1,0,0,0,1,0,0,0>: v:= <1$12>:
seq(u^%T . T^i . v, i = 0 .. 50); # Robert Israel, Apr 19 2023

Empirical: a(n) = 9*a(n-1) - 33*a(n-2) + 66*a(n-3) - 84*a(n-4) + 75*a(n-5) - 47*a(n-6) + 21*a(n-7) - 6*a(n-8) + a(n-9).
Empirical g.f.: x*(1 - 2*x + x^2 - x^3)*(3 - 12*x + 18*x^2 - 14*x^3 + 5*x^4 - x^5) / (1 - 3*x + 2*x^2 - x^3)^3. - Colin Barker, Jan 25 2019
Empirical recurrence verified (see link). - Robert Israel, Apr 19 2023

A269651 Number of length-n 0..3 arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

4, 16, 64, 253, 988, 3816, 14596, 55344, 208196, 777582, 2885120, 10640607, 39027196, 142415244, 517252820, 1870499543, 6736838192, 24172420350, 86428990624, 308016010824, 1094338584760, 3876828265788, 13696906252336

Offset: 1

R. H. Hardin, Mar 02 2016

nonn

Column 3 of A269656.

			Some solutions for n=8
..3. .0. .1. .1. .1. .1. .0. .1. .2. .2. .2. .2. .3. .3. .1. .3
..1. .2. .1. .2. .3. .3. .3. .0. .0. .2. .1. .0. .3. .1. .3. .3
..3. .2. .0. .3. .1. .2. .3. .3. .2. .2. .2. .1. .0. .1. .0. .1
..2. .3. .0. .0. .0. .0. .2. .1. .3. .1. .1. .2. .2. .1. .1. .0
..1. .0. .3. .2. .1. .3. .2. .3. .1. .0. .3. .0. .1. .3. .1. .0
..3. .1. .3. .0. .2. .1. .3. .3. .3. .1. .1. .2. .1. .1. .2. .0
..2. .3. .0. .0. .0. .0. .2. .2. .3. .1. .3. .2. .0. .2. .3. .1
..2. .3. .0. .3. .2. .0. .0. .1. .3. .3. .3. .0. .3. .3. .1. .0

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

Cf. A269656.

Empirical: a(n) = 16*a(n-1) -108*a(n-2) +408*a(n-3) -986*a(n-4) +1704*a(n-5) -2312*a(n-6) +2560*a(n-7) -2355*a(n-8) +1832*a(n-9) -1208*a(n-10) +672*a(n-11) -314*a(n-12) +120*a(n-13) -36*a(n-14) +8*a(n-15) -a(n-16)

A269652 Number of length-n 0..4 arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

5, 25, 125, 621, 3065, 15036, 73348, 355921, 1718569, 8259567, 39522507, 188338770, 894017122, 4228223130, 19928007170, 93614937601, 438406639797, 2047055696994, 9531639358730, 44264054228691, 205038988521635, 947492030288164

Offset: 1

R. H. Hardin, Mar 02 2016

nonn

Column 4 of A269656.

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

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

Cf. A269656.

Empirical: a(n) = 25*a(n-1) -270*a(n-2) +1665*a(n-3) -6595*a(n-4) +18220*a(n-5) -38090*a(n-6) +64915*a(n-7) -94320*a(n-8) +119745*a(n-9) -134854*a(n-10) +136350*a(n-11) -124755*a(n-12) +103755*a(n-13) -78640*a(n-14) +54403*a(n-15) -34320*a(n-16) +19685*a(n-17) -10220*a(n-18) +4770*a(n-19) -1977*a(n-20) +715*a(n-21) -220*a(n-22) +55*a(n-23) -10*a(n-24) +a(n-25)

A269653 Number of length-n 0..5 arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

6, 36, 216, 1291, 7686, 45590, 269472, 1587450, 9321628, 54569340, 318513432, 1853885152, 10761305560, 62304937944, 359833051280, 2073213227007, 11917692236634, 68357220234784, 391252708773072, 2234836878469971

Offset: 1

R. H. Hardin, Mar 02 2016

nonn

Column 5 of A269656.

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

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

Cf. A269656.

Empirical: a(n) = 36*a(n-1) -570*a(n-2) +5244*a(n-3) -31353*a(n-4) +130248*a(n-5) -398332*a(n-6) +954000*a(n-7) -1897824*a(n-8) +3269288*a(n-9) -5011092*a(n-10) +6954336*a(n-11) -8842060*a(n-12) +10392360*a(n-13) -11368380*a(n-14) +11633264*a(n-15) -11177154*a(n-16) +10110960*a(n-17) -8629656*a(n-18) +6958872*a(n-19) -5305806*a(n-20) +3825752*a(n-21) -2608020*a(n-22) +1679472*a(n-23) -1020104*a(n-24) +583128*a(n-25) -312780*a(n-26) +156800*a(n-27) -73068*a(n-28) +31416*a(n-29) -12340*a(n-30) +4368*a(n-31) -1365*a(n-32) +364*a(n-33) -78*a(n-34) +12*a(n-35) -a(n-36)

A269654 Number of length-n 0..6 arrays with no adjacent pair x,x+1 repeated.

Original entry on oeis.org

7, 49, 343, 2395, 16681, 115902, 803434, 5556909, 38350583, 264117327, 1815254345, 12451509292, 85246693192, 582545874403, 3973788262533, 27059842131395, 183956317982661, 1248518868323747, 8460336000063413, 57241697970038501

Offset: 1

R. H. Hardin, Mar 02 2016

nonn

Column 6 of A269656.

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

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

Cf. A269656.

Empirical: a(n) = 49*a(n-1) -1071*a(n-2) +13804*a(n-3) -117159*a(n-4) +695709*a(n-5) -3021382*a(n-6) +10023853*a(n-7) -26667942*a(n-8) +59661994*a(n-9) -116448220*a(n-10) +203589267*a(n-11) -324797263*a(n-12) +479027591*a(n-13) -659300067*a(n-14) +852873532*a(n-15) -1042800234*a(n-16) +1210501656*a(n-17) -1338796662*a(n-18) +1414739998*a(n-19) -1431668679*a(n-20) +1390012356*a(n-21) -1296744764*a(n-22) +1163755845*a(n-23) -1005616962*a(n-24) +837248601*a(n-25) -671927508*a(n-26) +519927877*a(n-27) -387912963*a(n-28) +279014001*a(n-29) -193401978*a(n-30) +129119305*a(n-31) -82960080*a(n-32) +51242821*a(n-33) -30387210*a(n-34) +17270445*a(n-35) -9387672*a(n-36) +4867688*a(n-37) -2399859*a(n-38) +1120399*a(n-39) -492772*a(n-40) +202839*a(n-41) -77471*a(n-42) +27132*a(n-43) -8568*a(n-44) +2380*a(n-45) -560*a(n-46) +105*a(n-47) -14*a(n-48) +a(n-49)

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A269657 Number of length-4 0..n arrays with no adjacent pair x,x+1 repeated.

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A269658 Number of length-5 0..n arrays with no adjacent pair x,x+1 repeated.

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A269659 Number of length-6 0..n arrays with no adjacent pair x,x+1 repeated.

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A269660 Number of length-7 0..n arrays with no adjacent pair x,x+1 repeated.

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A269649 Number of length-n 0..n arrays with no adjacent pair x,x+1 repeated.

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A269650 Number of length-n 0..2 arrays with no adjacent pair x,x+1 repeated.

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A269651 Number of length-n 0..3 arrays with no adjacent pair x,x+1 repeated.

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A269652 Number of length-n 0..4 arrays with no adjacent pair x,x+1 repeated.

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A269653 Number of length-n 0..5 arrays with no adjacent pair x,x+1 repeated.

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