A220062 -id:A220062 - cp's OEIS frontend

A090995 Number of meaningful differential operations of the n-th order on the space R^10.

10, 18, 32, 58, 104, 188, 338, 610, 1098, 1980, 3566, 6428, 11580, 20870, 37602, 67762, 122096, 220018, 396448, 714388, 1287266, 2319594, 4179738, 7531660, 13571542, 24455124, 44066548, 79405254, 143083226, 257827186, 464588384

Offset: 1

Branko Malesevic, Feb 29 2004

Also (starting 6,10,...) the number of zig-zag paths from top to bottom of a rectangle of width 6. - Joseph Myers, Dec 23 2008

Number of walks of length n on the path graph P_6. - Andrew Howroyd, Apr 17 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000
B. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
Branko Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Cf. A090989-A090994.

Column 6 of A220062.

a:=[10,18,32];; for n in [4..30] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Feb 02 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3) )); // G. C. Greubel, Feb 02 2019

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 10; # <- DIMENSION Fun := (i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity := (i,j)->piecewise(i=j,1,0); v := matrix(1,n,1); A := piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:

a[n_ /; n <= 6] := {10, 18, 32, 58, 104, 188}[[n]]; a[n_] := a[n] = 5*a[n-2] - 6*a[n-4] + a[n-6]; Array[a, 31] (* Jean-François Alcover, Oct 07 2017 *)
2*LinearRecurrence[{1,2,-1}, {5,9,16}, 40] (* G. C. Greubel, Feb 02 2019 *)

my(x='x+O('x^40)); Vec(2*x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3)) \\ G. C. Greubel, Feb 02 2019

a=(2*x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019

Equals 2 * A090990.
a(k+6) = 5*a(k+4) - 6*a(k+2) + a(k).
From Colin Barker, May 03 2012: (Start)
a(n) = a(n-1) + 2*a(n-2) - a(n-3).
G.f.: 2*x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3). (End)

More terms from Joseph Myers, Dec 23 2008

A188866 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally, vertically or antidiagonally.

Original entry on oeis.org

2, 4, 3, 8, 7, 4, 16, 17, 10, 5, 32, 41, 26, 13, 6, 64, 99, 68, 35, 16, 7, 128, 239, 178, 95, 44, 19, 8, 256, 577, 466, 259, 122, 53, 22, 9, 512, 1393, 1220, 707, 340, 149, 62, 25, 10, 1024, 3363, 3194, 1931, 950, 421, 176, 71, 28, 11, 2048, 8119, 8362, 5275, 2658, 1193, 502, 203, 80, 31, 12

Offset: 1

R. H. Hardin, Apr 12 2011

Number of 0..n strings of length k and adjacent elements differing by one or less. (See link for bijection.) Equivalently, number of base (n+1) k digit numbers with adjacent digits differing by one or less. - Andrew Howroyd, Mar 30 2017
All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions. - Andrew Howroyd, Apr 15 2017
Equivalently, the number of walks of length k-1 on the path graph P_{n+1} with a loop added at each vertex. - Pontus von Brömssen, Sep 08 2021

			Table starts:
   2  4  8  16  32   64  128   256   512   1024   2048    4096    8192    16384
   3  7 17  41  99  239  577  1393  3363   8119  19601   47321  114243   275807
   4 10 26  68 178  466 1220  3194  8362  21892  57314  150050  392836  1028458
   5 13 35  95 259  707 1931  5275 14411  39371 107563  293867  802859  2193451
   6 16 44 122 340  950 2658  7442 20844  58392 163594  458356 1284250  3598338
   7 19 53 149 421 1193 3387  9627 27383  77923 221805  631469 1797957  5119593
   8 22 62 176 502 1436 4116 11814 33942  97582 280676  807574 2324116  6689624
   9 25 71 203 583 1679 4845 14001 40503 117263 339699  984515 2854281  8277153
  10 28 80 230 664 1922 5574 16188 47064 136946 398746 1161634 3385486  9869934
  11 31 89 257 745 2165 6303 18375 53625 156629 457795 1338779 3916897 11463989
Some solutions for 5 X 3:
  1 1 1   1 1 1   1 1 1   1 1 1   0 0 0   1 1 1   1 1 1
  1 1 1   0 0 1   0 1 1   1 1 1   0 0 0   1 0 0   1 0 1
  0 0 0   0 0 0   0 0 1   1 1 1   0 0 0   0 0 0   0 0 0
  0 0 0   0 0 0   0 0 0   1 1 0   0 0 0   0 0 0   0 0 0
  0 0 0   0 0 0   0 0 0   0 0 0   0 0 0   0 0 0   0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..1741
Andrew Howroyd, Bijection with 0..n strings of length k.
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi and Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

Columns 2..8 are A016777, A017257(n-1), A188861-A188865.
Rows 2..31 are A001333(n+1), A126358, A057960(n+1), A126360, A002714, A126362-A126386.
Main diagonal is A188860.
Cf. A276562, A220062.

rows = 11; rowGf[n_, x_] = 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 + ChebyshevU[n-1, (1-x)/(2*x)])/ChebyshevU[n, (1-x)/(2*x)])/(1-3*x)^2;
row[n_] := rowGf[n+1, x] + O[x]^(rows+1) // CoefficientList[#, x]& // Rest; T = Array[row, rows]; Table[T[[n-k+1, k]], {n, 1, rows}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 07 2017, after Andrew Howroyd *)

\\ from Knopfmacher et al.
RowGf(k, x='x) = my(z=(1-x)/(2*x)); 1 + (x*(k-(3*k+2)*x) + (2*x^2)*(1+polchebyshev(k-1, 2, z))/polchebyshev(k, 2, z))/(1-3*x)^2;
T(n,k) = {polcoef(RowGf(n+1) + O(x*x^k),k)}
for(n=1, 10, print(Vec(RowGf(n+1) + O(x^11)))) \\ Andrew Howroyd, Apr 15 2017 [updated Mar 13 2021]

Empirical: T(n,1) = n + 1.
Empirical: T(n,2) = 3*n + 1.
Empirical: T(n,3) = 9*n - 1.
Empirical: T(n,4) = 27*n - 13 for n > 1.
Empirical: T(n,5) = 81*n - 65 for n > 2.
Empirical: T(n,6) = 243*n - 265 for n > 3.
Empirical: T(n,7) = 729*n - 987 for n > 4.
Empirical: T(n,8) = 2187*n - 3495 for n > 5.
Empirical: T(1,k) = 2*T(1,k-1).
Empirical: T(2,k) = 2*T(2,k-1) + T(2,k-2).
Empirical: T(3,k) = 3*T(3,k-1) - T(3,k-2).
Empirical: T(4,k) = 3*T(4,k-1) - 2*T(4,k-3).
Empirical: T(5,k) = 4*T(5,k-1) - 3*T(5,k-2) -   T(5,k-3).
Empirical: T(6,k) = 4*T(6,k-1) - 2*T(6,k-2) - 4*T(6,k-3) +   T(6,k-4).
Empirical: T(7,k) = 5*T(7,k-1) - 6*T(7,k-2) -   T(7,k-3) + 2*T(7,k-4).
Empirical: T(8,k) = 5*T(8,k-1) - 5*T(8,k-2) - 5*T(8,k-3) + 5*T(8,k-4) + T(8,k-5).

A102699 Number of strings of length n, using as symbols numbers from the set {1, 2, ..., n}, in which consecutive symbols differ by exactly 1.

Original entry on oeis.org

1, 1, 2, 6, 16, 42, 104, 252, 592, 1370, 3112, 6996, 15536, 34244, 74832, 162616, 351136, 754938, 1615208, 3443940, 7314928, 15493676, 32714992, 68918856, 144815456, 303703972, 635554064, 1327816392, 2769049312, 5766417480, 11989472672, 24897569648

Offset: 0

Don Rogers (donrogers42(AT)aol.com), Feb 07 2005

nonn

Equally, number of different n-digit numbers, using only the digits 1 through n, where consecutive digits differ by 1. It is assumed that there are n different digits available even when n > 9.
Number of endomorphisms of a path P_n. - N. J. A. Sloane, Sep 20 2009
a(n) is also the number of distinct paths of length n starting from the bottom row of an n X n chess board and ending at the top row, such that all the n squares traversed in the path are of the same color. - Kiran Ananthpur Bacche, Oct 25 2022

			For example, a(4)=16: the 16 strings are 1212, 1232, 1234, 2121, 2123, 2321, 2323, 2343, 3212, 3232, 3234, 3432, 3434, 4321, 4323, 4343.
G.f. = x + 2*x^2 + 6*x^3 + 16*x^4 + 42*x^5 + 104*x^6 + 252*x^7 + 592*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..3311 (terms n = 1..300 from T. D. Noe)
Sr. Arworn, An algorithm for the number of endomorphisms on paths, Disc. Math., 309 (2009), 94-103 (see p. 95).
Zhicong Lin and Jiang Zeng, On the number of congruence classes of paths, arXiv preprint arXiv:1112.4026 [math.CO], 2011.
M. A. Michels and U. Knauer, The congruence classes of paths and cycles, Discr. Math., 309 (2009), 5352-5359. See p. 5356. [From _N. J. A. Sloane_, Sep 20 2009]
Joseph Myers, BMO 2008-2009 Round 1 Problem 1-Generalisation

Cf. A110971, A152086.

Main diagonal of A220062. - Alois P. Heinz, Dec 03 2012

p:= 0; paths := proc(m, n, s, t) global p; if(((t+1) <= m) and s <= (n)) then paths(m,n,s+1,t+1); end if; if(((t-1) > 0) and s <= (n)) then paths(m,n,s+1,t-1); end if; if(s = n) then p:=p+1; end if; end proc; sumpaths:=proc(j) global p; p:=0; sp:=0; for h from 1 to j do p:=0; paths(j,j,1,h); sp:=sp+ p ; end do; sp; end proc; for l from 1 to 50 do sumpaths(l); end do; # Ben Paul Thurston, Oct 04 2006
# second Maple program:
a:= proc(n) option remember;
      `if`(n<5, [1, 1, 2, 6, 16][n+1], ((2*n^2-6*n-4) *a(n-1)
      +(56-32*n+4*n^2) *a(n-2) -8*(n-3)^2 *a(n-3))/ ((n-1)*(n-4)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 23 2012

a[n_] := a[n] = If[n <= 4, n*((n-3)*n+4)/2, ((2*n^2 - 6*n - 4)*a[n-1] + (4*n^2 - 32*n + 56)*a[n-2] - 8*(n-3)^2*a[n-3])/((n-1)*(n-4))]; Table[ a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *)

x='x+O('x^55); Vec(x*(2*(1-x)-sqrt(1-4*x^2))/(1-2*x)^2) \\ Altug Alkan, Nov 10 2015

from math import comb
def A102699(n): return ((n+1<>1) if n&1 else (n-1)*comb(n-2,n-2>>1)<<2)) if n else 1 # Chai Wah Wu, Oct 28 2024

It appears that the limit of a(n)/a(n-1) is decreasing towards 2. - Ben Paul Thurston, Oct 04 2006
a(n) = (n+1)2^(n-1) - 4(n-1)binomial(n-2,(n-2)/2) for n even, a(n) = (n+1)2^(n-1) - (2n-1)binomial(n-1,(n-1)/2) for n odd. - Joseph Myers, Dec 23 2008
a(n) = 2 * Sum_{k=1..n-1} k*A110971(n,k). - N. J. A. Sloane, Sep 20 2009
G.f.: x * (2*(1 - x) - sqrt(1 - 4*x^2)) / (1 - 2*x)^2. - Michael Somos, Mar 17 2014
0 = a(n) * 8*n^2 - a(n+1) * 4*(n^2 - 2*n - 1) - a(n+2) * 2*(n^2 + 3*n - 2) + a(n+3) * (n-1)*(n+2) for n>0. - Michael Somos, Mar 17 2014
0 = a(n) * (16*a(n+1) - 16*a(n+2) + 4*a(n+3)) + a(n+1) * (-16*a(n+1) + 20*a(n+2) - 4*a(n+3)) + a(n+2) * (-4*a(n+2) + a(n+3)) for n>0. - Michael Somos, Mar 17 2014

More terms from Ben Paul Thurston, Oct 04 2006
a(20) onwards from David Wasserman, Apr 26 2008
Edited by N. J. A. Sloane, Jan 03 2009 and Sep 23 2010
a(0)=1 prepended by Alois P. Heinz, Apr 17 2017

A090993 Number of meaningful differential operations of the n-th order on the space R^8.

Original entry on oeis.org

8, 14, 24, 42, 72, 126, 216, 378, 648, 1134, 1944, 3402, 5832, 10206, 17496, 30618, 52488, 91854, 157464, 275562, 472392, 826686, 1417176, 2480058, 4251528, 7440174, 12754584, 22320522, 38263752, 66961566, 114791256, 200884698

Offset: 1

Branko Malesevic, Feb 29 2004

Also (starting 5,8,...) the number of zig-zag paths from top to bottom of a rectangle of width 5. - Joseph Myers, Dec 23 2008

Number of walks of length n on the path graph P_5. - Andrew Howroyd, Apr 17 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000
B. Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
Branko Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A090989-A090995.

Column 5 of A220062.

a:=[8,14];; for n in [3..40] do a[n]:=3*a[n-2]; od; a; # G. C. Greubel, Feb 02 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(4+7*x)/(1-3*x^2) )); // G. C. Greubel, Feb 02 2019

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 8; # <- DIMENSION Fun := (i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity := (i,j)->piecewise(i=j,1,0); v := matrix(1,n,1); A := piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:

LinearRecurrence[{0, 3}, {8, 14}, 32] (* Jean-François Alcover, Jul 01 2018 *)

my(x='x+O('x^40)); Vec(2*x*(4+7*x)/(1-3*x^2)) \\ G. C. Greubel, Feb 02 2019

a=(2*x*(4+7*x)/(1-3*x^2)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019

a(n+4) = 4*a(n+2) - 3*a(n).
From Colin Barker, May 03 2012: (Start)
a(n) = 3*a(n-2).
G.f.: 2*x*(4+7*x)/(1-3*x^2). (End)
a(n) = (11+3*(-1)^n) * 3^floor((n-1)/2). - Ralf Stephan, Jul 19 2013

More terms from Joseph Myers, Dec 23 2008

A208671 T(n,k) = number of 2n-bead necklaces labeled with numbers 1..k allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 4, 1, 0, 5, 7, 8, 6, 1, 0, 6, 9, 12, 14, 8, 1, 0, 7, 11, 16, 23, 24, 13, 1, 0, 8, 13, 20, 32, 44, 47, 18, 1, 0, 9, 15, 24, 41, 65, 97, 89, 30, 1, 0, 10, 17, 28, 50, 86, 152, 212, 187, 46, 1, 0, 11, 19, 32, 59, 107, 208, 360, 512, 396, 78, 1, 0, 12, 21, 36

Offset: 1

R. H. Hardin, Feb 29 2012

Table starts
.0.1..2...3...4...5....6....7....8....9...10..11..12..13.14.15.16
.0.1..3...5...7...9...11...13...15...17...19..21..23..25.27.29
.0.1..4...8..12..16...20...24...28...32...36..40..44..48.52
.0.1..6..14..23..32...41...50...59...68...77..86..95.104
.0.1..8..24..44..65...86..107..128..149..170.191.212
.0.1.13..47..97.152..208..264..320..376..432.488
.0.1.18..89.212.360..514..669..824..979.1134
.0.1.30.187.512.937.1398.1866.2335.2804

			All solutions for n=4, k=3:
..1....1....1....1....1....2
..2....2....2....2....2....3
..3....1....1....1....3....2
..2....2....2....2....2....3
..1....3....1....1....3....2
..2....2....2....2....2....3
..3....3....3....1....3....2
..2....2....2....2....2....3

R. H. Hardin, Table of n, a(n) for n = 1..148

Column 3 is A000029.

Cf. A208727, A220062.

T(n,k) = (2*A208727(n) + A220062(n+1,k))/4. - Andrew Howroyd, Mar 19 2017

A276562 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 1 or less.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 10, 5, 1, 32, 35, 22, 13, 6, 1, 64, 83, 54, 29, 16, 7, 1, 128, 199, 134, 73, 36, 19, 8, 1, 256, 479, 340, 185, 92, 43, 22, 9, 1, 512, 1155, 872, 481, 236, 111, 50, 25, 10, 1, 1024, 2787, 2254, 1265, 622, 287, 130, 57, 28, 11

Offset: 1

Andrew Howroyd, Apr 15 2017

All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions.

			Array starts:
   1  1  1   1   1    1    1    1     1     1 ...
   2  4  8  16  32   64  128  256   512  1024 ...
   3  7 15  35  83  199  479 1155  2787  6727 ...
   4 10 22  54 134  340  872 2254  5854 15250 ...
   5 13 29  73 185  481 1265 3361  8993 24193 ...
   6 16 36  92 236  622 1658 4468 12132 33146 ...
   7 19 43 111 287  763 2051 5575 15271 42099 ...
   8 22 50 130 338  904 2444 6682 18410 51052 ...
   9 25 57 149 389 1045 2837 7789 21549 60005 ...
  10 28 64 168 440 1186 3230 8896 24688 68958 ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

Rows 3-32 are A124696-A124719, A124726, A124783, A124784, A124799, A124803, A124804.

Cf. A188866, A220062, A285280, A285281, A208777, A208721.

T[m_, n_] := Sum[(1 + 2*Cos[j*Pi/(m+1)])^n, {j, 1, m}] // FullSimplify;
Table[T[m-n+1, n], {m, 1, 11}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Jun 06 2017 *)

\\ from Knopfmacher et al.
ChebyshevU(n,x) = sum(i=0, n/2, 2*poltchebi(n-2*i,x)) + (n%2-1);
RowGf(k,x) = 1 + (k*x*(1+3*x) - 2*(k+1)*x*subst(ChebyshevU(k-1,z)/ChebyshevU(k,z),z,(1-x)/(2*x)))/((1+x)*(1-3*x));
a(m,n)=Vec(RowGf(m,x)+O(x^(n+1)))[n+1];
for(m=1, 10, print(RowGf(m,x)));
for(m=1, 10, for(n=1, 9, print1( a(m,n), ", ") ); print(); );

T(m, n) = Sum_{j=1..m} (1 + 2*cos(j*pi/(m+1)))^n. - Andrew Howroyd, Apr 15 2017

A153362 Number of zig-zag paths from top to bottom of a rectangle of width 9 with n rows.

Original entry on oeis.org

9, 16, 30, 56, 106, 200, 380, 720, 1370, 2600, 4950, 9400, 17900, 34000, 64750, 123000, 234250, 445000, 847500, 1610000, 3066250, 5825000, 11093750, 21075000, 40137500, 76250000, 145218750, 275875000, 525406250, 998125000, 1900937500

Offset: 1

Joseph Myers, Dec 24 2008

Number of words of length n using a 9 symbol alphabet where neighboring letters are neighbors in the alphabet. - Andrew Howroyd, Apr 17 2017

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation

Column 9 of A220062.

Cf. A153363, A153364, A153365, A153366, A153367.

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n - 1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jul 01 2018, after Alois P. Heinz *)

Empirical G.f.: x*(9+16*x-15*x^2-24*x^3+x^4)/(1-5*x^2+5*x^4). [Colin Barker, Sep 02 2012]

A153368 Number of zig-zag paths from top to bottom of a rectangle of width 11 with n rows.

Original entry on oeis.org

11, 20, 38, 72, 138, 264, 508, 976, 1882, 3624, 6996, 13488, 26054, 50264, 97124, 187440, 362250, 699240, 1351492, 2609008, 5042950, 9735768, 18818772, 36332016, 70229066, 135588200, 262091348, 506012592, 978124038, 1888445784, 3650380228

Offset: 1

Joseph Myers, Dec 24 2008

Heuristically, a(n) = +6*a(n-2) -9*a(n-4) +2*a(n-6). - R. J. Mathar, Jun 16 2011

Number of words of length n using a 11 symbol alphabet where neighboring letters are neighbors in the alphabet. - Andrew Howroyd, Apr 17 2017

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -9, 0, 2).

Column 11 of A220062.

Cf. A153369, A153370, A153371, A153372 (bisection), A153373.

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n - 1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jul 01 2018, after Alois P. Heinz *)

Empirical G.f.: x*(11+20*x-28*x^2-48*x^3+9*x^4+12*x^5)/((1-2*x^2)*(1-4*x^2+x^4)). - Colin Barker, Apr 17 2012

a(n) = A153369(n) + A153370(n). - Andrew Howroyd, Apr 17 2017

A153340 Number of zig-zag paths from top to bottom of a rectangle of width 8 with n rows.

Original entry on oeis.org

8, 14, 26, 48, 90, 168, 316, 592, 1114, 2090, 3932, 7382, 13884, 26076, 49032, 92110, 173170, 325360, 611618, 1149248, 2160212, 4059360, 7629882, 14338290, 26949004, 50644750, 95185300, 178883252, 336200648, 631835054, 1187485194, 2231705808

Offset: 1

Joseph Myers, Dec 24 2008

Number of words of length n using an 8-symbol alphabet where neighboring letters are neighbors in the alphabet. - Andrew Howroyd, Apr 17 2017

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

Column 8 of A220062.

Twice A090992.

LinearRecurrence[{1, 3, -2, -1}, {8, 14, 26, 48}, 32] (* Jean-François Alcover, Oct 08 2017 *)

G.f.: 2*x*(4+3*x-6*x^2-2*x^3)/((1-x)*(1-3*x^2-x^3)). - Colin Barker, May 10 2012

A153360 Number of zig-zag paths from top to bottom of a rectangle of width 10 with n rows.

Original entry on oeis.org

10, 18, 34, 64, 122, 232, 444, 848, 1626, 3112, 5972, 11442, 21964, 42106, 80832, 155010, 297570, 570760, 1095620, 2101752, 4034252, 7739690, 14855342, 28501710, 54703004, 104959000, 201439550, 386516750, 741790648, 1423365002, 2731617694

Offset: 1

Joseph Myers, Dec 24 2008

Number of words of length n using a 10 symbol alphabet where neighboring letters are neighbors in the alphabet. - Andrew Howroyd, Apr 17 2017

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (1, 4, -3, -3, 1).

Column 10 of A220062.

Twice A090994.

LinearRecurrence[{1, 4, -3, -3, 1}, {10, 18, 34, 64, 122}, 31] (* Jean-François Alcover, Jul 01 2018 *)

G.f.: 2*x*(5+4*x-12*x^2-6*x^3+3*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5) [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.

cp's OEIS Frontend

A090995 Number of meaningful differential operations of the n-th order on the space R^10.

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A188866 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally, vertically or antidiagonally.

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A102699 Number of strings of length n, using as symbols numbers from the set {1, 2, ..., n}, in which consecutive symbols differ by exactly 1.

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A090993 Number of meaningful differential operations of the n-th order on the space R^8.

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A208671 T(n,k) = number of 2n-bead necklaces labeled with numbers 1..k allowing reversal, with neighbors differing by exactly 1.

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A276562 Array read by antidiagonals: T(m,n) = number of m-ary words of length n with cyclically adjacent elements differing by 1 or less.

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