A090995 -id:A090995 - cp's OEIS frontend

A090989 Number of meaningful differential operations of the n-th order on the space R^4.

4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728, 4194304, 6291456, 8388608

Offset: 1

Branko Malesevic, Feb 29 2004

G. C. Greubel, Table of n, a(n) for n = 1..1000
Branko 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.
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A063759, A029744.

Cf. A090990, A090991, A090992, A090993, A090994, A090995.

a:=[4,6];; for n in [3..40] do a[n]:=2*a[n-2]; od; a; # G. C. Greubel, Feb 02 2019

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

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 4; # <- 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[{0,2}, {4,6}, 40] (* G. C. Greubel, Feb 02 2019 *)

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

(2*(2+3*x)/(1-2*x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 02 2019

a(k+2) = 2*a(k).
a(n) = b(n+3) where b(n) = gcdConv(c(n)) = Sum_{k=0..n} gcd(c(k),c(n-k)) and c(k)=A000079(k) for k>0 and c(0)=1. - Tilman Neumann, Jan 11 2009 [Updated by Sean A. Irvine, Jan 15 2025]
G.f.: 2*x*(2+3*x)/(1-2*x^2). - Colin Barker, May 03 2012
a(n) = 2*A164090(n). - R. J. Mathar, Jan 25 2023
a(n) = (sqrt(2))^n*(3/2 + sqrt(2) + (-1)^n*(3/2 - sqrt(2))). - Taras Goy, Jan 04 2025

More terms from Tilman Neumann, Feb 06 2009

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0

Offset: 0

Alois P. Heinz, Dec 03 2012

Equivalently, the number of walks of length n-1 on the path graph P_k. - Andrew Howroyd, Apr 17 2017

			A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,   1,   1, ...
  0,  1,  2,  3,  4,   5,   6,   7, ...
  0,  0,  2,  4,  6,   8,  10,  12, ...
  0,  0,  2,  6, 10,  14,  18,  22, ...
  0,  0,  2,  8, 16,  24,  32,  40, ...
  0,  0,  2, 12, 26,  42,  58,  74, ...
  0,  0,  2, 16, 42,  72, 104, 136, ...
  0,  0,  2, 24, 68, 126, 188, 252, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0, 2-10 give: A000007, A040000, A029744(n+2) for n>0, A006355(n+3) for n>0, A090993(n+1) for n>0, A090995(n-1) for n>2, A129639, A153340, A153362, A153360.
Rows 0-6 give: A000012, A001477, A005843(k-1) for k>0, A016825(k-2) for k>1, A008590(k-2) for k>2, A113770(k-2) for k>3, A063164(k-2) for k>4.
Main diagonal gives: A102699.
Cf. A198632, A188866, A276562, A208727, A208671.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-1, j, k), j=1..k),
      `if`(i>1, b(n-1, i-1, k), 0)+
      `if`(i b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

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, Jan 19 2015, after Alois P. Heinz *)

TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i)));
ColGf(m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)==1, j->1, z);
a(n,k)=Vec(ColGf(k,x) + O(x^(n+1)))[n+1];
for(n=0, 7, for(k=0, 7, print1( a(n,k), ", ") ); print(); );
\\ Andrew Howroyd, Apr 17 2017

A090994 Number of meaningful differential operations of the n-th order on the space R^9.

Original entry on oeis.org

9, 17, 32, 61, 116, 222, 424, 813, 1556, 2986, 5721, 10982, 21053, 40416, 77505, 148785, 285380, 547810, 1050876, 2017126, 3869845, 7427671, 14250855, 27351502, 52479500, 100719775, 193258375, 370895324, 711682501, 1365808847, 2620797529

Offset: 1

Branko Malesevic, Feb 29 2004

nonn

Also number of meaningful compositions of the n-th order of the differential operations and Gateaux directional derivative on the space R^8. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007
Also (starting 5,9,...) the number of zig-zag paths from top to bottom of a rectangle of width 10, whose color is that of the top right corner. [From Joseph Myers, Dec 23 2008]
Also, number of n-digit terms in A033075 (stated without proof in A033075). - Zak Seidov, Feb 02 2011

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.
B. Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux Directional Derivative, arxiv:0706.0249 [math.CO], 2007.
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (1, 4, -3, -3, 1).

Cf. A090989-A090995, A000079, A007283, A020701, A020714, A129638, A033075.

a:=[9,17,32,61,116];; for n in [6..40] do a[n]:=a[n-1]+4*a[n-2] - 3*a[n-3]-3*a[n-4]+a[n-5]; od; a; # G. C. Greubel, Feb 02 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5) )); // G. C. Greubel, Feb 02 2019

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 9; # <- 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[{1, 4, -3, -3, 1}, {9, 17, 32, 61, 116}, 31] (* Jean-François Alcover, Nov 20 2017 *)

my(x='x+O('x^40)); Vec(x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2 +3*x^3+3*x^4-x^5)) \\ G. C. Greubel, Feb 02 2019

a=(x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019

a(k+5) = a(k+4) + 4*a(k+3) - 3*a(k+2) - 3*a(k+1) + a(k).

G.f.: x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; corrected by R. J. Mathar, Sep 16 2009

More terms from Joseph Myers, Dec 23 2008

A090991 Number of meaningful differential operations of the n-th order on the space R^6.

Original entry on oeis.org

6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 1

Branko Malesevic, Feb 29 2004

Apparently a(n) = A054886(n+2) for n=1..1000. - Georg Fischer, Oct 06 2018

Indranil Ghosh, Table of n, a(n) for n = 1..4772
Tanya Khovanova, Recursive Sequences
Branko J. 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 J. Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A054886, A055389, A068922, A090989-A090995.

Essentially the same as A006355, A047992 and A078642.

a:=[6,10];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # Muniru A Asiru, Oct 06 2018

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

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 6; # <- 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:

CoefficientList[Series[2*(3+2z)/(1-z-z^2), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

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

(2*(3+2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 02 2019

a(k+4) = 3*a(k+2) - a(k).
a(k) = 2*Fibonacci(k+3).
From Philippe Deléham, Nov 19 2008: (Start)
a(n) = a(n-1) + a(n-2), n>2, where a(1)=6, a(2)=10.
G.f.: 2*x*(3+2*x)/(1-x-x^2). (End)
E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 4. - Stefano Spezia, Apr 18 2022

A090990 Number of meaningful differential operations of the n-th order on the space R^5.

Original entry on oeis.org

5, 9, 16, 29, 52, 94, 169, 305, 549, 990, 1783, 3214, 5790, 10435, 18801, 33881, 61048, 110009, 198224, 357194, 643633, 1159797, 2089869, 3765830, 6785771, 12227562, 22033274, 39702627, 71541613, 128913593, 232294192, 418579765

Offset: 1

Branko Malesevic, Feb 29 2004

nonn

Also number of meaningful compositions of the n-th order of the differential operations and Gateaux directional derivative on the space R^4. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Branko 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 composition on the space R^n, arXiv:0704.0750 [math.DG], 2007.
Branko Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux Directional Derivative, arXiv:0706.0249 [math.CO], 2007.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Cf. A090989, A090991, A090992, A090993, A090994, A090995.

Cf. A000079, A007283, A020701, A020714, A129638.

a:=[5,9,16];; 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!(  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 := 5; # <- 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[{1, 2, -1}, {5, 9, 16}, 32] (* Jean-François Alcover, Nov 22 2017 *)

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

a=(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

a(n+3) = a(n+2) + 2*a(n+1) - a(n).

G.f.: x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3). - Ralf Stephan, Aug 19 2004

More terms from Ralf Stephan, Aug 19 2004

More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007

A090992 Number of meaningful differential operations of the n-th order on the space R^7.

Original entry on oeis.org

7, 13, 24, 45, 84, 158, 296, 557, 1045, 1966, 3691, 6942, 13038, 24516, 46055, 86585, 162680, 305809, 574624, 1080106, 2029680, 3814941, 7169145, 13474502, 25322375, 47592650, 89441626, 168100324, 315917527, 593742597, 1115852904, 2097145317

Offset: 1

Branko Malesevic, Feb 29 2004

Also number of meaningful compositions of the n-th order of the differential operations and Gateaux directional derivative on the space R^6. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007

Also (starting 4,7,...) the number of zig-zag paths from top to bottom of a rectangle of width 8, whose color is that of the top right corner. - Joseph Myers, Dec 23 2008

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.
B. Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux DirectionalDerivative, arXiv:0706.0249 [math.CO], 2007.
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
2008/9 British Mathematical Olympiad Round 1: Thursday, 4 December 2008, Problem 1 [From _Joseph Myers_, Dec 23 2008]
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

Cf. A090989-A090995.
Partial sums of pairwise sums of A065455.
Cf. A000079, A007283, A020701, A020714, A129638.

a:=[7,13,24,45];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2] - 2*a[n-3] - a[n-4]; od; a; # G. C. Greubel, Feb 02 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3)) )); // G. C. Greubel, Feb 02 2019

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 7; # <- 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[{1, 3, -2, -1}, {7, 13, 24, 45}, 32] (* Jean-François Alcover, Nov 25 2017 *)

my(x='x+O('x^40)); Vec(x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3))) \\ G. C. Greubel, Feb 02 2019

a=(x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3))).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019

a(n+4) = a(n+3) + 3*a(n+2) - 2*a(n+1) - a(n).

G.f.: x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3)). - Colin Barker, Mar 08 2012

More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007

More terms from Joseph Myers, Dec 23 2008

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

A129638 Number of meaningful differential operations of the k-th order on the space R^11.

Original entry on oeis.org

11, 21, 40, 77, 148, 286, 552, 1069, 2068, 4010, 7768, 15074, 29225, 56736, 110055, 213705, 414676, 805314, 1562977, 3035514, 5892257, 11443768, 22215753, 43146726, 83766396, 162686691, 315860810, 613439352, 1191054193, 2313133481

Offset: 11

Branko Malesevic, May 31 2007

Also number of meaningful compositions of the k-th order of the differential operations and Gateaux directional derivative on the space R^10. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 20 2007

Also (starting 6,11,...) the number of zig-zag paths from top to bottom of a rectangle of width 12, whose color is that of the top right corner. [Joseph Myers, Dec 23 2008]

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.
B. Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux Directional Derivative , arXiv:0706.0249 [math.CO], 2007.
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-6,3,1).

Cf. A090989-A090995.

Cf. A000079, A007283, A020701, A020714.

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n:=11; # <- 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[{1, 5, -4, -6, 3, 1}, {11, 21, 40, 77, 148, 286}, 30] (* Jean-François Alcover, Oct 10 2017 *)

a(k+6) = a(k+5) +5*a(k+4) -4*a(k+3) -6*a(k+2) +3*a(k+1) +a(k).

G.f.: -x^11*(6*x^5+21*x^4-24*x^3-36*x^2+10*x+11)/(x^6+3*x^5-6*x^4-4*x^3+5*x^2+x-1). [Colin Barker, Jul 08 2012]

More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 20 2007

More terms from Joseph Myers, Dec 23 2008

A129639 Number of meaningful differential operations of the k-th order on the space R^12.

Original entry on oeis.org

12, 22, 40, 74, 136, 252, 464, 860, 1584, 2936, 5408, 10024, 18464, 34224, 63040, 116848, 215232, 398944, 734848, 1362080, 2508928, 4650432, 8566016, 15877568, 29246208, 54209408, 99852800, 185082496, 340918784, 631911168, 1163969536

Offset: 12

Branko Malesevic, May 31 2007

nonn

Also (starting 7,12,...) the number of zig-zag paths from top to bottom of a rectangle of width 7. [Joseph Myers, Dec 23 2008]

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

Cf. A090989, A090990, A090991, A090992, A090993, A090994, A090995.

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n:=12; # <- 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:

f[k_] := f[k] = If[k <= 17, {12, 22, 40, 74, 136, 252}[[k-11]], 6 f[k-2] - 10 f[k-4] + 4 f[k-6]];
f /@ Range[12, 42] (* Jean-François Alcover, Apr 21 2020 *)

f(k+6) = 6*f(k+4)-10*f(k+2)+4*f(k).

Empirical G.f.: 2*x^12*(6+11*x-4*x^2-7*x^3)/(1-4*x^2+2*x^4). [Colin Barker, May 07 2012]

More terms from Joseph Myers, Dec 23 2008

A116183 Array T(k,n) = number of meaningful differential operations of the n-th order on the space R^(3+k), for k=>0, n>0, read by antidiagonals.

Original entry on oeis.org

3, 4, 5, 5, 6, 8, 6, 9, 8, 13, 7, 10, 16, 12, 21, 8, 13, 16, 29, 16, 34, 9, 14, 24, 26, 52, 24, 55, 10, 17, 24, 45, 42, 94, 32, 89, 11, 18, 32, 42, 84, 68, 169

Offset: 1

Jonathan Vos Post, Apr 08 2007

Two more rows can be obtained from A129638 and A129639.

			Table begins:
k=0.|.3..5..8.13..21..34..55..89..144..233..377..610..987.1597...
k=1.|.4..6..8.12..16..24..32..48...64...96..128..192..256..384...
k=2.|.5..9.16.29..52..94.169.305..549..990.1783.3214.5790...
k=3.|.6.10.16.26..42..68.110.178..288..466..754.1220.1974...
k=4.|.7.13.24.45..84.158.296.557.1045.1966.3691.6942.13038...
k=5.|.8.14.24.42..72.126.216.378..648.1134.1944.3402..5832...
k=6.|.9.17.32.61.116.222.424.813.1556.2986.5721.10982...
k=7.|10.18.32.58.104.188.338.610.1098.1980.3566.6428...

Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33; arXiv:0704.0750 [math.DG], 2007.

k=0 row is A020701. k=1 row is A090989. k=2 row is A090990. k=3 row is A090991. k=4 row is A090992. k=5 row is A090993. k=6 row is A090994. k=7 row is A090995.

Diagonal: A127935.

cp's OEIS Frontend

A090989 Number of meaningful differential operations of the n-th order on the space R^4.

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A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A090994 Number of meaningful differential operations of the n-th order on the space R^9.

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A090991 Number of meaningful differential operations of the n-th order on the space R^6.

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A090990 Number of meaningful differential operations of the n-th order on the space R^5.

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A090992 Number of meaningful differential operations of the n-th order on the space R^7.

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