A129638 -id:A129638 - cp's OEIS frontend

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

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

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

A377000 Array read by ascending antidiagonals: T(n,k) = number of n-esthetic numbers with k digits.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 8, 6, 1, 6, 9, 12, 13, 8, 1, 7, 11, 16, 21, 21, 12, 1, 8, 13, 20, 29, 36, 34, 16, 1, 9, 15, 24, 37, 52, 63, 55, 24, 1, 10, 17, 28, 45, 68, 94, 108, 89, 32, 1, 11, 19, 32, 53, 84, 126, 169, 189, 144, 48, 1, 12, 21, 36, 61, 100, 158, 232, 305, 324, 233, 64, 1

Offset: 2

Paolo Xausa, Oct 12 2024

A number is n-esthetic if, when written in base n, adjacent digits differ by 1: see De Koninck and Doyon (2009), where T(n,k) is denoted by N_q(r).

			Array begins (cf. De Koninck and Doyon (2009), table on p. 155):
  n\k| 1   2   3   4    5    6    7    8     9    10  ...
  -------------------------------------------------------
   2 | 1,  1,  1,  1,   1,   1,   1,   1,    1,    1, ... = A000012
   3 | 2,  3,  4,  6,   8,  12,  16,  24,   32,   48, ... = A029744 (from n = 2)
   4 | 3,  5,  8, 13,  21,  34,  55,  89,  144,  233, ... = A000045 (from n = 4)
   5 | 4,  7, 12, 21,  36,  63, 108, 189,  324,  567, ... = A228879
   6 | 5,  9, 16, 29,  52,  94, 169, 305,  549,  990, ...
   7 | 6, 11, 20, 37,  68, 126, 232, 430,  792, 1468, ...
   8 | 7, 13, 24, 45,  84, 158, 296, 557, 1045, 1966, ...
   9 | 8, 15, 28, 53, 100, 190, 360, 685, 1300, 2475, ...
  10 | 9, 17, 32, 61, 116, 222, 424, 813, 1556, 2986, ... = A090994
  ...                                               \______ A152086 (main diagonal)

Paolo Xausa, Table of n, a(n) for n = 2..11326 (first 150 antidiagonals, flattened).
Jean-Marie De Koninck and Nicolas Doyon, Esthetic Numbers, Ann. Sci. Math. Québec 33 (2009), No. 2, pp. 155-164.
Giovanni Resta, Esthetic Numbers, Numbers Aplenty, 2013.
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 (arXiv version, arXiv:0704.0750 [math.DG], 2007).

Cf. A000012 (row n = 2), A029744 (row n = 3), A000045 (row n = 4), A228879 (row n = 5), A090994 (row n = 10).
Cf. A102699, A152086 (main diagonal).
Diagonal above the main diagonal appears to be A206603.

A377000[n_, k_] := Round[2^k/(n+1)*Sum[If[m != (n+1)/2, Cos[#]^k*(Cot[#] + Csc[#])^2 & [Pi*m/(n+1)], 0], {m, 1, n, 2}]];
Table[A377000[n-k+1, k], {n, 2, 15}, {k, n-1}]

from itertools import count, islice
from functools import lru_cache
@lru_cache(maxsize=None)
def A377000_N(q,r,i):
    if r==1 and i==0: return 0
    if r==1: return 1
    if q==2: return r+i&1^1
    if i == 0: return A377000_N(q,r-1,1)
    if i == q-1: return A377000_N(q,r-1,q-2)
    return A377000_N(q,r-1,i-1)+A377000_N(q,r-1,i+1)
def A377000_T(n,k): return sum(A377000_N(n,k,i) for i in range(n))
def A377000_gen(): # generator of terms
    for n in count(2):
        for k in range(1,n):
            yield A377000_T(n-k+1,k)
A377000_list = list(islice(A377000_gen(),100)) # Chai Wah Wu, Oct 21 2024

All of the following formulas are taken from De Koninck and Doyon (2009).
T(n,k) = 2^k/(n+1) * Sum_{m=1..n, m odd, m != (n+1)/2} cos(p)^k*(cot(p) + csc(p))^2, where p = Pi*m/(n+1).
T(n,1) = n - 1.
T(2,k) = 1.
T(3,k) = 2^((k+1)/2) if k is odd, 3*2^((k-2)/2) if k is even = A029744(k+1).
T(4,k) = A000045(k+3).
T(5,k) = 4*3^((k-1)/2) if k is odd, 7*3^((k-2)/2) if k is even = A228879(k-1).
Conjectures from Chai Wah Wu, Oct 21 2024: (Start)
Conjecture 1: For even n, T(n,k) is the number of meaningful differential operations of the k-th order on the space R^(n-1).
Conjecture 2: For each n, the row T(n,k) satisfies a linear recurrence. For example:
T(6,k) = T(6,k-1) + 2*T(6,k-2) - T(6,k-3) for k > 3 (A090990).
T(7,k) = 4*T(7,k-2) - 2*T(7,k-4) for k > 4.
T(8,k) = T(8,k-1) + 3*T(8,k-2) - 2*T(8,k-3) - T(8,k-4) for k > 4 (A090992).
T(9,k) = 5*T(9,k-2) - 5*T(9,k-4) for k > 4.
T(10,k) = T(10,k-1) + 4*T(10,k-2) - 3*T(10,k-3) - 3*T(10,k-4) + T(10,k-5) for k > 5.
T(11,k) = 6*T(11,k-2) - 9*T(11,k-4) + 2*T(11,k-6) for k > 6.
T(12,k) = T(12,k-1) + 5*T(12,k-2) - 4*T(12,k-3) - 6*T(12,k-4) + 3*T(12,k-5) + T(12,k-6) for k > 6 (A129638).
...
Note that for even n, Conjecture 1 implies Conjecture 2 due to (Malesevic, 1998).
Conjecture 3: T(n,n-2) = A182555(n-2). (End)

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.

A127935 Number of meaningful differential operations of the n-th order on the space R^(2+n).

Original entry on oeis.org

3, 6, 16, 26, 84, 126, 424, 610, 2068, 2936, 9816, 13884, 45608, 64750, 208336, 297570, 938676, 1351492, 4181752, 6071028, 18454648, 27023598, 80796336, 119300636, 351331464, 522981328, 1518742384, 2278188504, 6531607248, 9869753934, 27963677600, 42547990626

Offset: 1

Jonathan Vos Post, Apr 09 2007, Jun 08 2007

nonn

			a(1) = 3 = A020701(1) is number of meaningful differential operations of the first order on the space R^3, namely {div, grad, curl}.
a(2) = 6 = A090989(2) is number of meaningful differential operations of the 2nd order on the space R^4 (some of them are identically zero though).
a(3) = 16 = A090990(3) is number of meaningful differential operations of the 3rd order on the space R^5.

R. Bott, L. W. Tu, Differential forms in algebraic topology, New York: Springer, 1982.

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.

Main diagonal of A116183.

Cf. A020701, A090989-A090995, A129638, A129639.

r[n_] := Table[Boole[j == i + 1 || i + j == n + 1], {i, n}, {j, n}];
Table[Total@Total@If[n == 1, IdentityMatrix[3], MatrixPower[r[n+2], n-1]], {n, 10}]
(* Andrey Zabolotskiy, Apr 30 2021 *)

Corrected from 8th term onwards. It appears the 8th and 9th terms listed were incorrectly taken from A000045 with two numbers concatenated together, whereas the 8th, 9th and 10th terms should have been the 8th term of A090995, the 9th of A129638 and the 10th of A129639. Joseph Myers, Dec 23 2008

Name and examples corrected, terms a(11) and beyond added by Andrey Zabolotskiy, Apr 30 2021

A153361 Number of zig-zag paths from top to bottom of a rectangle of width 12 with n rows.

Original entry on oeis.org

12, 22, 42, 80, 154, 296, 572, 1104, 2138, 4136, 8020, 15536, 30148, 58450, 113472, 220110, 427410, 829352, 1610628, 3125954, 6071028, 11784514, 22887536, 44431506, 86293452, 167532792, 325373382, 631721620, 1226878704, 2382108386

Offset: 1

Joseph Myers, Dec 24 2008

Number of words of length n using a 12-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 (1,5,-4,-6,3,1).

Column 12 of A220062.

Twice A129638.

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[i < k, b[n - 1, i + 1, k], 0]]]; a[n_] := b[n, 0, 12]; Array[a, 30] (* Jean-François Alcover, Oct 10 2017, after Alois P. Heinz *)

G.f.: -2*x*(3*x^5 + 12*x^4 - 12*x^3 - 20*x^2 + 5*x + 6)/(x^6 + 3*x^5 - 6*x^4 - 4*x^3 + 5*x^2 + x - 1). - Colin Barker, Sep 02 2012

cp's OEIS Frontend

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

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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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A377000 Array read by ascending antidiagonals: T(n,k) = number of n-esthetic numbers with k digits.

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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.

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A127935 Number of meaningful differential operations of the n-th order on the space R^(2+n).

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