A090990 -id:A090990 - 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

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

Original entry on oeis.org

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

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

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

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

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

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

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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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