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

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

2,2

Comments

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).

Links

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).

Formula

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)

Example

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)

Mathematica

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}]

Prog

(Python)
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

Crossrefs

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.

Sequence in context: A208777 A104732 A132108 * A210489 A344821 A125175

Adjacent sequences: A376997 A376998 A376999 * A377001 A377002 A377003

Keyword

nonn,tabl,base

Author

Paolo Xausa, Oct 12 2024

Status

approved