A206603 -id:A206603 - cp's OEIS frontend

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

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)

A206604 Number of integers in the smallest interval containing both minimal and maximal possible apex values of an addition triangle whose base is a permutation of n+1 consecutive integers.

Original entry on oeis.org

1, 1, 3, 9, 27, 73, 189, 465, 1115, 2601, 5973, 13489, 30149, 66641, 146233, 318369, 689403, 1484137, 3181797, 6790641, 14445101, 30617841, 64724553, 136426849, 286926757, 601999633, 1260707529, 2634831585, 5497982025, 11452601761, 23823827825, 49484904257

Offset: 0

Alois P. Heinz, Feb 10 2012

nonn

For n>0 the base row of the addition triangle may contain a permutation of any set {b+k, k=0..n} where b is an integer or a half-integer. Each number in a higher row is the sum of the two numbers directly below it. Rows above the base row contain only integers.

a(n) = 3 (mod 4) if n = 2^m with m > 0 and a(n) = 1 (mod 4) else.

			a(3) =  9:   max:   20          min:   12
                  9   11             7   5
                3   6   5          5   2   3
             1/2 5/2 7/2 3/2    7/2 3/2 1/2 5/2
[12, 13, ..., 20] contains 20-12+1 = 9 integers.
a(4) = 27:   max:   13          min:  -13
                   5  8              -5 -8
                 0  5  3            0 -5 -3
              -2  2  3  0         2 -2 -3  0
            -2  0  2  1 -1      2  0 -2 -1  1
[-13, -12, ..., 13] contains 13-(-13)+1 = 27 integers.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A189390, A189391, A206603.

a:= n-> 1 +add(binomial(n, floor(k/2))*(2*k-n), k=0..n):
seq(a(n), n=0..40);
# second Maple program
a:= proc(n) option remember; `if`(n<3, 1+n*(n-1),
      (3*n^2-6*n+6+(2*n^2-6)*a(n-1)+4*(n-1)*(n-4)*a(n-2)
      -8*(n-1)*(n-2)*a(n-3)) / (n*(n-2)))
    end:
seq(a(n), n=0..40); # Alois P. Heinz, Apr 25 2013

a = DifferenceRoot[Function[{y, n}, {(-2n^2 - 12n - 12) y[n+2] - 3n^2 + 8(n+1)(n+2) y[n] - 4(n-1)(n+2) y[n+1] + (n+1)(n+3) y[n+3] - 12n - 15 == 0, y[0] == 1, y[1] == 1, y[2] == 3, y[3] == 9}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) = 1 + sum(k=0, n, binomial(n, k\2)*(2*k-n)); \\ Michel Marcus, Dec 20 2020

from math import comb
def A206604(n): return sum(comb(n,k>>1)*((k<<1)-n) for k in range(n+1))+1 # Chai Wah Wu, Oct 28 2024

a(n) = 1 + Sum_{k=0..n} C(n,floor(k/2)) * (2*k-n).
G.f.:  1/(1-x) + (1-sqrt(1-4*x^2)) / (2*x-1)^2.
a(n) = 1 + 2*A206603(n).
a(n) = 1 + A189390(n)-A189391(n).
a(n) ~ n*2^n * (1-2*sqrt(2)/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 15 2014

cp's OEIS Frontend

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

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