This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377000 #44 Oct 23 2024 08:20:59
%S A377000 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,
%T A377000 20,29,36,34,16,1,9,15,24,37,52,63,55,24,1,10,17,28,45,68,94,108,89,
%U A377000 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
%N A377000 Array read by ascending antidiagonals: T(n,k) = number of n-esthetic numbers with k digits.
%C A377000 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).
%H A377000 Paolo Xausa, <a href="/A377000/b377000.txt">Table of n, a(n) for n = 2..11326</a> (first 150 antidiagonals, flattened).
%H A377000 Jean-Marie De Koninck and Nicolas Doyon, <a href="https://www.labmath.uqam.ca/~annales/volumes/33-2/PDF/155-164.pdf">Esthetic Numbers</a>, Ann. Sci. Math. Québec 33 (2009), No. 2, pp. 155-164.
%H A377000 Giovanni Resta, <a href="https://www.numbersaplenty.com/set/esthetic_number/">Esthetic Numbers</a>, Numbers Aplenty, 2013.
%H A377000 Branko J. Malesevic, <a href="https://www.jstor.org/stable/43666958">Some combinatorial aspects of differential operation composition on the space R^n</a>, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33 (<a href="http://arxiv.org/abs/0704.0750">arXiv version</a>, arXiv:0704.0750 [math.DG], 2007).
%F A377000 All of the following formulas are taken from De Koninck and Doyon (2009).
%F A377000 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).
%F A377000 T(n,1) = n - 1.
%F A377000 T(2,k) = 1.
%F A377000 T(3,k) = 2^((k+1)/2) if k is odd, 3*2^((k-2)/2) if k is even = A029744(k+1).
%F A377000 T(4,k) = A000045(k+3).
%F A377000 T(5,k) = 4*3^((k-1)/2) if k is odd, 7*3^((k-2)/2) if k is even = A228879(k-1).
%F A377000 Conjectures from _Chai Wah Wu_, Oct 21 2024: (Start)
%F A377000 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).
%F A377000 Conjecture 2: For each n, the row T(n,k) satisfies a linear recurrence. For example:
%F A377000 T(6,k) = T(6,k-1) + 2*T(6,k-2) - T(6,k-3) for k > 3 (A090990).
%F A377000 T(7,k) = 4*T(7,k-2) - 2*T(7,k-4) for k > 4.
%F A377000 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).
%F A377000 T(9,k) = 5*T(9,k-2) - 5*T(9,k-4) for k > 4.
%F A377000 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.
%F A377000 T(11,k) = 6*T(11,k-2) - 9*T(11,k-4) + 2*T(11,k-6) for k > 6.
%F A377000 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).
%F A377000 ...
%F A377000 Note that for even n, Conjecture 1 implies Conjecture 2 due to (Malesevic, 1998).
%F A377000 Conjecture 3: T(n,n-2) = A182555(n-2). (End)
%e A377000 Array begins (cf. De Koninck and Doyon (2009), table on p. 155):
%e A377000 n\k| 1 2 3 4 5 6 7 8 9 10 ...
%e A377000 -------------------------------------------------------
%e A377000 2 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... = A000012
%e A377000 3 | 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, ... = A029744 (from n = 2)
%e A377000 4 | 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... = A000045 (from n = 4)
%e A377000 5 | 4, 7, 12, 21, 36, 63, 108, 189, 324, 567, ... = A228879
%e A377000 6 | 5, 9, 16, 29, 52, 94, 169, 305, 549, 990, ...
%e A377000 7 | 6, 11, 20, 37, 68, 126, 232, 430, 792, 1468, ...
%e A377000 8 | 7, 13, 24, 45, 84, 158, 296, 557, 1045, 1966, ...
%e A377000 9 | 8, 15, 28, 53, 100, 190, 360, 685, 1300, 2475, ...
%e A377000 10 | 9, 17, 32, 61, 116, 222, 424, 813, 1556, 2986, ... = A090994
%e A377000 ... \______ A152086 (main diagonal)
%t A377000 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}]];
%t A377000 Table[A377000[n-k+1, k], {n, 2, 15}, {k, n-1}]
%o A377000 (Python)
%o A377000 from itertools import count, islice
%o A377000 from functools import lru_cache
%o A377000 @lru_cache(maxsize=None)
%o A377000 def A377000_N(q,r,i):
%o A377000 if r==1 and i==0: return 0
%o A377000 if r==1: return 1
%o A377000 if q==2: return r+i&1^1
%o A377000 if i == 0: return A377000_N(q,r-1,1)
%o A377000 if i == q-1: return A377000_N(q,r-1,q-2)
%o A377000 return A377000_N(q,r-1,i-1)+A377000_N(q,r-1,i+1)
%o A377000 def A377000_T(n,k): return sum(A377000_N(n,k,i) for i in range(n))
%o A377000 def A377000_gen(): # generator of terms
%o A377000 for n in count(2):
%o A377000 for k in range(1,n):
%o A377000 yield A377000_T(n-k+1,k)
%o A377000 A377000_list = list(islice(A377000_gen(),100)) # _Chai Wah Wu_, Oct 21 2024
%Y A377000 Cf. A000012 (row n = 2), A029744 (row n = 3), A000045 (row n = 4), A228879 (row n = 5), A090994 (row n = 10).
%Y A377000 Cf. A102699, A152086 (main diagonal).
%Y A377000 Diagonal above the main diagonal appears to be A206603.
%K A377000 nonn,tabl,base
%O A377000 2,2
%A A377000 _Paolo Xausa_, Oct 12 2024