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 A385590 #7 Jul 07 2025 20:01:36
%S A385590 1,2,3,4,6,8,5,7,9,11,10,13,16,19,22,12,15,18,21,24,27,14,17,20,23,26,
%T A385590 29,32,25,30,35,40,45,50,55,60,28,33,38,43,48,53,58,63,68,31,36,41,46,
%U A385590 51,56,61,66,71,76,34,39,44,49,54,59,64,69,74,79,84,37,42,47,52,57,62,67,72,77,82,87,92,65,73,81,89,97
%N A385590 Triangle read by rows, based on Fibonacci numbers: Let i > 1 be such that F(i) <= n < F(i+1); i.e., i = A130233(n). Then T(n, k) = F(i-1)^2 + 1 - (i-1) mod 2 + (n - F(i)) * F(i-2) + (k-1) * F(i-1) where F(k) = A000045(k).
%C A385590 Conjecture: This triangle yields a permutation of the natural numbers.
%F A385590 Conjecture: Sum_{k=1..n} (-1)^k * binomial(n-1, k-1) * T(n, k) = 0 for n > 2 and (-1)^n for n < 3.
%e A385590 Triangle T(n, k) for 1 <= k <= n starts:
%e A385590 n\ k : 1 2 3 4 5 6 7 8 9 10 11 12 13
%e A385590 ==========================================================
%e A385590 1 : 1
%e A385590 2 : 2 3
%e A385590 3 : 4 6 8
%e A385590 4 : 5 7 9 11
%e A385590 5 : 10 13 16 19 22
%e A385590 6 : 12 15 18 21 24 27
%e A385590 7 : 14 17 20 23 26 29 32
%e A385590 8 : 25 30 35 40 45 50 55 60
%e A385590 9 : 28 33 38 43 48 53 58 63 68
%e A385590 10 : 31 36 41 46 51 56 61 66 71 76
%e A385590 11 : 34 39 44 49 54 59 64 69 74 79 84
%e A385590 12 : 37 42 47 52 57 62 67 72 77 82 87 92
%e A385590 13 : 65 73 81 89 97 105 113 121 129 137 145 153 161
%e A385590 etc.
%o A385590 (PARI) T(n, k) = i=1; for(j=1,n,if(j==fibonacci(i+1),i=i+1)); (fibonacci(i-1))^2+1-(i-1)%2 + (n-fibonacci(i))*fibonacci(i-2) + (k-1)*fibonacci(i-1)
%Y A385590 Cf. A000045, A130233.
%K A385590 nonn,easy,tabl
%O A385590 1,2
%A A385590 _Werner Schulte_, Jul 03 2025