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).
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, 29, 32, 25, 30, 35, 40, 45, 50, 55, 60, 28, 33, 38, 43, 48, 53, 58, 63, 68, 31, 36, 41, 46, 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
Offset: 1
Examples
Triangle T(n, k) for 1 <= k <= n starts: n\ k : 1 2 3 4 5 6 7 8 9 10 11 12 13 ========================================================== 1 : 1 2 : 2 3 3 : 4 6 8 4 : 5 7 9 11 5 : 10 13 16 19 22 6 : 12 15 18 21 24 27 7 : 14 17 20 23 26 29 32 8 : 25 30 35 40 45 50 55 60 9 : 28 33 38 43 48 53 58 63 68 10 : 31 36 41 46 51 56 61 66 71 76 11 : 34 39 44 49 54 59 64 69 74 79 84 12 : 37 42 47 52 57 62 67 72 77 82 87 92 13 : 65 73 81 89 97 105 113 121 129 137 145 153 161 etc.
Programs
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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)
Formula
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.
Comments