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

1,2

Comments

Conjecture: This triangle yields a permutation of the natural numbers.

Links

Table of n, a(n) for n=1..83.

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.

Example

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.

Prog

(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)

Crossrefs

Cf. A000045, A130233.

Sequence in context: A327887 A339361 A166310 * A293030 A109852 A083197

Adjacent sequences: A385587 A385588 A385589 * A385591 A385592 A385593

Keyword

nonn,easy,tabl

Author

Werner Schulte, Jul 03 2025

Status

approved