A094565 -id:A094565 - cp's OEIS frontend

A271354 Products of two distinct Fibonacci numbers, both greater than 1.

6, 10, 15, 16, 24, 26, 39, 40, 42, 63, 65, 68, 102, 104, 105, 110, 165, 168, 170, 178, 267, 272, 273, 275, 288, 432, 440, 442, 445, 466, 699, 712, 714, 715, 720, 754, 1131, 1152, 1155, 1157, 1165, 1220, 1830, 1864, 1869, 1870, 1872, 1885, 1974, 2961, 3016

Offset: 1

Clark Kimberling, May 02 2016

For n > 5, the numbers F(i)*F(j) satisfying F(n-1) <= F(i)*F(j) <= F(n) also satisfy F(n-1) < F(i)*F(j) < F(n). They are the numbers for which i + j = n + 1, where 2 < i < j, so that the number of such F(i)*F(j) is floor(n/2) - 2. The least is 3*F(n-3) and the greatest is 2*F(n-2).

			2*3 = 6, 2*5 = 10, 3*5 = 15, 2*8 = 16.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.

Cf. A000045, A004526, A094565, A271356 (difference sequence), subsequence of A049997.

z = 200; f[n_] := Fibonacci[n];
Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], 100]
Times@@@Subsets[Fibonacci[Range[3,20]],{2}]//Union (* Harvey P. Dale, Jul 12 2025 *)

list(lim)=my(v=List,F=vector(A130233(lim\2),k,fibonacci(k)),t); for(i=2,#F, for(j=1,i-1, t=F[i]*F[j]; if(t>lim,break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Oct 07 2016

A004526(n) = number of numbers a(k) between F(n+3) and F(n+4), where F = A000045 (Fibonacci numbers).

A094566 Triangle of binary products of Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 9, 10, 13, 21, 24, 25, 26, 34, 55, 63, 64, 65, 68, 89, 144, 165, 168, 169, 170, 178, 233, 377, 432, 440, 441, 442, 445, 466, 610, 987, 1131, 1152, 1155, 1156, 1157, 1165, 1220, 1597, 2584, 2961, 3016, 3024, 3025, 3026, 3029, 3050, 3194, 4181

Offset: 1

Clark Kimberling, May 12 2004

For n>1, row n consists of n numbers, first F(2n-2) and last F(2n-1).
Central numbers: (1,4,9,25,64,...), essentially A081016.
Row sums: A027991. Alternating row sums: 1,1,4,4,30,30,203,203; the sequence b=(1,4,30,203,1394,...) is A094567.
In each row, the difference between neighboring terms is a Fibonacci number.

			Rows 1 to 4:
1
1 2
3 4 5
8 9 10 13

Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.

Cf. A000045, A094565, A094567, A094568.

pef(k, n) = fibonacci(2*k)*fibonacci(2*n-2*k);
pof(k, n) = fibonacci(2*n-2*k+1)*fibonacci(2*k-1);
tabl(nn) = {for (n=1, nn, if (n==1, print1(1, ", "), if (n % 2 == 0, for (k=1, n/2, print1(pef(k,n), ", ");); forstep (k=n/2, 1, -1, print1(pof(k,n), ", "););, for (k=1, n\2, print1(pef(k,n), ", ");); forstep (k=n\2+1, 1, -1, print1(pof(k,n), ", ");););); print(););} \\ Michel Marcus, May 04 2016

Row 1 is the single number 1. For m>=1, Row 2m: F(2)F(4m-2), F(4)F(4m-4), ..., F(2m)F(2m), F(2m+1)F(2m-1), F(2m+3)F(2m-3), ..., F(4m-1)F(1) Row 2m+1: F(2)F(4m), F(4)F(4m-2), ..., F(2m+1)F(2m+1), F(2m+3)F(2m-1), F(2m+5)F(2m-3), ..., F(4m+1)F(1)

A094568 Triangle of binary products of Fibonacci numbers.

Original entry on oeis.org

2, 3, 5, 8, 10, 13, 21, 24, 26, 34, 55, 63, 65, 68, 89, 144, 165, 168, 170, 178, 233, 377, 432, 440, 442, 445, 466, 610, 987, 1131, 1152, 1155, 1157, 1165, 1220, 1597, 2584, 2961, 3016, 3024, 3026, 3029, 3050, 3194, 4181, 6765, 7752, 7896, 7917, 7920, 7922

Offset: 1

Clark Kimberling, May 12 2004

Start with the triangle in A094566: starting with row 2, expel from each row the term that is a square of a Fibonacci number (A007598). The remaining triangle is this sequence.
In each row, the difference between neighboring terms is a Fibonacci number. For n>1, row n consists of n numbers, first F(2n) and last F(2n+1).
Central numbers: (2,10,65,442,...), essentially A064170.
Alternating row sums: 2,2,11,11,78,78,...; the sequence b=(2,11,78,...) is A094569.

			First four rows:
2
3 5
8 10 13
21 24 26 34

Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.

Cf. A000045, A007598, A094565, A094566, A094569.

pef(k, n) = fibonacci(2*k)*fibonacci(2*n-2*k);
pof(k, n) = fibonacci(2*n-2*k+1)*fibonacci(2*k-1);
isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ from A010056
isfib2(x) = issquare(x) && isfib(sqrtint(x));
tabl(nn) = {for (n=2, nn, if (n % 2 == 0, for (k=1, n/2, if (! isfib2(x = pef(k,n)), print1(x, ", "));); forstep (k=n/2, 1, -1, if (! isfib2(x = pof(k,n)), print1(x, ", "));), for (k=1, n\2, if (! isfib2(x = pef(k,n)), print1(x, ", "));); forstep (k=n\2+1, 1, -1, if (! isfib2(x = pof(k,n)), print1(x, ", ")););); print(););} \\ Michel Marcus, May 04 2016

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A271354 Products of two distinct Fibonacci numbers, both greater than 1.

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A094566 Triangle of binary products of Fibonacci numbers.

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A094568 Triangle of binary products of Fibonacci numbers.

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