A094568 -id:A094568 - cp's OEIS frontend

A094569 Associated with alternating row sums of array in A094568.

2, 11, 78, 532, 3649, 25008, 171410, 1174859, 8052606, 55193380, 378301057, 2592914016, 17772097058, 121811765387, 834910260654, 5722560059188, 39223010153665, 268838511016464, 1842646566961586, 12629687457714635, 86565165637040862, 593326472001571396

Offset: 0

Clark Kimberling, May 12 2004

			Obtain 11,78,532 from a(0)=2 and Fibonacci numbers 13,89,610: 11=13-2, 78=89-11, 532=610-78.

Colin Barker, Table of n, a(n) for n = 0..1000
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly, 42:1 (2004), pp. 28-35.
Index entries for linear recurrences with constant coefficients, signature (6,6,-1).

Cf. A000045, A094568, A094567.

LinearRecurrence[{6,6,-1},{2,11,78},30] (* Harvey P. Dale, Feb 15 2025 *)

Vec(-(x-2)/((x+1)*(x^2-7*x+1)) + O(x^100)) \\ Colin Barker, Nov 19 2014

vector(30, n, n--;(fibonacci(4*n+5) + (-1)^n)/3) \\ Michel Marcus, Nov 19 2014

a(n) = F(4n+3) - a(n-1) for n >= 1, where a(0) = 2.
a(n) = (Fib(4n+5) + (-1)^n )/3. - Ralf Stephan, Dec 04 2004
a(n) = (-1)^n * sum((-1)^k*Fibonacci(4*k+3), k=0..n). - Gary Detlefs, Jan 22 2013
a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3). - Colin Barker, Nov 19 2014
G.f.: -(x-2) / ((x+1)*(x^2-7*x+1)). - Colin Barker, Nov 19 2014

More terms from Colin Barker, Nov 19 2014

A094565 Triangle read by rows: binary products of Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 13, 15, 16, 21, 34, 39, 40, 42, 55, 89, 102, 104, 105, 110, 144, 233, 267, 272, 273, 275, 288, 377, 610, 699, 712, 714, 715, 720, 754, 987, 1597, 1830, 1864, 1869, 1870, 1872, 1885, 1974, 2584, 4181, 4791, 4880, 4893, 4895, 4896, 4901, 4935, 5168, 6765

Offset: 1

Clark Kimberling, May 12 2004

Row n consists of n numbers, first F(2n-1) and last F(2n).
Central numbers: (1,6,40,273,...) = A081016.
Row sums: A001870.
Alternating row sums: 1,1,7,7,48,48,329,329; the sequence b=(1,7,48,329,...) is A004187, given by b(n)=F(4n+2)-b(n-1) for n>=2, with b(1)=1.
In each row, the difference between neighboring terms is a Fibonacci number.

			Triangle begins:
   1;
   2,   3;
   5,   6    8;
  13,  15,  16,  21;
  34,  39,  40,  42,  55;
  89, 102, 104, 105, 110, 144; ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.

Cf. A000045, A094566, A094568.

Flat(List([1..12], n-> List([1..n], k-> Fibonacci(2*k)*Fibonacci(2*n-2*k+1) ))); # G. C. Greubel, Jul 15 2019

[Fibonacci(2*k)*Fibonacci(2*n-2*k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 15 2019

Table[Fibonacci[2*k]*Fibonacci[2*n-2*k+1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 15 2019 *)

row(n) = vector(n, k, fibonacci(2*k)*fibonacci(2*n-2*k+1));
tabl(nn) = for(n=1, nn, print(row(n))); \\ Michel Marcus, May 03 2016

[[fibonacci(2*k)*fibonacci(2*n-2*k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 15 2019

Row n: F(2)F(2n-1), F(4)F(2n-3), ..., F(2n)F(1).

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)

cp's OEIS Frontend

A094569 Associated with alternating row sums of array in A094568.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A094565 Triangle read by rows: binary products of Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A094566 Triangle of binary products of Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula