A119996 -id:A119996 - cp's OEIS frontend

A059840 a(n) = F(n)F(n-1) if n odd otherwise F(n)F(n-1)-1, where F = Fibonacci numbers A000045.

0, 0, 2, 5, 15, 39, 104, 272, 714, 1869, 4895, 12815, 33552, 87840, 229970, 602069, 1576239, 4126647, 10803704, 28284464, 74049690, 193864605, 507544127, 1328767775, 3478759200, 9107509824, 23843770274, 62423800997, 163427632719, 427859097159, 1120149658760, 2932589879120

Offset: 1

N. J. A. Sloane, Feb 26 2001

Harry J. Smith, Table of n, a(n) for n = 1..500
S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
H. Ohtsuka and S. Nakamura, On the sum of reciprocal sums of Fibonacci numbers, Fibonacci Quart. 46/47 (2008/2009), 153-159.
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A000045, A001654, A059248, A064831, A119996.

List([1..30],n->Sum([1..n-2],k->Fibonacci(k)*Fibonacci(k+2))); # Muniru A Asiru, Aug 09 2018

F:=Fibonacci; [(n mod 2) eq 0 select F(n)*F(n-1)-1 else F(n)*F(n-1): n in [1..30]]; // G. C. Greubel, Jul 23 2019

seq(coeff(series(x^3*(2-x)/((1-x^2)*(1-3*x+x^2)), x,n+1),x,n),n=1..30); # Muniru A Asiru, Aug 09 2018

Table[If[OddQ[n],Fibonacci[n]Fibonacci[n-1],Fibonacci[n] Fibonacci[n-1]-1],{n,30}]  (* Harvey P. Dale, Apr 20 2011 *)

a(n) = { fibonacci(n)*fibonacci(n-1) - (n%2 == 0) } \\ Harry J. Smith, Jun 29 2009

a=(x^3*(2-x)/((1-x^2)*(1-3*x+x^2))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 23 2019

G.f.: x^3*(2 - x)/((1 - x^2)*(1 - 3*x + x^2)). See a comment on A080144. - Wolfdieter Lang, Jul 30 2012
a(n) = Sum_{k=1..n-2} F(k)*F(k+2). - Alexander Adamchuk, May 17 2007
a(n+2) = (3*A001654(n) + A027941(n))/2, n >= 0. - Wolfdieter Lang, Jul 21 2012
a(n+2) = (3*(-1)^(n+1) - 5 + 2*Lucas(2*n + 3))/10, n >= 0. - Ehren Metcalfe, Aug 21 2017
a(n) = floor(1/(Sum_{k>=n} 1/Fibonacci(k)^2)) [Ohtsuka and Nakamura]. - Michel Marcus, Aug 09 2018
For n > 2, 2 * A000217(a(n)) = A228873(n-2). - Diego Rattaggi, Jan 27 2021

A202503 Fibonacci self-fission matrix, by antidiagonals.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 8, 9, 8, 8, 8, 13, 14, 15, 13, 13, 13, 21, 23, 24, 24, 21, 21, 21, 34, 37, 39, 39, 39, 34, 34, 34, 55, 60, 63, 64, 63, 63, 55, 55, 55, 89, 97, 102, 103, 104, 102, 102, 89, 89, 89, 144, 157, 165, 167, 168, 168, 165, 165, 144, 144, 144

Offset: 1

Clark Kimberling, Dec 20 2011

The Fibonacci self-fission matrix, F, is the fission P^^Q, where P and Q are the matrices given at A202502 and A202451.  See A193842 for the definition of fission.
antidiagonal sums:  (1, 3, 8, 18, 38, ...), A064831
diagonal (1,  5, 14,  39, ...), A119996
diagonal (2,  8, 23,  63, ...), A180664
diagonal (2,  5, 15,  39, ...), A059840
diagonal (3,  8, 24,  63, ...), A080097
diagonal (5, 13, 39, 102, ...), A080143
diagonal (8, 21, 63, 165, ...), A080144
All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.

			Northwest corner:
1....1....2....3....5.....8....13...21
2....3....5....8...13....21....34...55
3....5....9...14...23....37....60...97
5....8...15...24...39....63...102...165
8...13...24...39...64...103...167...270

Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Cf. A000045, A202451, A202453, A202502.

n = 14;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
Qt = Transpose[Q]; P1 = Qt - IdentityMatrix[n];
P = P1[[Range[2, n], Range[1, n]]];
F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202502 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202503 as a sequence *)
TableForm[P]  (* A202502, modified lower triangular Fibonacci array *)
TableForm[Q]  (* A202451, upper tri. Fibonacci array *)
TableForm[F]  (* A202503, Fibonacci fission array *)

A163154 Primes one less than a Golden rectangle number.

Original entry on oeis.org

5, 103, 3478759199, 116139356908771351, 37396512239913013823, 285687842248637730909432643746211633, 1391541769353191693086710038712557510379751, 1550980526109101915069808788349000570735950731617761605783

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2009

nonn

Primes of the form A001654(k)-1, generated at k = 3, 6, 24, 42, 48, 86, 102, 138, 182,....

Yet another way of stating the definition: primes of the form F(k)*F(k+1)-1, where F(k) is the k-th Fibonacci number (A000045). - Colin Barker, Apr 07 2016

			103 is in the sequence because 103 = 8*13-1 = F(6)*F(7)-1.

Cf. A001654, A000045, A119996, A163157, A271428.

q=0;lst={};Do[f=Fibonacci[n];If[PrimeQ[f*q-1],AppendTo[lst,f*q-1]];q=f, {n,6!}];lst
f[n_] := Fibonacci@ n Fibonacci[n + 1] - 1; f /@ Select[Range@ 180, PrimeQ[f@ #] &] (* Michael De Vlieger, Apr 07 2016 *)
Select[Times@@@Partition[Fibonacci[Range[150]],2,1]-1,PrimeQ] (* Harvey P. Dale, Jul 04 2019 *)

L=List(); for(k=1, 200, if(isprime(p=fibonacci(k)*fibonacci(k+1)-1), listput(L, p))); Vec(L) /* Colin Barker, Apr 07 2016 */

Definition reworded by R. J. Mathar, Sep 11 2009

a(8) from Colin Barker, Apr 07 2016

A202874 Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 5, 8, 8, 5, 8, 13, 14, 13, 8, 13, 21, 23, 23, 21, 13, 21, 34, 37, 39, 37, 34, 21, 34, 55, 60, 63, 63, 60, 55, 34, 55, 89, 97, 102, 103, 102, 97, 89, 55, 89, 144, 157, 165, 167, 167, 165, 157, 144, 89, 144, 233, 254, 267, 270, 272, 270, 267, 254

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=(1,2,3,5,8,13,...)=(F(k+1)), where F=A000045, and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202874 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202875 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....2....3....5....8....13
2....5....8....13...21...34
3....8....14...23...37...60
5....13...23...39...63...102
8....21...37...63...102..167

Cf. A202875.

s[k_] := Fibonacci[k + 1];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A001911 *)
Table[m[1, j], {j, 1, 12}]     (* A000045 *)
Table[m[j, j], {j, 1, 12}]     (* A119996 *)
Table[m[j, j + 1], {j, 1, 12}] (* A180664 *)
Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}]  (* A002940 *)

A203003 Symmetric matrix based on A007598(n+1), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 9, 17, 9, 25, 40, 40, 25, 64, 109, 98, 109, 64, 169, 281, 265, 265, 281, 169, 441, 740, 685, 723, 685, 740, 441, 1156, 1933, 1802, 1865, 1865, 1802, 1933, 1156, 3025, 5065, 4709, 4910, 4819, 4910, 4709, 5065, 3025, 7921, 13256, 12337, 12827

Offset: 1

Clark Kimberling, Dec 27 2011

Let s=A007598(n+1) (squared Fibonacci numbers, beginning with F(2)), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203003 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203004 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....4.....9....25....64
4....17....40...109...281
9....40....98...265...685
25...109...265..1865

Cf. A203004, A203001, A202453.

s[k_] := Fibonacci[k + 1]^2;
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];  (* A203003 *)
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]    (* A119996 *)
Table[m[1, j], {j, 1, 12}]       (* A007598(n+1) *)

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A059840 a(n) = F(n)*F(n-1) if n odd otherwise F(n)*F(n-1)-1, where F = Fibonacci numbers A000045.

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A202503 Fibonacci self-fission matrix, by antidiagonals.

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A163154 Primes one less than a Golden rectangle number.

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A202874 Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals.

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A203003 Symmetric matrix based on A007598(n+1), by antidiagonals.

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Mathematica

A059840 a(n) = F(n)F(n-1) if n odd otherwise F(n)F(n-1)-1, where F = Fibonacci numbers A000045.