A374375 -id:A374375 - cp's OEIS frontend

A049684 a(n) = Fibonacci(2n)^2.

0, 1, 9, 64, 441, 3025, 20736, 142129, 974169, 6677056, 45765225, 313679521, 2149991424, 14736260449, 101003831721, 692290561600, 4745030099481, 32522920134769, 222915410843904, 1527884955772561, 10472279279564025, 71778070001175616, 491974210728665289

Offset: 0

Clark Kimberling

This is the r=9 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.

Apparently, this sequence consists of those nonnegative integers k for which x*(x^2-1)*y*(y^2-1) = k*(k^2-1) has a solution in nonnegative integers x, y. If k = a(n), x = A000045(2*n-1) and y = A000045(2*n+1) are a solution. See A374375 for numbers k*(k^2-1) that can be written as a product of two or more factors of the form x*(x^2-1). - Pontus von Brömssen, Jul 14 2024

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 27.
H. J. H. Tuenter, Fibonacci summation identities arising from Catalan's identity, Fib. Q., 60:4 (2022), 312-319.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Pridon Davlianidze, Problem B-1264, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 1 (2020), p. 82; It's All About Catalan, Solution to Problem B-1264, ibid., Vol. 59, No. 1 (2021), pp. 87-88.
E. Kilic, Y. T. Ulutas, and N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, table 1, k=2.
R. Stephan, Boring proof of a nonlinearity
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

First differences give A033890.
First differences of A103434.
Bisection of A007598 and A064841.
a(n) = A064170(n+2) - 1 = (1/5) A081070.
Cf. A000032, A000045, A001622, A001906, A056854, A065563, A092184, A374375.

Join[{a=0, b=1}, Table[c=7*b-1*a+2; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
Fibonacci[Range[0, 40, 2]]^2 (* Harvey P. Dale, Mar 22 2012 *)
Table[Fibonacci[n - 1] Fibonacci[n + 1] - 1, {n, 0, 40, 2}] (* Bruno Berselli, Feb 12 2015 *)
LinearRecurrence[{8, -8, 1},{0, 1, 9},21] (* Ray Chandler, Sep 23 2015 *)

numlib::fibonacci(2*n)^2 $ n = 0..35; // Zerinvary Lajos, May 13 2008

PARI
```
a(n)=fibonacci(2*n)^2
```

[fibonacci(2*n)^2 for n in range(0, 21)] # Zerinvary Lajos, May 15 2009

G.f.: (x+x^2) / ((1-x)*(1-7*x+x^2)).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) with n>2, a(0)=0, a(1)=1, a(2)=9.
a(n) = 7*a(n-1) - a(n-2) + 2 = A001906(n)^2.
a(n) = (A000032(4*n)-2)/5. [This is in Koshy's book (reference under A065563) on p. 88, attributed to Lucas 1876.] - Wolfdieter Lang, Aug 27 2012
a(n) = 1/5*(-2 + ( (7+sqrt(45))/2 )^n + ( (7-sqrt(45))/2 )^n). - Ralf Stephan, Apr 14 2004
a(n) = 2*(T(n, 7/2)-1)/5 with twice the Chebyshev polynomials of the first kind evaluated at x=7/2: 2*T(n, 7/2)= A056854(n). - Wolfdieter Lang, Oct 18 2004
a(n) = F(2*n-1)*F(2*n+1)-1, see A064170 - Bruno Berselli, Feb 12 2015
a(n) = Sum_{i=1..n} F(4*i-2) for n>0. - Bruno Berselli, Aug 25 2015
From Peter Bala, Nov 20 2019: (Start)
Sum_{n >= 1} 1/(a(n) + 1) = (sqrt(5) - 1)/2.
Sum_{n >= 1} 1/(a(n) + 4) = (3*sqrt(5) - 2)/16. More generally, it appears that
Sum_{n >= 1} 1/(a(n) + F(2*k+1)^2) = ((2*k+1)*F(2*k+1)*sqrt(5) - Lucas(2*k+1))/ (2*F(2*k+1)*F(4*k+2)) for k = 0,1,2,....
Sum_{n >= 2} 1/(a(n) - 1) = (8 - 3*sqrt(5))/9. (End)
E.g.f.: (1/5)*(-2*exp(x) + exp((16*x)/(1 + sqrt(5))^4) + exp((1/2)*(7 + 3*sqrt(5))*x)). - Stefano Spezia, Nov 23 2019
Product_{n>=2} (1 - 1/a(n)) = phi^2/3, where phi is the golden ratio (A001622) (Davlianidze, 2020). - Amiram Eldar, Dec 01 2021
a(n) = A092521(n-1)+A092521(n). - R. J. Mathar, Nov 22 2024

Better description and more terms from Michael Somos

A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

Original entry on oeis.org

0, 3, 7, 13, 17, 18, 21, 31, 38, 43, 47, 57, 68, 73, 91, 99, 111, 117, 123, 132, 133, 157, 183, 211, 241, 242, 253, 255, 268, 273, 293, 302, 307, 313, 322, 327, 343, 381, 413, 421, 438, 443, 463, 487, 507, 515, 553, 557, 577, 593, 601, 651, 693, 697, 703, 707

Offset: 1

Pontus von Brömssen, Jun 19 2023

nonn

For the corresponding sequence for numbers of the form k^3+1 instead of k^2+1, the only terms known to me are 0 and 26, with 26^3+1 = (2^3+1)^2*(6^3+1).

			0 is a term because 0^2+1 = 1 equals the empty product.
3 is a term because 3^2+1 = 10 = 2*5 = (1^2+1)*(2^2+1).
38 is a term because 38^2+1 = 1445 = 5*17*17 = (2^2+1)*(4^2+1)^2. (This is the first term that requires more than two factors.)

David Trimas, Table of n, a(n) for n = 1..2260

Sequences that list those terms (or their indices or some other key) of a given sequence that are products of smaller terms of the same sequence (in other words, the nonprimitive terms of the multiplicative closure of the sequence):
  A018252 (A000027),
  A034878 (A000142),
  A068143 (A000217),
  A363492 (A000041),
  A363634 (A000959),
  A363635 (A003309),
  this sequence (A002522),
  A363637 (A005563),
  A363638 (A008864),
  A363750 (A006093),
  A364151 (A000292),
  A374372 (A000326),
  A374373 (A000384),
  A374374 (A002378),
  A374375 (A007531),
  A374500 (A000330),
  A374501 (A002411),
  A374502 (A002412).

g[lst_, p_] :=
  Module[{t, i, j},
   Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
      Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
    Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
multPartition[n_] :=
  Module[{i, j, p, e, lst = {{}}}, {p, e} =
    Transpose[FactorInteger[n]];
   Do[lst = g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
output = Join[{0}, Flatten[Position[Table[
     test = Sqrt[multPartition[n^2 + 1][[2 ;; All]] - 1];
     Count[AllTrue[#, IntegerQ] & /@ test, True] > 0
     , {n, 707}], True]]]
(* David Trimas, Jul 23 2023 *)

A374374 Oblong numbers that are products of smaller oblong numbers.

Original entry on oeis.org

12, 72, 240, 420, 600, 1056, 1260, 2352, 4032, 6480, 7140, 9120, 9900, 10920, 14280, 14520, 18360, 20592, 20880, 28392, 38220, 46872, 50400, 65280, 78120, 82656, 83232, 104652, 123552, 129960, 147840, 159600, 194040, 233772, 245520, 262656, 278256, 279312

Offset: 1

Pontus von Brömssen, Jul 07 2024

nonn

The oblong number (k^2+2*k)*(k^2+2*k+1) = A047928(k+2) is a term for all k >= 1, because it is the product of the oblong numbers k*(k+1) and (k+1)*(k+2).

			72 is a term because 72 = 8*9 = 6*12 = (2*3)*(3*4).
1056 is a term because 1056 = 32*33 = 2*2*2*132 = (1*2)*(1*2)*(1*2)*(11*12). (This is the first term that requires more than two factors.)

A188660 is a subsequence (only 2 factors allowed).

Cf. A002378, A047928, A068143, A374375.

cp's OEIS Frontend

A049684 a(n) = Fibonacci(2n)^2.

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A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

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Programs

Mathematica

A374374 Oblong numbers that are products of smaller oblong numbers.

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