proposed

approved

editing

proposed

a(n) = 4*A001109(n)^2. - G. C. Greubel, Aug 18 2022

4*ChebyshevU[Range[-1, 30], 3]^2 (* G. C. Greubel, Aug 18 2022 *)

(Magma) I:=[0, 4, 144]; [*Evaluate(ChebyshevU(n le ), 3 select I[n] else 35*Self(n-1) -35*Self(n-^2) +Self(n-3): n in [10..3130]]; // G. C. Greubel, Aug 18 2022

(SageMath) [4*chebyshev_U(n-1, 3)^2 for n in (0..30)] # _G. C. Greubel_, Aug 18 2022

@CachedFunction

def b(n): return n if (n<2) else 34*b(n-1) -b(n-2) +2 # b = A001110

def A084703(n): return 4*b(n)

[A084703(n) for n in (0..30)] # G. C. Greubel, Aug 18 2022

Cf. A000129, A001109, A001110, A001333, A001541, A001542, A001652, A001653, A014105.

Cf. A014105, A029549, A046090, A046176, A055792, A075114, A079291, A084702, A089928.

Cf. A089928, A204576.

proposed

editing

editing

proposed

Bisection of A079291. The squares 2*nk+1 are given in A055792.

Squares n k such that 2*nk+1 is also a square.

E. Kilic, Y. T. Ulutas, and N. Omur, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq. 14 (2011) #11.5.6, table 3, k=2.

proposed

editing

editing

proposed

a(n+1), n >= 0, is the perimeter squared (x(n) + y(n) + z(n))^2 of the ordered primitive Pythagorean triple (x(n), y(n) = x(n) + 1, z(n)). The first two terms are (x(0)=0, y(0)=1, z(0)=1), a(1) = 2^2, and (x(1)=3, y(1)=4, z(1)=5), a(2) = 12^2. - George F. Johnson, Nov 02 2012

For k>=n>=0, a(n) = A001653(k+n)*A001653(k-n) - A001653(k)^2, for k >= n >= 0; e.g. 144 = 5741*5 - 169^2. - Charlie Marion, Jul 16 2003

Empirical: for n>0, a(n) = A089928(4*n-2), for n > 0. - Alex Ratushnyak, Apr 12 2013

a[0] = 0; a[1] = 1; a[n_] := 34a[n - 1] - a[n - 2] + 2; Table[ 4a[n], {n, 0, 15}]

b[n_]:= b[n]= If[n<2, n, 34*b[n-1] -b[n-2] +2]; (* b=A001110 *)

a[n_]:= 4*b[n]; Table[a[n], {n, 0, 30}]

(Magma) I:=[0, 4, 144]; [n le 3 select I[n] else 35*Self(n-1) -35*Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Aug 18 2022

(SageMath)

@CachedFunction

def b(n): return n if (n<2) else 34*b(n-1) -b(n-2) +2 # b = A001110

def A084703(n): return 4*b(n)

[A084703(n) for n in (0..30)] # G. C. Greubel, Aug 18 2022

Cf. A000129, A001110, A055792, A075114, A079291, A084702, A204576A001333, A001541, A001542, A001652, A001653, A014105.

Cf. A029549, A046090, A046176, A055792, A075114, A079291, A084702, A089928.

Cf. A204576.

approved

editing

editing

approved

E. Kilic, Y. T. Ulutas, N. Omur, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq. 14 (2011) #11.5.6, table 3, k=2.

approved

editing