A082639 Numbers k such that 2*k*(k+2) is a square.

0, 2, 16, 98, 576, 3362, 19600, 114242, 665856, 3880898, 22619536, 131836322, 768398400, 4478554082, 26102926096, 152139002498, 886731088896, 5168247530882, 30122754096400, 175568277047522, 1023286908188736

Offset: 1

Author

James R. Buddenhagen, May 15 2003

Keywords

• easy
• nonn

Comments

Even-indexed terms are squares. Their square roots form sequence A005319. Odd-indexed terms divided by 2 are squares. Their square roots form the sequence A002315. (Index starts at 0.)

Lower of two terms with difference 2 in A088827. - Ophir Spector, Nov 06 2024

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Cf. A001541, A002315, A005319, A088827.

Programs

Mathematica

a[0] = 0; a[1] = 2; a[n_] := a[n] = 6a[n - 1] - a[n - 2] + 4; Table[ a[n], {n, 0, 20}]
LinearRecurrence[{7,-7,1},{0,2,16},30] (* Harvey P. Dale, Nov 21 2015 *)

Formula

a(n) = A001541(n) - 1.

a(n) = (1/2)*(s^n + t^n) - 1, where s = 3 + 2*sqrt(2), t = 3 - 2*sqrt(2). Note: s=1/t. a(n) = 6*a(n-1) - a(n-2) + 4, a(0)=0, a(1)=2.

a(n) = 1/kappa(sqrt(2)/A001542(n)); a(n) = 1/kappa(sqrt(8)/A005319(n)) where kappa(x) is the sum of successive remainders by computing the Euclidean algorithm for (1, x). - Thomas Baruchel, Nov 29 2003

G.f.: -2*x^2*(x+1)/((x-1)*(x^2-6*x+1)). - Colin Barker, Nov 22 2012

Extensions

More terms from Robert G. Wilson v, May 15 2003