A390325 Oblong numbers of the form 2^k*(2^k - 1) or 2^k*(2^k + 1).

2, 6, 12, 20, 56, 72, 240, 272, 992, 1056, 4032, 4160, 16256, 16512, 65280, 65792, 261632, 262656, 1047552, 1049600, 4192256, 4196352, 16773120, 16781312, 67100672, 67117056, 268419072, 268451840, 1073709056, 1073774592, 4294901760, 4295032832, 17179738112, 17180000256

Offset

1,1

Comments

Oblong numbers whose distance from a perfect square of the form 4^k (A000302) is 2^k (A000079), forming the interleaved pairs 4^k - 2^k (A020522) and 4^k + 2^k (A063376).

These are the oblong numbers for which no pair (x, y) with x odd satisfies x*y = m*(m + 1) and 2 <= y <= floor(x/4).

Links

Table of n, a(n) for n=1..34.

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-8).

Formula

a(2n - 1) = 2^n*(2^n - 1) = 4^n - 2^n = A020522(n).

a(2n) = 2^n*(2^n + 1) = 4^n + 2^n = A063376(n).

a(n) = 2^( ceiling(n/2) )*( 2^( ceiling(n/2) ) + (-1)^n ) = 4^( ceiling(n/2) ) + (-1)^n*2^( ceiling(n/2) ).

Complement of A390326 within A002378.

Example

For k = 3, (2^3-1)*2^3 = 7*8 = 56, and 2^3*(2^3+1) = 8*9 = 72. Then, 56 and 72 form the consecutive oblong pair surrounding the square (2^3)^2 = 64, and the square 64 is equidistant from 56 and 72 at a distance of 2^3.
For k = 4, (2^4-1)*2^4 = 15*16 = 240, and 2^4*(2^4+1) = 16*17 = 272. Then, 240 and 272 form the consecutive oblong pair surrounding the square (2^4)^2 = 256, and the square 256 is equidistant from 240 and 272 at a distance of 2^4.

Mathematica

a[n_]:=4^Ceiling[n/2]+(-1)^n*2^(Ceiling[n/2]); Array[a, 34] (* James C. McMahon, Nov 16 2025 *)
(* Alternative: *)
LinearRecurrence[{0, 6, 0, -8}, {2, 6, 12, 20}, 34] (* James C. McMahon, Nov 16 2025 *)

Prog

(Python)
def A390325(n): return (m:=1<<(n+1>>1))*(m+(-1 if n&1 else 1)) # Chai Wah Wu, Nov 17 2025

Crossrefs

Cf. A390326 (complementary oblong numbers), A002378 (oblong numbers), A000079 (powers of 2), A000302 (powers of 4), A020522 (oblongs 4^k - 2^k), A063376 (oblongs 4^k + 2^k).

Sequence in context: A290209 A286780 A099885 * A106372 A354895 A214916

Adjacent sequences: A390322 A390323 A390324 * A390326 A390327 A390328

Keyword

nonn,easy,changed

Author

Charles Kusniec, Nov 01 2025

Status

approved