A390325 - OEIS

%I #47 Jun 30 2026 22:31:38

%S 2,6,12,20,56,72,240,272,992,1056,4032,4160,16256,16512,65280,65792,

%T 261632,262656,1047552,1049600,4192256,4196352,16773120,16781312,

%U 67100672,67117056,268419072,268451840,1073709056,1073774592,4294901760,4295032832,17179738112,17180000256

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

%C 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).

%C 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).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-8).

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

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

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

%F Complement of A390326 within A002378.

%e 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.

%e 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.

%t a[n_]:=4^Ceiling[n/2]+(-1)^n*2^(Ceiling[n/2]);Array[a,34] (* _James C. McMahon_, Nov 16 2025 *)

%t (* Alternative: *)

%t LinearRecurrence[{0,6,0,-8},{2,6,12,20},34] (* _James C. McMahon_, Nov 16 2025 *)

%o (Python)

%o def A390325(n): return (m:=1<<(n+1>>1))*(m+(-1 if n&1 else 1)) # _Chai Wah Wu_, Nov 17 2025

%Y 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).

%K nonn,easy,changed

%O 1,1

%A _Charles Kusniec_, Nov 01 2025