A393680 Lower (1/2,1/3) midsequence of (n^2) and ((n+2)^2); see Comments.

1, 3, 7, 12, 20, 28, 39, 51, 65, 80, 98, 116, 137, 159, 183, 208, 236, 264, 295, 327, 361, 396, 434, 472, 513, 555, 599, 644, 692, 740, 791, 843, 897, 952, 1010, 1068, 1129, 1191, 1255, 1320, 1388, 1456, 1527, 1599, 1673, 1748, 1826, 1904, 1985, 2067, 2151

Offset

0,2

Comments

Suppose that s = (s(n)) and t = (t(n)) are sequences of numbers and h > 0 and k > 0. The lower (h, k)-midsequence of s and t is floor(h*s + k*t); the upper (h, k)-midsequence of s and t is ceiling(h*s + k*t).

Links

Table of n, a(n) for n=0..50.

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Formula

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), with (a(0),...,a(5)) = (1, 3, 7, 12, 20, 28).

G.f.: -(1 + x^2)*(1 + 2*x + 2*x^2)/((-1 + x)^3*(1 + x)*(1 + x + x^2)).

Example

s = (n^2) = A000290 = (0, 1, 4, 9, 16, 25, 36, ...).
t = ((n+2)^2) = (4, 9, 16, 25, 36, 49, 64, 81, ...).
u(n) = (1, 3, 7, 12, 20, 28, 39, 51, 65, 80, 98, 116, 137, ...).
v(n) = (2, 4, 8, 13, 20, 29, 40, 52, 66, 81, 98, 117, 138, ...).

Mathematica

f[n_] := n^2; g[n_] := (n+2)^2; r = 1/2; s = 1/3;
u[n_] := Floor[r*f[n] + s*g[n]];
v[n_] := Ceiling[r*f[n] + s*g[n]];
Table[u[n], {n, 0, 50}]   (* A393680 *)
Table[v[n], {n, 0, 50}]   (* A393681 *)
(* Also *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 3, 7, 12, 20, 28}, 50]    (* A393680 *)
LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 1}, {2, 4, 8, 13, 20, 29}, 50]    (* A393681 *)

Crossrefs

Cf. A000290, A000578, A393680, A393681.

Sequence in context: A091369 A036698 A279169 * A374352 A132273 A130050

Adjacent sequences: A393677 A393678 A393679 * A393681 A393682 A393683

Keyword

nonn,easy

Author

Clark Kimberling, Mar 20 2026

Status

approved