A393679 Upper (1/2,1/3) midsequence of (n^2) and ((n+1)^2); see Comments.

1, 2, 5, 10, 17, 25, 35, 46, 59, 74, 91, 109, 129, 150, 173, 198, 225, 253, 283, 314, 347, 382, 419, 457, 497, 538, 581, 626, 673, 721, 771, 822, 875, 930, 987, 1045, 1105, 1166, 1229, 1294, 1361, 1429, 1499, 1570, 1643, 1718, 1795, 1873, 1953, 2034, 2117

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 (2,-1,0,0,0,1,-2,1).

Formula

a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8), with (a(0),...,a(7)) = (1, 2, 5, 10, 17, 25, 35, 46).

G.f.: (-1 - 2*x^2 - 2*x^3 - 2*x^4 - x^5 - x^6 - x^7)/((-1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5)).

Example

s = (n^2) = A000290 = (0, 1, 4, 9, 16, 25, 36, ...).
t = ((n+1)^2) = (1, 4, 9, 16, 25, 36, 49, 64, ...).
u(n) = (0, 1, 5, 9, 16, 24, 34, 45, 59, 73, 90, 108, ...).
v(n) = (1, 2, 5, 10, 17, 25, 35, 46, 59, 74, 91, 109, ...).

Mathematica

f[n_] := n^2; g[n_] := (n+1)^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]];
v[n_] := Ceiling[r*f[n] + s*g[n]];
Table[u[n], {n, 0, 50}]   (* A393678 *)
Table[v[n], {n, 0, 50}]   (* A393679 *)
(* Also *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 1, 5, 9, 16, 24}, 50]  (* A393678 *)
LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 1}, {1, 2, 5, 10, 17, 25, 35, 46}, 50]  (* A393679 *)

Crossrefs

Cf. A000290, A000578, A393778, A393681.

Sequence in context: A003192 A380546 A018682 * A375703 A078393 A340045

Adjacent sequences: A393676 A393677 A393678 * A393680 A393681 A393682

Keyword

nonn,easy

Author

Clark Kimberling, Mar 20 2026

Status

approved