A393682 Lower (1/3,1/6) midsequence of (n^2) and ((n+2)^2); see Comments.

0, 1, 4, 7, 11, 16, 22, 29, 38, 47, 57, 68, 80, 93, 108, 123, 139, 156, 174, 193, 214, 235, 257, 280, 304, 329, 356, 383, 411, 440, 470, 501, 534, 567, 601, 636, 672, 709, 748, 787, 827, 868, 910, 953, 998, 1043, 1089, 1136, 1184, 1233, 1284, 1335, 1387

Offset

0,3

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 floor (h*s + k*t).

Links

Table of n, a(n) for n=0..52.

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)) = (0,1,4,7,11,16,22,29).

G.f. -((x (1 + 2 x + x^3 + x^4 + x^5))/((-1 + x)^3 (1 + x + x^2 + x^3 + x^4 + x^5))).

a(n) = n^2/2 + 2/3*n + O(1). - Charles R Greathouse IV, May 31 2026

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) = (0, 1, 4, 7, 11, 16, 22, 29, 38, 47, 57, 68, 80, 93, ...).
v(n) = (1, 2, 4, 8, 12, 17, 23, 30, 38, 48, 58, 69, 81, 94, ...).

Mathematica

f[n_] := n^2; g[n_] := (n+2)^2; r = 1/3; s = 1/6;
u[n_] := Floor[r*f[n] + s*g[n]];
v[n_] := Ceiling[r*f[n] + s*g[n]];
Table[u[n], {n, 0, z}]   (* A393682  *)
Table[v[n], {n, 0, z}]   (* A393683  *)
(* Also *)
LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 1}, {0, 1, 4, 7, 11, 16, 22, 29}, 30]  (* A393682 *)
LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 1}, {1, 2, 4, 8, 12, 17, 23, 30}, 30]  (* A393683 *)

Prog

(PARI) a(n)=(3*n^2+4*n+[0, -1, 4, 3, 2, 1][n%6+1])/6 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A000290, A000578, A393680, A393683.

Sequence in context: A310757 A231603 A134869 * A171377 A020683 A310758

Adjacent sequences: A393679 A393680 A393681 * A393683 A393684 A393685

Keyword

nonn,easy,changed

Author

Clark Kimberling, Mar 24 2026

Status

approved