A393683 Upper (1/3,1/6) midsequence of (n^2) and ((n+2)^2); see Comments.

1, 2, 4, 8, 12, 17, 23, 30, 38, 48, 58, 69, 81, 94, 108, 124, 140, 157, 175, 194, 214, 236, 258, 281, 305, 330, 356, 384, 412, 441, 471, 502, 534, 568, 602, 637, 673, 710, 748, 788, 828, 869, 911, 954, 998, 1044, 1090, 1137, 1185, 1234, 1284, 1336, 1388

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 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)) = (1, 2, 4, 8, 12, 17, 23, 30).

G.f. (-1 - x^2 - 2 x^3 - x^5 - x^7)/((-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

Mathematica

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, ...).
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+29600\6^(n%6)%6+4)/6 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A000290, A000578, A393682.

Sequence in context: A376256 A373728 A292060 * A152125 A338097 A305694

Adjacent sequences: A393680 A393681 A393682 * A393684 A393685 A393686

Keyword

nonn,easy

Author

Clark Kimberling, Mar 24 2026

Status

approved