A393681 Upper (1/2,1/3) midsequence of (n^2) and ((n+2)^2); see Comments.

2, 4, 8, 13, 20, 29, 40, 52, 66, 81, 98, 117, 138, 160, 184, 209, 236, 265, 296, 328, 362, 397, 434, 473, 514, 556, 600, 645, 692, 741, 792, 844, 898, 953, 1010, 1069, 1130, 1192, 1256, 1321, 1388, 1457, 1528, 1600, 1674, 1749, 1826, 1905, 1986, 2068, 2152

Offset

0,1

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) = 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).

G.f.: (-2 - 2*x^2 - x^3 - 2*x^4 - 2*x^5 - 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+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

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 3, 7, 12, 20, 28}, 30]    (* A393680 *)
LinearRecurrence[{2, -1, 0, 0, 0, 1, -2, 1}, {2, 4, 8, 13, 20, 29}, 30]    (* A393681 *)

Crossrefs

Cf. A000290, A000578, A393680.

Sequence in context: A164486 A084684 A011907 * A056133 A172131 A173721

Adjacent sequences: A393678 A393679 A393680 * A393682 A393683 A393684

Keyword

nonn,easy

Author

Clark Kimberling, Mar 22 2026

Status

approved