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URL: https://oeis.org/A389143

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A389143
Upper (1/3)-midsequence of square numbers and triangular numbers; see Comments.
2
0, 1, 3, 5, 9, 14, 19, 26, 34, 42, 52, 63, 74, 87, 101, 115, 131, 148, 165, 184, 204, 224, 246, 269, 292, 317, 343, 369, 397, 426, 455, 486, 518, 550, 584, 619, 654, 691, 729, 767, 807, 848, 889, 932, 976, 1020, 1066, 1113, 1160, 1209, 1259, 1309, 1361, 1414
OFFSET
0,3
COMMENTS
Suppose that s = (s(n)) and t = (t(n)) are sequences of numbers and r > 0. The lower (r)-midsequence of s and t is given by u = floor(r*(s + t)); the upper r-midsequence of s and t is given by v = ceiling(r*(s + t)). If s and t are linearly recurrent and r is rational, then u and v are linearly recurrent.
FORMULA
a(n) = ceiling((3*n^2 + n)/6).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
G.f.: -x*(1 + x + x^3)/((-1 + x)^3*(1 + x + x^2)).
EXAMPLE
s = (n^2) = A000290 = (0, 1, 4, 9, 16, 25, ...).
t = (n*(n+1)/2) = A000217 = (0, 1, 3, 6, 10, 15, ...).
u(n) = floor((1/3)(0+0, 1+1, 4+3, 9+6, 16+10, ...)) = (0, 0, 2, 5, 8, ...).
v(n) = ceiling((1/3)(0+0, 1+1, 4+3, 9+6, 16+10, ...)) = (0, 1, 3, 5, 9, ...).
MATHEMATICA
s[n_] := n^2; t[n_] := n (n + 1)/2; r = 1/3;
u[n_] := Floor[r*(s[n] + t[n])];
v[n_] := Ceiling[r*(s[n] + t[n])];
Table[u[n], {n, 0, 60}] (* A389142 *)
Table[v[n], {n, 0, 60}] (* A389143 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 29 2025
STATUS
approved