A343264 -id:A343264 - cp's OEIS frontend

A343291 a(n) = (n-2)*2^(n-1) + n + 2.

1, 2, 4, 9, 22, 55, 136, 329, 778, 1803, 4108, 9229, 20494, 45071, 98320, 213009, 458770, 983059, 2097172, 4456469, 9437206, 19922967, 41943064, 88080409, 184549402, 385875995, 805306396, 1677721629, 3489660958, 7247757343, 15032385568, 31138512929, 64424509474

Offset: 0

Alois P. Heinz, Apr 10 2021

a(n) is the cardinality of set s(n), where s(0) = {0} and s(n+1) = s(n) union {(i+j+1)/2 : i,j in s(n)}. s(4) = {0, 1/2, 3/4, 7/8, 15/16, 1, 17/16, 9/8, 19/16, 5/4, 21/16, 11/8, 23/16, 3/2, 25/16, 13/8, 27/16, 7/4, 29/16, 15/8, 31/16, 2} has cardinality a(4) = 22.

Total number of 0-bits in all numbers <= 2^n and for n >= 1 the total number of bits in all numbers <= 2^(n-1); similar to A048495. - Ruud H.G. van Tol, Apr 28 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Partial differences give A005183 (shifted).

Cf. A048495, A343264.

a:= n-> (n-2)*2^(n-1)+n+2:
seq(a(n), n=0..35);

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

cp's OEIS Frontend

A343291 a(n) = (n-2)*2^(n-1) + n + 2.

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