A295643 - OEIS

A295643

Squares repeated 4 times; a(n) = floor(n/4)^2.

0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 4, 4, 9, 9, 9, 9, 16, 16, 16, 16, 25, 25, 25, 25, 36, 36, 36, 36, 49, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 81, 100, 100, 100, 100, 121, 121, 121, 121, 144, 144, 144, 144, 169, 169, 169, 169, 196, 196, 196, 196, 225, 225, 225

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OFFSET

0,9

COMMENTS

a(n+1) is the sum of the smallest odd parts of the partitions of n into two distinct parts. For example, a(11) = 4; the partitions of 10 into two distinct parts are (9,1), (8,2), (7,3) and (6,4). The sum of the smallest odd parts in these partitions is then 1+3 = 4.

a(n+2) is the sum of the smallest odd parts of the partitions of n into two parts. For example, a(8) = 4; the partitions of 6 into two parts are (5,1), (4,2) and (3,3). The sum of the smallest odd parts is then 1+3 = 4.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

FORMULA

a(n) = A002265(n)^2.

a(2n) = a(2n+1) = floor(n/2)^2 = A004526(n)^2 = A008794(n).

a(4n) = A000290(n).

a(n) = Sum_{i=1..floor(n/2)-1} i * (i mod 2).

From Colin Barker, Nov 25 2017: (Start)

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

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n>8.