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

Offset: 0

Author

Wesley Ivan Hurt, Nov 25 2017

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000290, A002265, A004526, A008794.

See also the quarter-squares, A002620.

Programs

Magma

[Floor(n/4)^2 : n in [0..100]];

Maple

A295643:=n->floor(n/4)^2: seq(A295643(n), n=0..100);

Mathematica

Floor[Range[0, 80]/4]^2

PARI

concat(vector(4), Vec(x^4*(1 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2) + O(x^100))) \\ Colin Barker, Nov 25 2017

PARI

a(n) = (n\4)^2; \\ Altug Alkan, Dec 17 2017

Python

def A295643(n): return (n>>2)**2 # Chai Wah Wu, Feb 06 2023

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.

(End)

a(n) = (1/16)*(n-(3-(-1)^n-2*(-1)^((2*n-1+(-1)^n)/4))/2)^2. - Iain Fox, Dec 18 2017