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A295643
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Squares repeated 4 times; a(n) = floor(n/4)^2.
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1
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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
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0,9
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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
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MAPLE
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A295643:=n->floor(n/4)^2: seq(A295643(n), n=0..100);
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MATHEMATICA
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Floor[Range[0, 80]/4]^2
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PROG
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(MAGMA) [Floor(n/4)^2 : n in [0..100]];
(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
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CROSSREFS
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Cf. A000290, A002265, A004526, A008794.
See also the quarter-squares, A002620.
Sequence in context: A213083 A053187 A013189 * A190718 A035621 A046109
Adjacent sequences: A295640 A295641 A295642 * A295644 A295645 A295646
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KEYWORD
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nonn,easy,changed
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AUTHOR
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Wesley Ivan Hurt, Nov 25 2017
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STATUS
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approved
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