A184637 -id:A184637 - cp's OEIS frontend

A256320 Number of partitions of 4n into exactly 3 parts.

0, 1, 5, 12, 21, 33, 48, 65, 85, 108, 133, 161, 192, 225, 261, 300, 341, 385, 432, 481, 533, 588, 645, 705, 768, 833, 901, 972, 1045, 1121, 1200, 1281, 1365, 1452, 1541, 1633, 1728, 1825, 1925, 2028, 2133, 2241, 2352, 2465, 2581, 2700, 2821, 2945, 3072, 3201

Offset: 0

Colin Barker, Mar 24 2015

			For n=2 the 5 partitions of 4*2 = 8 are [1,1,6], [1,2,5], [1,3,4], [2,2,4] and [2,3,3].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A033428 (6n), A256321 (5n), A256322 (7n).

Cf. A184637.

Length /@ (Total /@ IntegerPartitions[4 #, {3}] & /@ Range[0, 49]) (* Michael De Vlieger, Mar 24 2015 *)
CoefficientList[Series[-x (x + 1)^3/((x - 1)^3 (x^2 + x + 1)), {x, 0, 49}], x] (* or *)
Table[2 (6 n^2 + Cos[(2 Pi n)/3] - 1)/9, {n, 0, 49}] (* Michael De Vlieger, Jun 06 2016 *)

concat(0, vector(40, n, k=0; forpart(p=4*n, k++, , [3,3]); k))

concat(0, Vec(-x*(x+1)^3/((x-1)^3*(x^2+x+1)) + O(x^100)))

a(n) = A184637(n) for n > 2.
a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5) for n>4.
G.f.: -x*(x+1)^3 / ((x-1)^3*(x^2+x+1)).
a(n) = 2*(6*n^2+cos((2*Pi*n)/3)-1)/9. - Colin Barker, Jun 06 2016

A184636 a(n) = floor(1/{(n^4+2*n)^(1/4)}), where {}=fractional part.

Original entry on oeis.org

3, 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, 578, 648, 722, 800, 882, 968, 1058, 1152, 1250, 1352, 1458, 1568, 1682, 1800, 1922, 2048, 2178, 2312, 2450, 2592, 2738, 2888, 3042, 3200, 3362, 3528, 3698, 3872, 4050, 4232, 4418, 4608, 4802, 5000, 5202, 5408, 5618, 5832, 6050, 6272, 6498, 6728, 6962, 7200, 7442, 7688, 7938, 8192, 8450, 8712, 8978, 9248, 9522, 9800

Offset: 1

Clark Kimberling, Jan 18 2011

Is a(n) = A001105(n) for n>=2 ? R. J. Mathar, Jan 28 2011

Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A184536, A184635, A184637.

p[n_]:=FractionalPart[(n^4+2*n)^(1/4)];
q[n_]:=Floor[1/p[n]];
Table[q[n], {n, 1, 80}]
FindLinearRecurrence[Table[q[n], {n, 1, 1000}]]
Join[{3},LinearRecurrence[{3,-3,1},{8,18,32},69]] (* Ray Chandler, Aug 02 2015 *)

a(n)=floor(1/{(n^4+2*n)^(1/4)}), where {}=fractional part.

It appears that a(n)=3a(n-1)-3a(n-2)+a(n-3) for n>=5, and that a(n)=2*n^2 for n>=2.

A222170 a(n) = n^2 + 2*floor(n^2/3).

Original entry on oeis.org

0, 1, 6, 15, 26, 41, 60, 81, 106, 135, 166, 201, 240, 281, 326, 375, 426, 481, 540, 601, 666, 735, 806, 881, 960, 1041, 1126, 1215, 1306, 1401, 1500, 1601, 1706, 1815, 1926, 2041, 2160, 2281, 2406, 2535, 2666, 2801, 2940, 3081, 3226, 3375, 3526, 3681, 3840

Offset: 0

Bruno Berselli, Aug 08 2013

Also, a(n) = n^2 + floor(2*n^2/3), since 2*floor(n^2/3) = floor(2*n^2/3).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tadeusz Dorozinskis, Pentagonpolyhedra Doro
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Subsequence of A008851.

Cf. A004773 (numbers of the type n+floor(n/3)), A008810 (numbers of the type n^2-2*floor(n^2/3)), A047220 (numbers of the type n+floor(2*n/3)), A184637 (numbers of the type n^2+floor(n^2/3), except the first two).

Magma
```
[n^2+2*Floor(n^2/3): n in [0..50]];
```

I:=[0,1,6,15,26]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Table[n^2 + 2 Floor[n^2/3], {n, 0, 50}]
CoefficientList[Series[x (1 + x) (1 + 3 x + x^2) / ((1 + x + x^2) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 6, 15, 26}, 50] (* Hugo Pfoertner, Jan 17 2023 *)

G.f.: x*(1+x)*(1 + 3*x + x^2)/((1 + x + x^2)*(1-x)^3).
a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(n) = floor(5*n^2/3). - Wesley Ivan Hurt, Mar 16 2015
a(n) = a(n-3) + 5*(2n-3) [Tadeusz Dorozinski]. - Eduard Baumann, Jan 18 2023

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A256320 Number of partitions of 4n into exactly 3 parts.

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A184636 a(n) = floor(1/{(n^4+2*n)^(1/4)}), where {}=fractional part.

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A222170 a(n) = n^2 + 2*floor(n^2/3).

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