[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a309881 -id:a309881
Displaying 1-3 of 3 results found.
page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A309882
Sum of the even parts appearing among the fourth largest parts of the partitions of n into 5 parts.
+0
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 10, 14, 18, 24, 30, 40, 52, 68, 88, 110, 136, 166, 198, 240, 286, 340, 404, 478, 560, 652, 754, 872, 1000, 1146, 1308, 1488, 1686, 1908, 2148, 2416, 2708, 3028, 3376, 3758, 4168, 4616, 5098, 5630, 6200, 6816, 7482, 8198
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
LINKS
Table of n, a(n) for n=0..53.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -1).
FORMULA
a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} k * ((k-1) mod 2).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 2*a(n-8) - 2*a(n-9) + 6*a(n-10) - 10*a(n-11) + 12*a(n-12) - 12*a(n-13) + 12*a(n-14) - 12*a(n-15) + 11*a(n-16) - 9*a(n-17) + 4*a(n-18) + 4*a(n-19) - 9*a(n-20) + 11*a(n-21) - 12*a(n-22) + 12*a(n-23) - 12*a(n-24) + 12*a(n-25) - 10*a(n-26) + 6*a(n-27) - 2*a(n-28) - 2*a(n-29) + 4*a(n-30) - 4*a(n-31) + 4*a(n-32) - 4*a(n-33) + 4*a(n-34) - 4*a(n-35) + 3*a(n-36) - a(n-37) for n > 36.
EXAMPLE
Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
1+1+1+1+10
1+1+1+2+9
1+1+1+3+8
1+1+1+4+7
1+1+1+5+6
1+1+1+1+9 1+1+2+2+8
1+1+1+2+8 1+1+2+3+7
1+1+1+3+7 1+1+2+4+6
1+1+1+4+6 1+1+2+5+5
1+1+1+5+5 1+1+3+3+6
1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
--------------------------------------------------------------------------
n | 10 11 12 13 14 ...
--------------------------------------------------------------------------
a(n) | 4 6 10 14 18 ...
--------------------------------------------------------------------------
MATHEMATICA
Table[Sum[Sum[Sum[Sum[k * Mod[k - 1, 2], {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]
LinearRecurrence[{3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12,
12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2,
4, -4, 4, -4, 4, -4, 3, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6,
10, 14, 18, 24, 30, 40, 52, 68, 88, 110, 136, 166, 198, 240, 286,
340, 404, 478, 560, 652, 754, 872, 1000, 1146, 1308}, 50]
CROSSREFS
Cf. A309879, A309880, A309881.
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 21 2019
STATUS
approved
A309880
Sum of the odd parts appearing among the fourth largest parts in the partitions of n into 5 parts.
+0
3
0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 13, 18, 26, 34, 46, 57, 72, 87, 110, 133, 165, 201, 246, 291, 349, 407, 481, 559, 653, 754, 875, 1003, 1154, 1309, 1496, 1690, 1913, 2152, 2423, 2707, 3032, 3373, 3763, 4169, 4627, 5109, 5643, 6204, 6825, 7473, 8197
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Table of n, a(n) for n=0..53.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -1).
FORMULA
a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} k * (k mod 2).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 2*a(n-8) - 2*a(n-9) + 6*a(n-10) - 10*a(n-11) + 12*a(n-12) - 12*a(n-13) + 12*a(n-14) - 12*a(n-15) + 11*a(n-16) - 9*a(n-17) + 4*a(n-18) + 4*a(n-19) - 9*a(n-20) + 11*a(n-21) - 12*a(n-22) + 12*a(n-23) - 12*a(n-24) + 12*a(n-25) - 10*a(n-26) + 6*a(n-27) - 2*a(n-28) - 2*a(n-29) + 4*a(n-30) - 4*a(n-31) + 4*a(n-32) - 4*a(n-33) + 4*a(n-34) - 4*a(n-35) + 3*a(n-36) - a(n-37) for n > 36.
EXAMPLE
Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
1+1+1+1+10
1+1+1+2+9
1+1+1+3+8
1+1+1+4+7
1+1+1+5+6
1+1+1+1+9 1+1+2+2+8
1+1+1+2+8 1+1+2+3+7
1+1+1+3+7 1+1+2+4+6
1+1+1+4+6 1+1+2+5+5
1+1+1+5+5 1+1+3+3+6
1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
--------------------------------------------------------------------------
n | 10 11 12 13 14 ...
--------------------------------------------------------------------------
a(n) | 5 7 8 13 18 ...
--------------------------------------------------------------------------
MATHEMATICA
LinearRecurrence[{3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -1}, {0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 13, 18, 26, 34, 46, 57, 72, 87, 110, 133, 165, 201, 246, 291, 349, 407, 481, 559, 653, 754, 875, 1003, 1154, 1309}, 50]
Table[Total[Select[IntegerPartitions[n, {5}][[;; , 4]], OddQ]], {n, 0, 60}] (* Harvey P. Dale, Oct 10 2024 *)
CROSSREFS
Cf. A309879, A309881, A309882.
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 21 2019
STATUS
approved
A309879
Number of odd parts appearing among the fourth largest parts of the partitions of n into 5 parts.
+0
3
0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 11, 14, 18, 22, 28, 33, 40, 47, 56, 65, 77, 89, 104, 119, 137, 155, 177, 199, 225, 252, 283, 315, 352, 389, 432, 476, 525, 576, 633, 691, 756, 823, 897, 973, 1057, 1143, 1237, 1334, 1439, 1547, 1665, 1786, 1917, 2052
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Table of n, a(n) for n=0..56.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1,0,0,1,-2,2,-3,3,-2,2,-1,0,0,-1,2,-1,1,-2,1).
FORMULA
a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} (k mod 2).
Conjectures from Colin Barker, Aug 22 2019: (Start)
G.f.: x^5*(1 - x + x^2)*(1 - x^3 + x^6) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x^4)*(1 + x + x^2 + x^3 + x^4)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) + a(n-8) - 2*a(n-9) + 2*a(n-10) - 3*a(n-11) + 3*a(n-12) - 2*a(n-13) + 2*a(n-14) - a(n-15) - a(n-18) + 2*a(n-19) - a(n-20) + a(n-21) - 2*a(n-22) + a(n-23) for n>22.
(End) [Conjectures verified by Wesley Ivan Hurt, Aug 24 2019]
EXAMPLE
Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
1+1+1+1+10
1+1+1+2+9
1+1+1+3+8
1+1+1+4+7
1+1+1+5+6
1+1+1+1+9 1+1+2+2+8
1+1+1+2+8 1+1+2+3+7
1+1+1+3+7 1+1+2+4+6
1+1+1+4+6 1+1+2+5+5
1+1+1+5+5 1+1+3+3+6
1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
--------------------------------------------------------------------------
n | 10 11 12 13 14 ...
--------------------------------------------------------------------------
a(n) | 5 7 8 11 14 ...
--------------------------------------------------------------------------
MATHEMATICA
LinearRecurrence[{2, -1, 1, -2, 1, 0, 0, 1, -2, 2, -3, 3, -2, 2, -1,
0, 0, -1, 2, -1, 1, -2, 1}, {0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8,
11, 14, 18, 22, 28, 33, 40, 47, 56, 65}, 50]
PROG
(PARI) Vec(x^5*(1-x+x^2)*(1-x^3+x^6)/((1-x)^5*(1+x)^2*(1+x^2)*(1+x+x^2)*(1-x+x^2-x^3+x^4)*(1+x^4)*(1+x+x^2+x^3+x^4)) + O(x^70)) \\ Jinyuan Wang, Feb 28 2020
CROSSREFS
Cf. A309880, A309881, A309882.
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 21 2019
STATUS
approved
page 1

Search completed in 0.007 seconds

Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified November 7 19:25 EST 2024. Contains 377536 sequences.
License Agreements, Terms of Use, Privacy Policy