login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A296161 Sum of the larger parts of the partitions of n into two parts such that the smaller part is odd. 0
0, 1, 2, 3, 4, 8, 10, 12, 14, 21, 24, 27, 30, 40, 44, 48, 52, 65, 70, 75, 80, 96, 102, 108, 114, 133, 140, 147, 154, 176, 184, 192, 200, 225, 234, 243, 252, 280, 290, 300, 310, 341, 352, 363, 374, 408, 420, 432, 444, 481, 494, 507, 520, 560, 574, 588, 602 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Sum of the lengths of the distinct rectangles with integer length and odd width such that L + W = n, W <= L. For example, a(6) = 8; the rectangles with odd width are 1 X 5 and 3 X 3, and the sum of their lengths gives 5 + 3 = 8. - Wesley Ivan Hurt, Dec 06 2017

LINKS

Table of n, a(n) for n=1..57.

Index entries for sequences related to partitions

FORMULA

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

Conjectures from Colin Barker, Dec 06 2017: (Start)

G.f.: x^2*(1 + x + x^2 + x^3 + 2*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>9.

(End)

a(n) = (1/64)*(12*n^2+4*n-6-4*(-1)^((2*n-3-(-1)^n)/4)+(4*n-2)*(-1)^n+(8*n-4)*(-1)^((2*n+3+(-1)^n)/4)). - Wesley Ivan Hurt, Dec 06 2017

EXAMPLE

a(10) = 21; the partitions of 10 into two parts are 9 + 1, 8 + 2, 7 + 3, 6 + 4 and 5 + 5. Of these, three have an odd smaller part and the sum of the larger parts of these partitions gives 9 + 7 + 5 = 21.

MATHEMATICA

Table[Sum[(n - i)  Mod[i, 2], {i, Floor[n/2]}], {n, 80}]

Table[Total[Select[IntegerPartitions[n, {2}], OddQ[#[[2]]]&][[All, 1]]], {n, 60}] (* Harvey P. Dale, Jan 21 2019 *)

CROSSREFS

Cf. A295286, A295287.

Sequence in context: A073465 A174816 A161494 * A121612 A061193 A346971

Adjacent sequences:  A296158 A296159 A296160 * A296162 A296163 A296164

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Dec 06 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 26 13:09 EST 2022. Contains 350598 sequences. (Running on oeis4.)