A268351 -id:A268351 - cp's OEIS frontend

A272104 Sum of the even numbers among the larger parts of the partitions of n into two parts.

0, 0, 0, 2, 2, 4, 4, 10, 10, 14, 14, 24, 24, 30, 30, 44, 44, 52, 52, 70, 70, 80, 80, 102, 102, 114, 114, 140, 140, 154, 154, 184, 184, 200, 200, 234, 234, 252, 252, 290, 290, 310, 310, 352, 352, 374, 374, 420, 420, 444, 444, 494, 494, 520, 520, 574, 574, 602

Offset: 0

Wesley Ivan Hurt, Apr 20 2016

Essentially, repeated values of A152749.

Sum of the lengths of the distinct rectangles with even length and integer width such that L + W = n, W <= L. For example, a(10) = 14; the rectangles are 2 X 8 and 4 X 6, so 8 + 6 = 14. - Wesley Ivan Hurt, Nov 04 2017

			a(5) = 4; the partitions of 5 into 2 parts are (4,1),(3,2) and the sum of the larger even parts is 4.
a(6) = 4; the partitions of 6 into 2 parts are (5,1),(4,2),(3,3) and the sum of the larger even parts is also 4.

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).

Cf. A000326, A001318, A026741, A049450, A049452, A109043, A126964, A152749, A268351, A272212.

[(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4)/2)/8 : n in [0..50]];

A272104:=n->(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4)/2)/8: seq(A272104(n), n=0..100);

Table[(1 + 3(2n-3-(-1)^n)/2 + 3(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8, {n, 0, 50}]
Table[Total@ Map[First, IntegerPartitions[n, {2}] /. {k_, } /; OddQ@ k -> Nothing], {n, 0, 57}] (* _Michael De Vlieger, Apr 20 2016, Version 10.2 *)

concat(vector(3), Vec(2*x^3*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x)^2*(1+x^2)^2) + O(x^50))) \\ Colin Barker, Apr 20 2016

a(n) = (1 + 3*(2n-3-(-1)^n)/2 + 3*(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8.
a(n) = Sum_{i=ceiling(n/2)..n-1} i * (i+1 mod 2).
a(n) = Sum_{i=1..floor(n/2)} (n-i) * (n-i+1 mod 2).
a(2n+1) = a(2n+2) = A152749(n) = 2*A001318(n).
G.f.: 2*x^3*(1-x+x^2)*(1+x+x^2) / ((1-x)^3*(1+x)^2*(1+x^2)^2). - Colin Barker, Apr 20 2016
From Wesley Ivan Hurt, Apr 22 2016, Apr 23 2016: (Start)
a(2n+2)-a(2n) = A109043(n) = 2*A026741(n).
a(4n) = A049450(n) = 2*A000326(n),
a(8n) = A126964(n) = 2*A049452(n),
a(12n) = 2*A268351(n).
a(n+1) = A001318(n) - A272212(n+1). (End)
E.g.f.: ((2 + 3*x*(1 + x))*cosh(x) - 2*(cos(x) + x*cos(x) + x*sin(x)) + (-1 + 3*(-1 + x)*x)*sinh(x))/16. - Ilya Gutkovskiy, Apr 29 2016

A298397 Pentagonal numbers divisible by 4.

Original entry on oeis.org

0, 12, 92, 176, 376, 532, 852, 1080, 1520, 1820, 2380, 2752, 3432, 3876, 4676, 5192, 6112, 6700, 7740, 8400, 9560, 10292, 11572, 12376, 13776, 14652, 16172, 17120, 18760, 19780, 21540, 22632, 24512, 25676, 27676, 28912, 31032, 32340, 34580, 35960, 38320, 39772, 42252

Offset: 1

Bruno Berselli, Jan 18 2018

If b(n) is the n-th octagonal number multiple of 32 then a(n) = b(n)/8.

			A000326(8) = 92 is in the sequence because 92 = 4*23.

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

Cf. A000326, A000567.
Subsequence of A047217, A047388.
Cf. pentagonal numbers divisible by k: A014633 (k=2), A268351 (k=3), this sequence (k=4), A117793 (k=5).

List([1..50], n -> 8*n*(3*n-7)-(6*n-7)*(-1)^n+33);

[8*n*(3*n-7)-(6*n-7)*(-1)^n+33: n in [1..50]];

P:=proc(n) local x; x:=n*(3*n-1)/2; if x mod 4=0 then x; fi; end:
seq(P(i),i=0..2*10^2); # Paolo P. Lava, Jan 19 2018

Table[8 n (3 n - 7) - (6 n - 7) (-1)^n + 33, {n, 1, 50}]
(* Second program (using definition): *)
Select[Table[k*(3*k - 1)/2, {k, 0, 200}], Divisible[#, 4]&] (* Jean-François Alcover, Jan 19 2018 *)

makelist(8*n*(3*n-7)-(6*n-7)*(-1)^n+33, n, 1, 50);

vector(50, n, nn; 8*n*(3*n-7)-(6*n-7)*(-1)^n+33)

concat(0, Vec(4*x^2*(3 + 20*x + 15*x^2 + 10*x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jan 20 2018

[8*n*(3*n-7)-(6*n-7)*(-1)^n+33 for n in (1..50)]

O.g.f.: 4*x^2*(3 + 20*x + 15*x^2 + 10*x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: (33 - 32*x + 24*x^2)*exp(x) + (7 + 6*x)*exp(-x) - 40.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = 8*n*(3*n - 7) - (6*n - 7)*(-1)^n + 33.
From Colin Barker, Jan 20 2018: (Start)
a(n) = 24*n^2 - 62*n + 40 for n even.
a(n) = 24*n^2 - 50*n + 26 for n odd. (End)

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A272104 Sum of the even numbers among the larger parts of the partitions of n into two parts.

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A298397 Pentagonal numbers divisible by 4.

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