A049798 a(n) = (1/2)*Sum_{k = 1..n} T(n,k), array T as in A049800.

0, 0, 0, 1, 0, 2, 2, 2, 3, 7, 2, 7, 10, 8, 8, 15, 11, 19, 16, 15, 22, 32, 19, 25, 34, 34, 33, 46, 33, 47, 47, 48, 61, 65, 45, 62, 77, 79, 68, 87, 74, 94, 97, 86, 105, 127, 98, 114, 120, 124, 129, 154, 141, 151, 142, 147, 172, 200, 151, 180

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

a(n) is the sum of the remainders after dividing each larger part by its corresponding smaller part for each partition of n+1 into two parts. - Wesley Ivan Hurt, Dec 20 2020

Examples

			From _Lei Zhou_, Mar 10 2014: (Start)
For n = 3, n+1 = 4, floor((n+1)/2) = 2, mod(4,2) = 0, and so a(3) = 0.
For n = 4, n+1 = 5, floor((n+1)/2) = 2, mod(5,2) = 1, and so a(4) = 1.
...
For n = 12, n+1 = 13, floor((n+1)/2) = 6, mod(13,2) = 1, mod(13,3) = 1, mod(13,4) = 1, mod(13,5) = 3, mod(13,6) = 1, and so a(12) = 1 + 1 + 1 + 3 + 1 = 7. (End)
		

Links

• Lei Zhou, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004125, A008611, A008805, A049797, A049799, A049801.

Half row sums of A049800.

Programs

GAP

List([1..60], n-> Sum([1..n], k-> (n+1) mod Int((k+1)/2))/2 ); # G. C. Greubel, Dec 09 2019

Magma

[ (&+[(n+1) mod Floor((k+1)/2): k in [1..n]])/2: n in [1..60]]; // G. C. Greubel, Dec 09 2019

Maple

seq( add( (n+1) mod floor((k+1)/2), k=1..n)/2, n=1..60); # G. C. Greubel, Dec 09 2019

Mathematica

Table[Sum[Mod[n+1, Floor[(k+1)/2]], {k,n}]/2, {n, 60}] (* G. C. Greubel, Dec 09 2019 *)

PARI

vector(60, n, sum(k=1,n, lift(Mod(n+1, (k+1)\2)) )/2 ) \\ G. C. Greubel, Dec 09 2019

Python

def A049798(n): return sum((n+1)%k for k in range(2,(n+1>>1)+1)) # Chai Wah Wu, Oct 20 2023

Sage

def a(n):
    return sum([(n+1)%k for k in range(2,floor((n+3)/2))])
# Ralf Stephan, Mar 14 2014

Formula

a(n) = Sum_{k=2..floor((n+1)/2)} ((n+1) mod k). - Lei Zhou, Mar 10 2014

a(n) = A004125(n+1) - A008805(n-2), for n >= 2. - Carl Najafi, Jan 31 2013

a(n) = Sum_{i = 1..ceiling(n/2)} ((n-i+1) mod i). - Wesley Ivan Hurt, Jan 05 2017