A354330 Distance from the sum of the first n positive triangular numbers to the nearest triangular number.
0, 0, 1, 0, 1, 1, 1, 6, 0, 6, 10, 10, 13, 10, 1, 14, 4, 21, 12, 4, 0, 1, 8, 22, 28, 1, 36, 1, 35, 30, 10, 4, 11, 10, 0, 20, 51, 41, 10, 71, 4, 62, 41, 6, 45, 75, 91, 88, 97, 85, 55, 10, 51, 100, 10, 99, 20, 124, 29, 56, 130, 90, 48, 20, 7, 10, 30, 68, 125, 136
Offset: 0
Examples
a(4) = 1 because the sum of the first 4 positive triangular numbers is 1 + 3 + 6 + 10 = 20, the nearest triangular number is 21 and 21 - 20 = 1.
Links
- Paolo Xausa, Table of n, a(n) for n = 0..9999
- Wikipedia, Triangular number.
Programs
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Mathematica
nterms=100;Table[ts=n(n+1)(n+2)/3;t=Floor[Sqrt[ts]];Abs[t^2+t-ts]/2,{n,0,nterms-1}] -
PARI
a(n)=my(ts=n*(n+1)*(n+2)/3,t=sqrtint(ts));abs(t^2+t-ts)/2; vector(100,n,a(n-1)) \\ Paolo Xausa, Jul 06 2022
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Python
from math import isqrt def A354330(n): return abs((m:=isqrt(k:=n*(n*(n + 3) + 2)//3))*(m+1)-k)>>1 # Chai Wah Wu, Jul 15 2022
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