Sequence \(t\)-balancing numbers (Q2853453)

A positive integer \(n\) is called \textit{co-balancing number} if NEWLINE\[NEWLINE 1+2+\dots+n=(n+1)+(n+2)+\dots+(n+r) NEWLINE\]NEWLINE for some positive integer \(r\) called the \textit{co-balancer} and \textit{balancing number} if NEWLINE\[NEWLINE 1+2+\dots+(n-1)=(n+1)+(n+2)+\dots+(n+r) NEWLINE\]NEWLINE where \(r\) is called \textit{balancer}.NEWLINENEWLINEThe authors defined a number \(n\) to be a \textit{\(t\)-balancing number} if NEWLINE\[NEWLINE 1+2+\dots+n=(n+1+t)+(n+2+t)+\dots+(n+r+t), NEWLINE\]NEWLINE \(r\) is called the \textit{\(t\)-balancer}. With this generalized definition, the 0-balancing numbers are the co-balancing numbers and 1-balancing numbers are one less than the balancing numbers.NEWLINENEWLINEFor a sequence of real numbers \(\{a_n\}^\infty_{n=1}\) G. K. Panda defined a number \(a_m\) of this sequence a \textit{sequence balancing numbers} if NEWLINE\[NEWLINE a_1+a_2+\cdots+a_{m-1}=a_{m+1}+a_{m+2}+\cdots+a_{m+r} NEWLINE\]NEWLINE for some natural number \(r\). Similarly, he defined \(a_m\) a \textit{sequence cobalancing number} if NEWLINE\[NEWLINE a_1+a_2+\cdots+a_{m}=a_{m+1}+a_{m+2}+\cdots+a_{m+r}. NEWLINE\]NEWLINE The authors generalized the above concepts as sequence \(t\)-balancing number.NEWLINENEWLINEThey called a number \(a_m\) a \textit{sequence \(t\)-balancing number} if NEWLINE\[NEWLINE a_1+a_2+\cdots+a_{m}=a_{m+t+1}+a_{m+t+2}+\cdots+a_{m+t+r}. NEWLINE\]NEWLINE Accordingly, the sequence 0-balancing number is the sequence cobalancing number and the sequence 1-balancing number is the next term of sequence balancing number.NEWLINENEWLINEThe authors made two theorems in this topic. These results are the following:\newline { Theorem 1.} The recurrence relation for the sequence \(t\)-balancing number in the sequence of odd natural numbers is \(a_{m_n}=6a_{m_{n-1}}-a_{m_{n-2}}+4(t+1)\).\newline { Theorem 2.} The only sequence \(t\)-balancing number in the Fibonacci sequence for \(t=0\) is \(F_2=1\).\newline For more details see this paper.