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Let $T(n, k)$ (A008292) be the number of permutations of length $n+1$ of distinct elements $p_1,p_2,\cdots, p_{n+1}$, where the elements belong to the set $\left\lbrace1, 2,\cdots, n+1\right\rbrace$, $ …

Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{b_1}(1+2^{b_2+1}(1+2^{b_3+1}(1+\cdots(1+2^{b_{\operatorname{wt}(n)-1}+1}(1+2^ …

Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n-1)$ fixed and reversing every n consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $PS(2)$, then apply …

Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $ …

Let $\operatorname{wt}(n)$ be A000120, i.e. the number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Also let's consider $$\ell(n)=\left\lfloor\log_{2} n\right\rfloor$$ and $$T(n, …

Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expan …

Let $$f(n,k)=n\operatorname{mod} k, g(n,k)=\left\lfloor\frac{n}{k}\right\rfloor$$ Let $T(n,k)$ be A072030, i.e., array read by antidiagonals: $T(n,k)$ = number of steps in simple Euclidean algorithm f …

Let $a_1(n)$ be A200714, i.e., Stolarsky representation interpreted as binary to decimal integers. The sequence begins with $$0, 1, 3, 2, 7, 5, 6, 15, 4, 11, 13, 14, 31, 10, 9, 23, 12, 27, 29$$ Let $a …

Let $T(n,k)$ be A035506, i.e., Stolarsky array read by antidiagonals. Here we consider that $T(n,k)=0$ for $n<1, k<1$. Let $a(n)$ be A200714, i.e., Stolarsky representation interpreted as binary to de …

Let $$\varphi=\frac{1+\sqrt{5}}{2}$$ Let $$a_1(n)=\left\lfloor n\varphi \right\rfloor, a_2(n)=n+a_1(n)$$ Let $\operatorname{tr}(n)$ be A007814, i.e., the number of trailing zeros in the binary represe …

Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$. Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with $$1, 2, 3, …

Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$. Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with $$1, 2, 3, …

Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ Let $$f(n)=n-2^{\ell(n)}$$ Let $a(n)$ be a permutation of the nonnegative integers such that $a(0)=0$, $a(n)=n$ if $n$ is a power of $2$ and subsequen …

Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor $$ Let $$ f(n) = 2^{\ell(n)} $$ Let $q_1(n)$ and $q_2(n)$ be an arbitrary self-inverse permutations of non-negative integers (that is, $q_i(q_i(n)) … Let $p_1(n)$ and $p_2(n)$ be a permutations of non-negative integers such that $p_i(n)<2^k$ iff $n<2^k$. …

Let $a(n)$ be A193231, blue code of $n$ i.e. self-inverse permutation of non-negative integers such that $a(n)<2^k$ iff $n<2^k$ and $$ a(n\operatorname{XOR}k) = a(n) \operatorname{XOR} a(k) $$ Let $$ …