All Questions
Tagged with integer-sequences co.combinatorics
160 questions
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55
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Sequences that sum up to sums of integer coefficients
Let
$$
T(n,k,p,q,r,s) = (q(k-1)+1)T(n-1,k,p,q,r,s) + s(n+r(k-1)+p-2)T(n-1,k-1,p,q,r,s), \\
T(n,1,p,q,r,s) = 1, \\
T(n,0,p,q,r,s) = T(0,k,p,q,r,s) = 0
$$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\...
- 4,948
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190
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On a A057985 without recursion
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0\to01, 1\to12, 2\to0$).
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\...
- 4,948
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63
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Pairs of permutations such that $p(n)<2^k$ iff $n<2^k$
Let $p(n)$ be an arbitrary permutation of natural numbers such that $p(n)<2^k$ iff $n<2^k$.
Let $q(n)$ be an inverse permutation of $p(n)$.
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
...
- 4,948
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0
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107
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Formula for individual term of the Proth numbers
Let $a(n)$ be A080075 i.e. Proth numbers: of the form $k2^m + 1$ for $k$ odd, $m \geqslant 1$ and $2^m > k$.
The sequence begins with
$$
3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129
$$...
- 4,948
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0
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61
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Stolarsky array and Stolarsky representation
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 ...
- 4,948
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94
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Closed form for the number of steps required to get $n$ balls in the last box
Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Then we have an integer sequence given by
$$a(n)=n(n+1)-\sum\limits_{k=0}^{n}\...
- 4,948
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315
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Number Theory and p-Power-Partitioned Numbers
Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, we're going to define its digits-partition as the next set $D_{N} = \bigcup_{j=1}^{n}\bigcup_{k=1}^{p(a_{j})}\{(P_{k},j)\}$, where each pair $(...
user89024
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540
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Number Theory and d-Self-Contained Numbers
Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...
user89024
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0
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224
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Classic question on integer partitions (with distinct summands)
I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem:
Denote $R(N,L)$ ...
- 1
-4
votes
1
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250
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What are the patterns of the sequence of polynomials? [closed]
In my research, I obtained a sequence of polynomials (I am only able to compute the first 4 of them):
\begin{align}
& f(2) = 1+t, \\
& f(3) = 1+4t+3t^2, \\
& f(4) = 1+6t+12t^2+7t^3, \\
&...
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