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Search: a279385 -id:a279385
Displaying 1-7 of 7 results found. page 1
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A280223 Precipice of n: descending by the main diagonal of the pyramid described in A245092, a(n) is the height difference between the n-th level (starting from the top) and the level of the next terrace. +10
8
1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 3, 2, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 5, 4, 3, 2, 1, 3, 2, 1, 1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 1, 5, 4, 3, 2, 1, 2, 1, 1, 1, 4, 3, 2, 1, 4, 3, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the n-th level of the pyramid are also the parts of the symmetric representation of sigma(n).
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
Note that if a(n) > 1 then the next k terms are the first k positive integers in decreasing order, where k = a(n) - 1.
For more information about the precipices see A277437 and A280295.
a(n) is also the number of numbers >= n whose largest Dyck paths of the symmetric representation of sigma share the same point at the main diagonal of the diagram. For more information see A237593.
LINKS
Table of n, a(n) for n=1..107.
Omar E. Pol, Perspective view of the pyramid (first 16 levels)
EXAMPLE
Descending by the main diagonal of the stepped pyramid, for the levels 9, 10 and 11 we have that the next terrace is in the 12th level, so a(9) = 12 - 9 = 3, a(10) = 12 - 10 = 2, and a(11) = 12 - 11 = 1.
CROSSREFS
Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A277437, A279286, A279385, A280295.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 29 2016
EXTENSIONS
More terms from Omar E. Pol, Jan 02 2017
STATUS
approved
A277437 Square array read by antidiagonals upwards in which T(n,k) is the n-th number j such that, descending by the main diagonal of the pyramid described in A245092, the height difference between the level j (starting from the top) and the level of the next terrace is equal to k. +10
7
1, 3, 2, 5, 4, 9, 7, 6, 12, 20, 8, 10, 21, 36, 72, 11, 13, 25, 50, 91, 144, 14, 16, 32, 56, 112 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This is a permutation of the natural numbers.
Column k lists the numbers with precipice k. For more information about the precipices see A280223 and A280295.
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
If a number m is in the column k and k > 1 then m + 1 is the column k - 1.
The largest Dyck path of the symmetric representations of next k - 1 positive integers greater than T(n,k) shares the middle point of the largest Dyck path of the symmetric representation of sigma(T(n,k)). For more information see A237593.
LINKS
Table of n, a(n) for n=1..26.
Omar E. Pol, Perspective view of the pyramid (first 16 levels)
Index entries for sequences that are permutations of the natural numbers
FORMULA
T(n,1) = A071562(n+1) - 1.
EXAMPLE
The corner of the square array begins:
1, 2, 9, 20, 72, 144,
3, 4, 12, 36, 91,
5, 6, 21, 50,
7, 10, 25,
8, 13,
11,
...
T(1,6) = 144 because it is the smallest number with precipice 6.
CROSSREFS
Row 1 gives A280295.
Column 1 gives A276112.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223.
KEYWORD
nonn,tabl,more
AUTHOR
Omar E. Pol, Dec 29 2016
EXTENSIONS
a(20)-a(26) from Omar E. Pol, Jan 02 2017
STATUS
approved
A280295 Smallest number with precipice n. Descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to n. +10
7
1, 2, 9, 20, 72, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k), k >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the precipices see A277437 and A280223.
Is this sequence infinite?
LINKS
Table of n, a(n) for n=1..6.
Omar E. Pol, Perspective view of the pyramid (first 16 steps)
EXAMPLE
a(3) = 9 because descending by the main diagonal of the pyramid, the height difference between the level 9 and the level of the next terrace is equal to 3, and 9 is the smallest number with this property.
CROSSREFS
Row 1 of A277437.
Cf. A000203, A196020, A235791, A236104, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A276112, A277437, A279286, A279385, A280223.
KEYWORD
nonn,more
AUTHOR
Omar E. Pol, Dec 31 2016
EXTENSIONS
a(6) from Omar E. Pol, Jan 02 2017
STATUS
approved
A276112 Numbers with precipice 1: descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to 1. +10
6
1, 3, 5, 7, 8, 11, 14, 15, 17, 19, 23, 24, 27, 29, 31, 34, 35, 39, 41, 44, 47, 48, 49, 53, 55, 59, 62, 63, 65, 69, 71, 76, 79, 80, 83, 87, 89, 90, 95, 97, 98, 99, 103, 107, 109, 111, 116, 119, 120, 125, 127, 129, 131, 134, 139, 142, 143, 149, 152, 153, 155, 159 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k).
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the precipices see A277437, A280223 and A280295.
From Hartmut F. W. Hoft, Feb 02 2022: (Start)
Also partial sums of A280919.
a(n) is also the largest number of a Dyck path that crosses the diagonal at point A282131(n) which also is the rightmost number in each nonzero row of the irregular triangle in A279385. (End)
LINKS
Table of n, a(n) for n=1..62.
Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)
FORMULA
a(n) = A071562(n+1) - 1.
a(n) = Sum_{i=1..n} A280919(i), n >= 1. - Hartmut F. W. Hoft, Feb 02 2022
EXAMPLE
From Hartmut F. W. Hoft, Feb 02 2022: (Start)
n: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 index.
A282131: 1 2 3 5 6 7 9 11 12 13 15 17 18 20 position on diagonal.
A276112: 1 3 5 7 8 11 14 15 17 19 23 24 27 29 max index of Dyck path.
A280919: 1 2 2 2 1 3 3 1 2 2 4 1 3 2 paths at diag position.
(End)
MATHEMATICA
(* last computed value of a280919[ ] is dropped to avoid a potential undercount of crossings *)
a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]
a280919[n_] := Most[Map[Length, Split[Map[a240542, Range[n]]]]]
A276112[160] (* Hartmut F. W. Hoft, Feb 02 2022 *)
CROSSREFS
Column 1 of A277437.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223, A280295.
Cf. A280919, A282131.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 02 2017
STATUS
approved
A299472 a(n) is the sum of all divisors of all numbers k whose associated largest Dyck path contains the point (n,n) in the diagram of the symmetric representation of sigma(k) described in A237593, or 0 if no such k exists. +10
1
1, 7, 13, 0, 20, 15, 43, 0, 66, 0, 24, 49, 59, 0, 134, 0, 60, 113, 0, 86, 0, 104, 165, 0, 48, 245, 0, 132, 0, 224, 0, 198, 0, 124, 57, 317, 0, 192, 0, 350, 0, 326, 0, 104, 211, 0, 434, 0, 216, 0, 0, 647, 0, 344, 0, 186, 331, 0, 584, 0, 270, 0, 234, 0, 672, 0, 350, 171, 0, 156, 639, 0, 672, 0, 390, 0, 368, 0, 956 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..79.
CROSSREFS
Row sums of A299693.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A259179, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 19 2018
STATUS
approved
A299482 Numbers m such that in the diagram of the symmetric representation of sigma(k) described in A237593 there is no Dyck path that contains the point (m,m), where both k and m are positive integers. +10
1
4, 8, 10, 14, 16, 19, 21, 24, 27, 29, 31, 33, 37, 39, 41, 43, 46, 48, 50, 51, 53, 55, 58, 60, 62, 64, 66, 69, 72, 74, 76, 78, 80, 82, 83, 84, 87, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 114, 116, 119, 121, 123, 124, 125, 127, 129, 131, 133, 135, 138, 141, 143, 145, 147, 149, 151, 153 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Indices of the rows that contain a zero in the triangle A279385.
a(n) is the index of the n-th zero in A259179; i.e. A259179(a(n)) = 0. - Hartmut F. W. Hoft, Aug 07 2020
LINKS
Table of n, a(n) for n=1..69.
MATHEMATICA
a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a299482[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; Flatten[Position[t, 0]]]
a299482[153] (* Hartmut F. W. Hoft, Aug 07 2020 *)
CROSSREFS
Complement of A282131.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A244050, A245092, A262626, A279385.
Cf. A240542, A259179.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 19 2018
STATUS
approved
A299693 Irregular triangle read by rows in which row n lists the total sum of the divisors of all numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n); or row n is 0 if no such k exists. +10
1
1, 3, 4, 7, 6, 0, 12, 8, 15, 13, 18, 12, 0, 28, 14, 24, 0, 24, 31, 18, 39, 20, 0, 42, 32, 36, 24, 0, 60, 31, 42, 40, 0, 56, 30, 0, 72, 32, 63, 48, 54, 0, 48, 91, 38, 60, 56, 0, 90, 42, 0, 96, 44, 84, 0, 78, 72, 48, 0, 124, 57, 93, 72, 98, 54, 0, 120, 72, 0, 120, 80, 90, 60, 0, 168, 62, 96, 0, 104, 127, 84, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..82.
FORMULA
T(n,m) = A000203(A279385(n,m) if A279385(n,m) > 0, otherwise T(n,m) = 0.
EXAMPLE
Triangle begins:
1;
3, 4;
7, 6;
0;
12, 8;
15;
13, 18, 12;
0;
28, 14, 24;
0;
24;
31, 18;
39, 20;
0;
42, 32, 36, 24;
0;
...
CROSSREFS
Nonzero terms give A000203.
Row sums give A299472.
Cf. A259179(n) is the number of positive terms in row n.
Cf. A071562, A196020, A235791, A236104, A237048, A237591, A237593, A240542, A244050, A245092, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Feb 19 2018
STATUS
approved
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Last modified August 30 00:57 EDT 2024. Contains 375520 sequences. (Running on oeis4.)