A332662 -id:A332662 - cp's OEIS frontend

A332667 Permutation of N = {0, 1, 2, ...} induced by the enumeration of N X N in A332662.

0, 1, 2, 4, 5, 3, 10, 11, 6, 7, 20, 21, 12, 13, 8, 35, 36, 22, 23, 14, 9, 56, 57, 37, 38, 24, 15, 16, 84, 85, 58, 59, 39, 25, 26, 17, 120, 121, 86, 87, 60, 40, 41, 27, 18, 165, 166, 122, 123, 88, 61, 62, 42, 28, 19, 220, 221, 167, 168, 124, 89, 90, 63, 43, 29, 30

Offset: 0

Peter Luschny, Feb 19 2020

Motivated by the question how a sequence of regular integer triangles can be stored in linear memory (see A332662).

			a(n) can be seen as the triangle read by rows:
[0]  0;
[1]  1,  2;
[2]  4,  5,  3;
[3] 10, 11,  6,  7;
[4] 20, 21, 12, 13, 8;
[5] 35, 36, 22, 23, 14, 9;
[6] 56, 57, 37, 38, 24, 15, 16;
[7] 84, 85, 58, 59, 39, 25, 26, 17;
...
a(n) can also be seen as the rectangular array read by upwards antidiagonals (with flat rows):
(A) [ 0], [ 2,  3], [ 7,  8,  9], [16, 17, 18, 19], [30, 31, 32, 33, 34],...
(B) [ 1], [ 5,  6], [13, 14, 15], [26, 27, 28, 29], ...
(C) [ 4], [11, 12], [23, 24, 25], ...
(D) [10], [21, 22], ...
(E) [20], ...
...

Cf. A332662, A000292 (first column), A332699 (main diagonal).

F := L -> ListTools:-Flatten(L): b := n -> floor((sqrt(8*n+1)-1)/2):
S := (n, k) -> [seq(binomial(n+k+2,3) + binomial(k+1,2)+j, j=0..k)]:
A332667 := (n, k) -> F([seq(S(n-k,j), j=0..b(k))])[k+1]:
seq(seq(A332667(n, k), k=0..n), n=0..10);

A332663 Even bisection of A332662: the x-coordinates of an enumeration of N X N.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

Offset: 0

Peter Luschny, Feb 19 2020

nonn

Cf. A332662, A002260, A014370, A000292 (positions of 0).

Maple
```
# See A332662.
```

A332699 First row of A332662, also main diagonal of A332667.

Original entry on oeis.org

0, 2, 3, 7, 8, 9, 16, 17, 18, 19, 30, 31, 32, 33, 34, 50, 51, 52, 53, 54, 55, 77, 78, 79, 80, 81, 82, 83, 112, 113, 114, 115, 116, 117, 118, 119, 156, 157, 158, 159, 160, 161, 162, 163, 164, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 275, 276, 277, 278

Offset: 0

Peter Luschny, Feb 25 2020

nonn

Cf. A332667, A332662.

a := n -> A332667(n, n): seq(a(n), n=0..58);

A056559 Tetrahedron with T(t,n,k) = t - n; succession of growing finite triangles with declining values per row.

Original entry on oeis.org

0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2

Offset: 0

Henry Bottomley, Jun 26 2000

			First triangle: [0]; second triangle: [1; 0 0]; third triangle: [2; 1 1; 0 0 0]; ...

Peter Luschny, Table of n, a(n) for n = 0..10000 (first 105 terms by Henry Bottomley).

Together with A056558 and A056560 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analog of A002262 and A025581 with square arrays.

Bisection (y-coordinates) of A332662.

function a_list(N)
    a = Int[]
    for n in 1:N
        for j in ((k:-1:1) for k in 1:n)
            t = n - j[1]
            for m in j
                push!(a, t)
end end end; a end
A = a_list(10) # Peter Luschny, Feb 19 2020

from math import isqrt, comb
from sympy import integer_nthroot
def A056559(n): return (m:=integer_nthroot(6*(n+1),3)[0])-(a:=nChai Wah Wu, Dec 11 2024

a(n) = A056556(n) - A056557(n).

A331987 a(n) = ((n + 1) - 9(n + 1)^2 + 8(n + 1)^3)/6.

Original entry on oeis.org

0, 5, 23, 62, 130, 235, 385, 588, 852, 1185, 1595, 2090, 2678, 3367, 4165, 5080, 6120, 7293, 8607, 10070, 11690, 13475, 15433, 17572, 19900, 22425, 25155, 28098, 31262, 34655, 38285, 42160, 46288, 50677, 55335, 60270, 65490, 71003, 76817, 82940, 89380, 96145

Offset: 0

Peter Luschny, Feb 19 2020

nonn

The start values of the partial rows on the main diagonal of A332662 in the representation in the example section.

Apparently the sum of the hook lengths over the partitions of 2*n + 1 with exactly 2 parts (cf. A180681).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Apparently a bisection of A049779 and of A024862.

Cf. A180681, A332662.

[n*(n+1)*(8*n+7)/6: n in [0..50]]; // G. C. Greubel, Apr 19 2023

a := n -> ((n+1) - 9*(n+1)^2 + 8*(n+1)^3)/6: seq(a(n), n=0..41);
gf := (x*(3*x + 5))/(x - 1)^4: ser := series(gf, x, 44):
seq(coeff(ser, x, n), n=0..41);

LinearRecurrence[{4,-6,4,-1}, {0,5,23,62}, 42]
Table[(n-9n^2+8n^3)/6,{n,50}] (* Harvey P. Dale, Apr 11 2024 *)

def A331987(n): return n*(n+1)*(8*n+7)/6
[A331987(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

a(n) = [x^n] (x*(5 + 3*x)/(1 - x)^4).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2).
From G. C. Greubel, Apr 19 2023: (Start)
a(n) = 3*binomial(n+1,1) - 11*binomial(n+2,2) + 8*binomial(n+3,3).
a(n) = n*binomial(8*n+8,2)/24.
a(n) = n*(n+1)*(8*n+7)/6.
E.g.f.: (1/6)*x*(30 + 39*x + 8*x^2)*exp(x). (End)

A332698 a(n) = (8n^3 + 15n^2 + 13*n)/6.

Original entry on oeis.org

0, 6, 25, 65, 134, 240, 391, 595, 860, 1194, 1605, 2101, 2690, 3380, 4179, 5095, 6136, 7310, 8625, 10089, 11710, 13496, 15455, 17595, 19924, 22450, 25181, 28125, 31290, 34684, 38315, 42191, 46320, 50710, 55369, 60305, 65526, 71040, 76855, 82979, 89420, 96186

Offset: 0

Peter Luschny, Feb 20 2020

The end values of the partial rows on the main diagonal of A332662 in the representation in the example section.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A332662, A331987.

a := n -> (8*n^3 + 15*n^2 + 13*n)/6: seq(a(n), n=0..41);
gf := (x*(x^2 + x + 6))/(x - 1)^4: ser := series(gf, x, 44):
seq(coeff(ser, x, n), n=0..41);

LinearRecurrence[{4, -6, 4, -1}, {0, 6, 25, 65}, 42]
Table[(8n^3+15n^2+13n)/6,{n,0,50}] (* Harvey P. Dale, Sep 13 2024 *)

a(n) = [x^n] (x*(x^2 + x + 6))/(x - 1)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2) + binomial(n, 1) = A331987(n) + n.

cp's OEIS Frontend

A332667 Permutation of N = {0, 1, 2, ...} induced by the enumeration of N X N in A332662.

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Maple

A332663 Even bisection of A332662: the x-coordinates of an enumeration of N X N.

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Maple

A332699 First row of A332662, also main diagonal of A332667.

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Maple

A056559 Tetrahedron with T(t,n,k) = t - n; succession of growing finite triangles with declining values per row.

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Julia

Python

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A331987 a(n) = ((n + 1) - 9*(n + 1)^2 + 8*(n + 1)^3)/6.

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Magma

Maple

Mathematica

SageMath

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A332698 a(n) = (8*n^3 + 15*n^2 + 13*n)/6.

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Maple

Mathematica

Formula

A331987 a(n) = ((n + 1) - 9(n + 1)^2 + 8(n + 1)^3)/6.

A332698 a(n) = (8n^3 + 15n^2 + 13*n)/6.