A140090 -id:A140090 - cp's OEIS frontend

A363231 Number of partitions of n with rank 4 or higher (the rank of a partition is the largest part minus the number of parts).

0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 209, 267, 343, 435, 551, 693, 870, 1084, 1351, 1672, 2066, 2542, 3121, 3815, 4658, 5664, 6875, 8319, 10049, 12102, 14553, 17452, 20894, 24959, 29766, 35420, 42089, 49911, 59100, 69856, 82452, 97152, 114324, 134315

Offset: 1

Seiichi Manyama, May 22 2023

nonn

In general, for r>=0, Sum_{k>=1} (-1)^(k-1) * p(n - k*(3*k + 2*r - 1)/2) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + (12*r-5)*Pi/144) / sqrt(n/6)), where p() is the partition function. - Vaclav Kotesovec, May 26 2023

			a(7) = 2 counts these partitions: 7, 6+1.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

With rank r or higher: A064174 (r=0), A064173 (r=1), A123975 (r=2), A363230 (r=3), this sequence (r=4).

Cf. A000041, A140090, A363213.

Table[Count[IntegerPartitions[n],?(#[[1]]-Length[#]>3&)],{n,60}] (* _Harvey P. Dale, Jul 29 2024 *)

a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+7)/2));

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+7)/2).
a(n) = p(n-5) - p(n-13) + p(n-24) - ... + (-1)^(k-1) * p(n-k*(3*k+7)/2) + ..., where p() is A000041().
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 43*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

A370238 a(n) = n(3n + 23)/2.

Original entry on oeis.org

0, 13, 29, 48, 70, 95, 123, 154, 188, 225, 265, 308, 354, 403, 455, 510, 568, 629, 693, 760, 830, 903, 979, 1058, 1140, 1225, 1313, 1404, 1498, 1595, 1695, 1798, 1904, 2013, 2125, 2240, 2358, 2479, 2603, 2730, 2860, 2993, 3129, 3268, 3410, 3555, 3703, 3854, 4008

Offset: 0

Torlach Rush, Feb 12 2024

a(a(1)) = A000566(a(1)). This is also true for each of the sequences provided in the formulae below; e.g., A151542(A151542(1)) = A000566(A151542(1)).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A000566, A008594, A005449, A045943, A059845, A115067, A140090, A140091, A140672, A140673, A140674, A140675, A151542, A277976.

Table[n(3n+23)/2,{n,0,48}] (* James C. McMahon, Feb 20 2024 *)

Python
```
def a(n): return n*(3*n+23)//2
```

a(n) = n*(3*n + 23)/2 = A277976(n)/2.
G.f.: x*(13-10*x)/(1-x)^3.
a(n) = A151542(n) + n.
a(n) = A140675(n) + 2*n.
a(n) = A140674(n) + 3*n.
a(n) = A140673(n) + 4*n.
a(n) = A140672(n) + 5*n.
a(n) = A059845(n) + 6*n.
a(n) = A140091(n) + 7*n.
a(n) = A140090(n) + 8*n.
a(n) = A115067(n) + 9*n.
a(n) = A045943(n) + 10*n.
a(n) = A005449(n) + 11*n.
a(n) = A000326(n) + A008594(n).
Sum_{n>=1} 1/a(n) = 823467/2769844 + sqrt(3)*Pi/69 -3*log(3)/23 = 0.2328608... - R. J. Mathar, Apr 23 2024
E.g.f.: exp(x)*x*(26 + 3*x)/2. - Stefano Spezia, Apr 26 2024

A119789 T(n, k) = floor(log_{goldenratio}(Fibonacci(n)*Fibonacci(k))), with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n > 2, triangle read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 3, 4, 5, 6, 4, 4, 4, 5, 6, 7, 8, 5, 5, 5, 6, 7, 8, 9, 10, 6, 6, 6, 7, 8, 9, 10, 11, 12, 7, 7, 7, 8, 9, 10, 11, 12, 13, 14, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset: 0

Roger L. Bagula, Jul 30 2006

			Triangle begins as:
  0;
  0, 0;
  0, 0, 0;
  1, 1, 1, 2;
  2, 2, 2, 3, 4;
  3, 3, 3, 4, 5, 6;
  4, 4, 4, 5, 6, 7, 8;
  5, 5, 5, 6, 7, 8, 9, 10;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000045, A001477, A016789, A033627, A035517, A064455, A140090.

A119789:= func< n,k | n le 2 select 0 else k le 1 select n-2 else n+k-4 >;
[A119789(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 17 2022

f[n_, k_]= If[n<3, 0, If[k==0, n-2, Floor[Log[GoldenRatio, Fibonacci[n]*Fibonacci[k]]]]];
Table[f[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<3, 0, If[k<2, n-2, n+k-4]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 17 2022 *)

def A119789(n,k):
    if (n<3): return 0
    elif (k<2): return n-2
    else: return n+k-4
flatten([[A119789(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 17 2022

T(n, k) = floor(log_{goldenratio}(Fibonacci(n)*Fibonacci(k))), with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n > 2.
From G. C. Greubel, Dec 17 2022: (Start)
T(n, k) = n+k-4, with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n >= 3.
T(n, n) = 2*T(n, 0).
T(2*n, n) = 0*[n<2] + A016789(n-2)*[n>1].
T(2*n, n+1) = 3*A001477(n-1), for n > 0.
T(2*n, n-1) = A033627(n) - [n=1].
T(3*n, n) = n*[n<2] + 4*A000027(n-2)*[n>1].
Sum_{k=0..n} T(n, k) = 0*[n<2] + A140090(n-2)*[n>1].
Sum_{k=0..n} (-1)^k * T(n, k) = 0*[n<2] + (-1)^n*A064455(n-2)*[n>1]. (End)

Edited by G. C. Greubel, Dec 17 2022

A342138 Array T(n,k) = (n+k)(3n+3k-5)/2 + (3k+1), read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 3, 2, 5, 8, 7, 10, 13, 16, 15, 18, 21, 24, 27, 26, 29, 32, 35, 38, 41, 40, 43, 46, 49, 52, 55, 58, 57, 60, 63, 66, 69, 72, 75, 78, 77, 80, 83, 86, 89, 92, 95, 98, 101, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156

Offset: 0

Michel Marcus, Mar 01 2021

This is an instance of a storing function on N^2 (injective) with density 1/3.

			Array begins:
   1  3   8  16  27 ...
   0  5  13  24  38 ...
   2 10  21  35  52 ...
   7 18  32  49  69 ...
  15 29  46  66  89 ...
  ...

John S. Lew and Arnold L. Rosenberg, Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials, Journal of Number Theory, 10(2):215-243, 1978. See Corollary 5.2 p. 211.

Cf. A005449 (first column), A104249 (first row), A140090 (second row), A201279 (diagonal).

T(n,k) = (n+k)*(3*n+3*k-5)/2 + (3*k+1);
matrix(8, 8, n, k, T(n-1, k-1))

cp's OEIS Frontend

A363231 Number of partitions of n with rank 4 or higher (the rank of a partition is the largest part minus the number of parts).

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A370238 a(n) = n*(3*n + 23)/2.

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A119789 T(n, k) = floor(log_{goldenratio}(Fibonacci(n)*Fibonacci(k))), with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n > 2, triangle read by rows.

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A342138 Array T(n,k) = (n+k)*(3*n+3*k-5)/2 + (3*k+1), read by ascending antidiagonals.

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PARI

A370238 a(n) = n(3n + 23)/2.

A342138 Array T(n,k) = (n+k)(3n+3k-5)/2 + (3k+1), read by ascending antidiagonals.