A004247 -id:A004247 - cp's OEIS frontend

A353128 Antidiagonal sums of A353109.

0, 0, 1, 4, 10, 20, 35, 20, 39, 48, 57, 40, 61, 58, 68, 92, 59, 96, 105, 114, 79, 118, 106, 116, 149, 98, 153, 162, 171, 118, 175, 154, 164, 206, 137, 210, 219, 228, 157, 232, 202, 212, 263, 176, 267, 276, 285, 196, 289, 250, 260, 320, 215, 324, 333, 342, 235, 346

Offset: 0

Stefano Spezia, Apr 24 2022

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

Cf. A004247, A010888, A353109, A353933.

LinearRecurrence[{0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1},{0,0,1,4,10,20,35,20,39,48,57,40,61,58,68,92,59,96,105},58]

G.f.: x^2*(1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4 + 20*x^5 + 39*x^6 + 48*x^7 + 55*x^9 + 32*x^10 + 41*x^11 + 18*x^12 - 2*x^13 - 19*x^15 + 105*x^17 + x^18 + 3*x^19 + 6*x^20 + 10*x^21 + 15*x^22 + 89*x^23 + 19*x^24 + 9*x^25 - 48*x^26)/((1 - x)^2*(1 + x + x^2)^2*(1 + x^3 + x^6)^2).

a(n) = 2*a(n-9) - a(n-18) for n > 18.

A317095 a(n) = 40*n.

Original entry on oeis.org

0, 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, 440, 480, 520, 560, 600, 640, 680, 720, 760, 800, 840, 880, 920, 960, 1000, 1040, 1080, 1120, 1160, 1200, 1240, 1280, 1320, 1360, 1400, 1440, 1480, 1520, 1560, 1600, 1640, 1680, 1720, 1760, 1800, 1840, 1880

Offset: 0

Felix Fröhlich, Sep 07 2018

a(n) is equal to the freshwater zone below sea level for a water table of elevation n above sea level in a simplified freshwater-saltwater interface in a coastal water-table aquifer (cf. Barlow, 2003, p. 14, eq. (2) and p. 15, Fig. B-1 and B-2).
From Bruno Berselli, Sep 10 2018: (Start)
After 0, subsequence of A065607: 1/a(n)^2 + 1/(30*n)^2 = 1/(24*n)^2, with n > 0 and a(n) > 30*n.
Also, all positive terms belong to A049094: 2^(40*n)-1 = 1024^(4*n)-1 and (25*41-1)^(4*n)-1 is divisible by 25. (End)

P. M. Barlow, Ground Water in Freshwater-Saltwater Environments of the Atlantic Coast, Circular 1262 (2003).
Wikipedia, Aquifer
Wikipedia, Saltwater intrusion
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A007395, A008586, A008592, A008602, A010692, A049094, A065607.
Row n = 40 of A004247. Intersection of A008587 and A008590.
After 0, subsequence of A005101.

Table[40 n, {n, 0, 50}] (* or *)
LinearRecurrence[{2, -1}, {0, 40}, 50] (* or *)
CoefficientList[Series[40*x/(1 - x)^2, {x, 0, 50}], x] (* Stefano Spezia, Sep 07 2018 *)

PARI
```
a(n) = 40*n
```

a(n) = if(n==0, 0, if(n==1, 40, 2*a(n-1)-a(n-2)))

PARI
```
concat(0, Vec(40*x/(1-x)^2 + O(x^60)))
```

O.g.f.: 40*x/(1 - x)^2.
E.g.f.: 40*x*exp(x). - Bruno Berselli, Sep 10 2018
a(n) = 2*a(n - 1) - a(n - 2) for n > 1. - Stefano Spezia, Sep 07 2018
a(n) = A008586(A008592(n)) = 4*A008592(n).
a(n) = A010692(n)*A008586(n) = 10*A008586(n).
a(n) = A008602(A005843(n)) = 20*A005843(n).
a(n) = A007395(n)*A008602(n) = 2*A008602(n).

A069971 Table by antidiagonals of variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -k for the first time.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 8, 8, 8, 0, 0, 20, 22, 22, 20, 0, 0, 40, 48, 48, 48, 40, 0, 0, 70, 90, 92, 92, 90, 70, 0, 0, 112, 152, 160, 160, 160, 152, 112, 0, 0, 168, 238, 258, 260, 260, 258, 238, 168, 0, 0, 240, 352, 392, 400, 400, 400, 392, 352, 240, 0, 0, 330, 498

Offset: 0

Henry Bottomley, Apr 29 2002

Expected time to reach one of the boundaries at +n or -k for the first time is n*k, i.e. A004247.

			Rows start 0,0,0,0,0,0,0,...; 0,0,2,8,20,40,70...; 0,2,8,22,48,90,152...; 0,8,22,48,92,160,258...; etc.

T(n, k) =nk(n^2+k^2-2)/3 =T(n+1, k-1)/2+T(n-1, k+1)/2+(n-k)^2 with T(n, 0)=T(0, k)=0. T(n, n)=n^2*(n^2-1)*2/3=8*A002415(n).

A141429 Triangle T(n, k) = (k+1)*(n-k+1), read by rows.

Original entry on oeis.org

2, 4, 3, 6, 6, 4, 8, 9, 8, 5, 10, 12, 12, 10, 6, 12, 15, 16, 15, 12, 7, 14, 18, 20, 20, 18, 14, 8, 16, 21, 24, 25, 24, 21, 16, 9, 18, 24, 28, 30, 30, 28, 24, 18, 10, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 24, 33, 40, 45, 48, 49, 48, 45, 40, 33, 24, 13

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 06 2008

			Triangle begins as:
   2;
   4,  3;
   6,  6,  4;
   8,  9,  8,  5;
  10, 12, 12, 10,  6;
  12, 15, 16, 15, 12,  7;
  14, 18, 20, 20, 18, 14,  8;
  16, 21, 24, 25, 24, 21, 16,  9;
  18, 24, 28, 30, 30, 28, 24, 18, 10;
  20, 27, 32, 35, 36, 35, 32, 27, 20, 11;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A003991, A004247, A005581, A144329, A158823.

[(k+1)*(n-k+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 30 2021

A141429 := proc(n,k)
        (k+1)*(n-k+1) ;
end proc:
seq(seq(A141429(n,m),m=1..n),n=1..14) ; # R. J. Mathar, Nov 10 2011

Table[(k+1)*(n-k+1), {n,15}, {k,n}]//Flatten

flatten([[(k+1)*(n-k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 30 2021

T(n, k) = (k+1)*(n-k+1).
T(n, k) = A158823(n+2, k+2).
Sum_{k=1..n} T(n, k) = A005581(n+1).

Edited by G. C. Greubel, Mar 30 2021

A141432 Triangle T(n,k) = (k+1)*(n-k-1) read by rows.

Original entry on oeis.org

-2, 0, -3, 2, 0, -4, 4, 3, 0, -5, 6, 6, 4, 0, -6, 8, 9, 8, 5, 0, -7, 10, 12, 12, 10, 6, 0, -8, 12, 15, 16, 15, 12, 7, 0, -9, 14, 18, 20, 20, 18, 14, 8, 0, -10, 16, 21, 24, 25, 24, 21, 16, 9, 0, -11

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 06 2008

			Triangle begins as:
  -2;
   0, -3;
   2,  0, -4;
   4,  3,  0, -5;
   6,  6,  4,  0, -6;
   8,  9,  8,  5,  0, -7;
  10, 12, 12, 10,  6,  0, -8;
  12, 15, 16, 15, 12,  7,  0, -9;
  14, 18, 20, 20, 18, 14,  8,  0, -10;
  16, 21, 24, 25, 24, 21, 16,  9,   0, -11;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A003991, A004247.

[(k+1)*(n-k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021

A141432:= (n,k) -> (k+1)*(n-k-1); seq(seq(A141432(n,k), k=1..n), n=1..12); # G. C. Greubel, Apr 01 2021

Table[(k+1)*(n-k-1), {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 01 2021 *)

flatten([[(k+1)*(n-k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

T(n,k) = (k+1)*(n-k-1).
Sum_{k=1..n} T(n, k) = n*(n^2 - 13)/6.
G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k = (2*x-1-y)/((1-y)^3*(x-1)^2). - R. J. Mathar, Feb 19 2020

A349039 Square array T(n, k) read by antidiagonals, n, k >= 0; T(n, k) = n^2 - n*k + k^2.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 9, 3, 3, 9, 16, 7, 4, 7, 16, 25, 13, 7, 7, 13, 25, 36, 21, 12, 9, 12, 21, 36, 49, 31, 19, 13, 13, 19, 31, 49, 64, 43, 28, 19, 16, 19, 28, 43, 64, 81, 57, 39, 27, 21, 21, 27, 39, 57, 81, 100, 73, 52, 37, 28, 25, 28, 37, 52, 73, 100, 121, 91, 67, 49, 37, 31, 31, 37, 49, 67, 91, 121

Offset: 0

Rémy Sigrist, Nov 06 2021

T(n, k) is the norm of the Eisenstein integer n + k*w (where w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity).

All terms belong to A003136.

			Array T(n, k) begins:
  n\k|    0   1   2   3   4   5   6   7   8   9   10
  ---+----------------------------------------------
    0|    0   1   4   9  16  25  36  49  64  81  100
    1|    1   1   3   7  13  21  31  43  57  73   91
    2|    4   3   4   7  12  19  28  39  52  67   84
    3|    9   7   7   9  13  19  27  37  49  63   79
    4|   16  13  12  13  16  21  28  37  48  61   76
    5|   25  21  19  19  21  25  31  39  49  61   75
    6|   36  31  28  27  28  31  36  43  52  63   76
    7|   49  43  39  37  37  39  43  49  57  67   79
    8|   64  57  52  49  48  49  52  57  64  73   84
    9|   81  73  67  63  61  61  63  67  73  81   91
   10|  100  91  84  79  76  75  76  79  84  91  100

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Wikipedia, Eisenstein integers: Euclidean domain

Cf. A003136, A004247, A048147, A073254.

T[n_, k_] := n^2 - n*k + k^2; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 08 2021 *)

PARI
```
T(n,k) = n^2 - n*k + k^2
```

T(n, k) = T(k, n).
T(n, 0) = T(n, n) = n^2.
T(n, k) = A048147(n, k) - A004247(n, k).
G.f.: (x - 5*x*y + y*(1 + y) + x^2*(1 + y^2))/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Nov 08 2021

A375577 Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

Original entry on oeis.org

2, 1, 2, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 7, 2, 1, 6, 15, 16, 9, 2, 1, 7, 25, 37, 25, 11, 2, 1, 8, 43, 94, 77, 36, 13, 2, 1, 9, 77, 259, 273, 141, 49, 15, 2, 1, 10, 143, 748, 1045, 646, 235, 64, 17, 2, 1, 11, 273, 2209, 4121, 3151, 1321, 365, 81, 19, 2

Offset: 0

Stefano Spezia, Aug 19 2024

			Array begins:
  2, 2,  2,   2,    2,     2, ...
  1, 3,  5,   7,    9,    11, ...
  1, 4,  9,  16,   25,    36, ...
  1, 5, 15,  37,   77,   141, ...
  1, 6, 25,  94,  273,   646, ...
  1, 7, 43, 259, 1045,  3151, ...
  1, 8, 77, 748, 4121, 15656, ...
  ...

Cf. A000290, A004247, A004248, A005408 (n=1), A005491 (n=3), A007395 (n=0), A054977 (k=0), A176691 (k=2), A176805 (k=3), A176916 (k=5), A176972 (k=7), A214647.

Cf. A375578 (antidiagonal sums).

A[0,0]=2; A[n_,k_]:=k^n+k*n+1;Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

G.f. for the k-th column: (2*x^2 - 3*x - k^2 + k + 1)/((x - 1)^2*(x - k)).
E.g.f. for the k-th column: exp(x)*(1 + exp((k-1)*x) + k*x).
A(n,1) = n + 2.
A(2,n) = A000290(n+1).
A(n,n) = 2*A214647(n) + 1.

A363436 Array read by ascending antidiagonals: A(n, k) = k*n^2, with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 2, 0, 0, 9, 8, 3, 0, 0, 16, 18, 12, 4, 0, 0, 25, 32, 27, 16, 5, 0, 0, 36, 50, 48, 36, 20, 6, 0, 0, 49, 72, 75, 64, 45, 24, 7, 0, 0, 64, 98, 108, 100, 80, 54, 28, 8, 0, 0, 81, 128, 147, 144, 125, 96, 63, 32, 9, 0, 0, 100, 162, 192, 196, 180, 150, 112, 72, 36, 10, 0

Offset: 0

Stefano Spezia, Jul 08 2023

			The array begins:
  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5,   6, ...
  0,  4,  8,  12,  16,  20,  24, ...
  0,  9, 18,  27,  36,  45,  54, ...
  0, 16, 32,  48,  64,  80,  96, ...
  0, 25, 50,  75, 100, 125, 150, ...
  0, 36, 72, 108, 144, 180, 216, ...
  ...

Cf. A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12),  A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), A244630 (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).
Cf. A001477 (n = 1), A008586 (n = 2), A008591 (n = 3), A008598 (n = 4), A008607 (n = 5), A044102 (n = 6), A152691 (n = 8).
Cf. A000007 (n = 0 or k = 0), A000578 (main diagonal), A002415 (antidiagonal sums), A004247.

A[n_,k_]:=k n^2; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

O.g.f.: x*y*(1 + x)/((1 - x)^3*(1 - y)^2).
E.g.f.: x*y*(1 + x)*exp(x + y).
A(n, k) = n*A004247(n, k).

cp's OEIS Frontend

A353128 Antidiagonal sums of A353109.

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A317095 a(n) = 40*n.

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A069971 Table by antidiagonals of variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -k for the first time.

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A141429 Triangle T(n, k) = (k+1)*(n-k+1), read by rows.

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A141432 Triangle T(n,k) = (k+1)*(n-k-1) read by rows.

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A349039 Square array T(n, k) read by antidiagonals, n, k >= 0; T(n, k) = n^2 - n*k + k^2.

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A375577 Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

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A363436 Array read by ascending antidiagonals: A(n, k) = k*n^2, with k >= 0.

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