A081436 -id:A081436 - cp's OEIS frontend

A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

1, 1, 6, 22, 55, 111, 196, 316, 477, 685, 946, 1266, 1651, 2107, 2640, 3256, 3961, 4761, 5662, 6670, 7791, 9031, 10396, 11892, 13525, 15301, 17226, 19306, 21547, 23955, 26536, 29296, 32241, 35377, 38710, 42246, 45991, 49951, 54132, 58540, 63181

Offset: 0

Paul Barry, Mar 21 2003

Diagonal in array of n-gonal numbers A081422.

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

Cf. A005564, A081422, A081436, A081437, A081438.

List([0..40], n-> (2*n^3-n^2-n+2)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3-n^2-n+2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

a:= n-> (2*n^3-n^2-n+2)/2: seq(a(n), n=0..50); # Zerinvary Lajos, Sep 13 2006

Table[(2n^3-n^2-n+2)/2,{n,0,40}] (* Harvey P. Dale, May 29 2012 *)
CoefficientList[Series[(1 - 4 x + 11 x^2 - 8 x^3) / (1 - x)^5, {x, 0, 50}],x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (2*n^3-n^2-n+2)/2) \\ G. C. Greubel, Aug 14 2019

[(2*n^3-n^2-n+2)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

G.f.: (1 -4*x +11*x^2 -8*x^3)/(1-x)^5.
a(n) = (n + 1)*(2*n^2 - 3*n + 2)/2 = (n-1)*A005564(n+1) - n*A005564(n), where A005564(0..2) = 0, -1, 0. - Bruno Berselli, May 19 2015
E.g.f.: (2 + 5*x^2 + 2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A235355 0 followed by the sum of (1),(2), (3,4),(5,6), (7,8,9),(10,11,12) from the natural numbers.

Original entry on oeis.org

0, 1, 2, 7, 11, 24, 33, 58, 74, 115, 140, 201, 237, 322, 371, 484, 548, 693, 774, 955, 1055, 1276, 1397, 1662, 1806, 2119, 2288, 2653, 2849, 3270, 3495, 3976, 4232, 4777, 5066, 5679, 6003, 6688, 7049, 7810, 8210, 9051, 9492, 10417, 10901, 11914, 12443, 13548

Offset: 0

Paul Curtz, Jan 07 2014

Difference table for 0 followed by a(n):
  0,  0,  1,   2,   7,  11,  24,  33,...
  0,  1,  1,   5,   4,  13,   9,  25,... =A147685(n)
  1,  0,  4,  -1,   9,  -4,  16,  -9,... =interleave A000290(n+1),-A000290(n)
  -1, 4, -5,  10, -13,  20, -25,  34,...
  5, -9, 15, -23,  33, -45,  59, -75,... =(-1)^n*A027688(n+2).
a(-n) = -a(n-1).
From the second row, signature (0,3,0,-3,0,1).
Consider a(n+2k+1)+a(2k-n):
  1,     2,   6,   9,  17,  22,  34,...
  9,    12,  24,  33,  57,  72, 108,...
  35,   40,  60,  75, 115, 140, 200,...
  91,   98, 126, 147, 203, 238, 322,...
  189, 198, 234, 261, 333, 378, 486,... .
The first column is A005898(n).
The rows are successively divisible by 2*k+1. Hence
  1,   2,  6,  9, 17, 22, 34,...
  3,   4,  8, 11, 19, 24, 36,...
  7,   8, 12, 15, 23, 28, 40,...
  13, 14, 18, 21, 29, 34, 46,...
  21, 22, 26, 29, 37, 42, 54,...
The first column is A002061(n+1).
The main diagonal is A212965(n).
The first difference of every row is A022998(n+1).
Compare to the (2k+1)-sections of A061037 in A165943.

			a(1)=1, a(2)=2, a(3)=3+4=7, a(4)=5+6=11, a(5)=7+8+9=24, a(6)=10+11+12=33.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A075356, A234305.

LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,2,7,11,24,33},50] (* Harvey P. Dale, Nov 22 2014 *)

Vec(x*(x^2+1)*(x^2+x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Jan 20 2014

a(n) = 4*a(n-2) -6*a(n-4) +4*a(n-6) -a(n-8), n>7.
a(2n) = 0 followed by A085786(n). a(2n+1) =  A081436(n).
a(2n) + a(2n+1) = A005898(n).
a(2n-1) + a(2n) = A061317(n).
a(n) = (-1)*((-1+(-1)^n-2*n)*(2+n+n^2))/16. a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). G.f.: x*(x^2+1)*(x^2+x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Jan 20 2014

More terms from Colin Barker, Jan 20 2014

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

Original entry on oeis.org

1, 5, 2, 13, 8, 3, 25, 18, 11, 4, 41, 32, 23, 14, 5, 61, 50, 39, 28, 17, 6, 85, 72, 59, 46, 33, 20, 7, 113, 98, 83, 68, 53, 38, 23, 8, 145, 128, 111, 94, 77, 60, 43, 26, 9, 181, 162, 143, 124, 105, 86, 67, 48, 29, 10, 221, 200, 179, 158, 137, 116, 95, 74, 53, 32, 11

Offset: 1

Stefano Spezia, Dec 20 2019

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).

			n\k|   0   1   2   3   4   5
---+------------------------
1  |   1
2  |   5   2
3  |  13   8   3
4  |  25  18  11   4
5  |  41  32  23  14   5
6  |  61  50  39  28  17   6
...
For n = 3 the matrix M is
      1, 2, 3
      2, 4, 6
      3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A005408, A330287.

Cf. A000027: diagonal; A001105: 2nd column; A001844: 1st column; A016789: 1st subdiagonal; A016885: 2nd subdiagonal; A017029: 3rd subdiagonal; A017221: 4th subdiagonal; A017461: 5th subdiagonal; A081436: row sums; A132209: 3rd column; A164284: 7th subdiagonal; A269044: 6th subdiagonal.

Flatten[Table[1+k-2n-2k*n+2n^2,{n,1,11},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]

O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]

A281725 Triangular array T(n,k) is the sum of elements in an n X k matrix that will be assigned the same value whether the integers from 1 to n*k are assigned to elements in row-major order or column-major order.

Original entry on oeis.org

1, 3, 5, 6, 7, 15, 10, 9, 13, 34, 15, 11, 24, 21, 65, 21, 13, 19, 25, 31, 111, 28, 15, 33, 58, 54, 43, 175, 36, 17, 25, 33, 41, 49, 57, 260, 45, 19, 42, 37, 115, 55, 96, 73, 369, 55, 21, 31, 82, 51, 61, 142, 81, 91, 505, 66, 23, 51, 45, 84, 201, 117, 89, 150, 111, 671

Offset: 1

Michel Marcus, Jan 28 2017

			For n=2 and k=3, the matrix will be
1 2 3  and  1 3 5
4 5 6       2 4 6
and there are 2 identical terms, 1 and 6, whose sum is 7.
The triangle begins
   1;
   3,  5;
   6,  7, 15;
  10,  9, 13, 34;
  15, 11, 24, 21, 65;
  21, 13, 19, 25, 31, 111;
  ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Ana Rechtman, 3ème Défi du Calendrier Mathématique, Images des Mathématiques, CNRS, January 2017.

Cf. A281726.
Main diagonal gives A006003.
Column k=1 gives A000217.
T(2n,n) gives A058331 for n>0.
T(2n+1,n+1) gives A081436.

T:= (n, k)-> add(add(`if`(j+k*(i-1)=
    i+n*(j-1), j+k*(i-1), 0), i=1..n), j=1..k):
seq(seq(T(n,k), k=1..n), n=1..20);  # Alois P. Heinz, Jan 28 2017

a(n, k) = {ml = matrix(n, k, i, j, ((i-1)*k+j)); mc = matrix(n, k, i, j, ((j-1)*n+i)); sum(i=1, n, sum(j=1, k, ml[i,j]*(ml[i,j] == mc[i,j])));}

A129312 A minimal 2 X 2 subdeterminant array.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 13, 11, 5, 6, 14, 18, 18, 14, 6, 7, 17, 23, 25, 23, 17, 7, 8, 20, 28, 32, 32, 28, 20, 8, 9, 23, 33, 39, 41, 39, 33, 23, 9, 10, 26, 38, 46, 50, 50, 46, 38, 26, 10, 11, 29, 43, 53, 59, 61, 59, 53, 43, 29, 11, 12, 32, 48, 60, 68, 72, 72, 68, 60

Offset: 1

Clark Kimberling, Apr 09 2007

Given that row 1 and column 1 are the sequence (1,2,3,4,...), T is the array of minimal positive subdeterminants in the sense that for each 2 X 2 submatrix
a b
c d,
d is the least integer for which the resulting
determinant is positive; indeed, the determinant is 1.
T(n,n)=A001844(n).
SUM{T(n,k): k=1,2,...,n}=A081436(n).
When T is written as the triangle
1
2 2
3 5 3
4 8 8 4
5 11 13 11 5, etc.,
the row sums are A006527 and the alternating row sums are 1,0,1,0,1,0,1,0,... (A059841).
The underlying function T is the same as in A244418, but this triangle's rows hold n+k constant, while in A244418, n is held constant on each row, and k <= n.
T(n,k) can be interpreted as a figurate number, with an (n-1) x (k-1) rectangle of dots interleaved with an n x k rectangle. The American flag illustrates T(5,6).

			Northwest corner:
1 2 3 4 5 6
2 5 8 11 14 17
3 8 13 18 23 28
4 11 18 25 32 39
T(2,2)=5 because 5 is the least positive integer x for which the determinant of the 2 X 2 matrix below is positive:
1 2
2 x

Cf. A001844, A081436, A006527, A059841.

Cf. A244418 (different triangle for the same function T).

(* Array version: *)
Grid[Table[SeriesCoefficient[Series[(n + (n - 1)*x)/(1 - x)^2, {x, 0, 12}], k], {n, 12}, {k, 0, 12}]] (* L. Edson Jeffery, Aug 23 2014 *)
(* Triangle version: *)
Grid[Table[SeriesCoefficient[Series[(n - k + (n - k - 1)*x)/(1 - x)^2, {x, 0, 12}], k], {n, 12}, {k, 0, n - 1}]] (* L. Edson Jeffery, Aug 23 2014 *)

T(n,k)=(2n-1)*k-n+1.

Connection to A244418 and interpretation as figurate numbers from Allan C. Wechsler, Nov 18 2018

A259549 Triangle T(n,k) with rows of length 2n-1 filled with consecutive integers, each appearing twice except for the last term, T(n,2n-1) = n(n+1)/2.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38

Offset: 1

Craig Knecht, Jun 30 2015

This sequence is derived from the water retention heights between a group of hexagonally packed, vertically oriented cylinders. The cylinders fill an equilateral triangle. The cylinder height is specified by sequential natural numbers.
This entry is motivated by A258445 and A259052. The water retention on mathematical surfaces concept uses the integer values to specify the height of cylinders. Three touching cylinders define the retention unit and thus a relationship between 3 integers. Taking the minimum, maximum or sum of these three integers provides the data points to construct a new triangle. The water retention between three cylinders is the minimum of the three heights.
The row sums for the minimum of three terms in this triangle are A081436.
The retention values between rows can be thought of having a wave form with peak and trough values. These peak and trough values correspond to A006002 and A006003. This water retention business shows that A081436(n) = A006002(n) + A006003(n+1).
In general any 2D arrangement of numbers can be subdivided by up and down retention triples.
The terms of the "flattened" sequence (concatenation of the rows) are the positive integers, repeated twice except for (1, 3, 6, 10, ...) = triangular numbers A000217, which are only listed once. - M. F. Hasler, Aug 11 2015

			The irregular triangle T(n,k) starts:
n/k 1  2  3  4  5  6  7  8  9 10 11
1:  1
2:  2  2  3
3:  4  4  5  5  6
4:  7  7  8  8  9  9 10
5: 11 11 12 12 13 13 14 14 15
6: 16 16 17 17 18 18 19 19 20 20 21

Craig Knecht, Micro and macro equilateral triangles.
Craig Knecht, Wave peak and trough.
Wikipedia, Water Retention on Mathematical Surfaces

Partial sums of A071028.

a259549(nmax) = { /* Give the first nmax rows */
  my(L=List(), t);
  for(n=1, nmax,
    t=(n^2-n+2)/2;
    for(k=0, n-2,
      listput(L, t+k); listput(L, t+k)
    );
    listput(L, n*(n+1)/2)
  );
  Vec(L)
}
a259549(6) \\ Colin Barker, Jul 04 2015

A259549(n,k)=(n^2-n+k+1)\2 \\ M. F. Hasler, Aug 11 2015

a(n) = (1/4)*(2*t(n) + 1 - (-1)^t(n)), where t(n) = n + floor(sqrt(n)). - Ridouane Oudra, Jun 08 2019

More terms from Colin Barker, Jul 04 2015

A360665 Square array T(n, k) = k((2n-1)*k+1)/2 read by rising antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, -1, 0, 2, 3, -3, 0, 3, 7, 6, -6, 0, 4, 11, 15, 10, -10, 0, 5, 15, 24, 26, 15, -15, 0, 6, 19, 33, 42, 40, 21, -21, 0, 7, 23, 42, 58, 65, 57, 28, -28, 0, 8, 27, 51, 74, 90, 93, 77, 36, -36, 0, 9, 31, 60, 90, 115, 129, 126, 100, 45, -45

Offset: 0

Paul Curtz, Mar 17 2023

			By rows:
   0,   0,  -1,  -3,  -6,  -10,  -15,  -21,  -28, ...   = -A161680
   0,   1,   3,   6,  10,   15,   21,   28,   36, ...   =  A000217
   0,   2,   7,  15,  26,   40,   57,   77,  100, ...   =  A005449
   0,   3,  11,  24,  42,   65,   93,  126,  164, ...   =  A005475
   0,   4,  15,  33,  58,   90,  129,  175,  228, ...   =  A022265
   0,   5,  19,  42,  74,  115,  165,  224,  292, ...   =  A022267
   0,   6,  23,  51,  90,  140,  201,  273,  356, ...   =  A022269
   ... .

Cf. Antidiagonal sums: A034827(n+1).
Cf. A000217, A000290, A005449, A005475, A022265.
Cf. A022267, A022269, A059270, A081436, A101986.
Cf. A161680, A162254, A179297, A360962, A361226.

T[n_, k_] := ((2*n - 1)*k^2 + k)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 31 2023 *)

T(n, k) = ((2*n-1)*k^2+k)/2 \\ Thomas Scheuerle, Mar 17 2023

T(n,k) = T(n,k-1)+k^2.
T(n,n) = A081436(n-1).
T(n,n+1) = A059270(n).
T(n,n+4) = -3*A179297(n+4).
T(n+3,n) = A162254(n).
T(n+5,n) = 3*A101986(n).
From Stefano Spezia, Mar 31 2023: (Start)
O.g.f.: (x*y - y^2 + 2*x*y^2)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2*x - y + 2*x*y)/2. (End)

cp's OEIS Frontend

A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

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A235355 0 followed by the sum of (1),(2), (3,4),(5,6), (7,8,9),(10,11,12) from the natural numbers.

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2*n - 2*k*n + 2*n^2, with 0 <= k < n.

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A281725 Triangular array T(n,k) is the sum of elements in an n X k matrix that will be assigned the same value whether the integers from 1 to n*k are assigned to elements in row-major order or column-major order.

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A129312 A minimal 2 X 2 subdeterminant array.

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A259549 Triangle T(n,k) with rows of length 2n-1 filled with consecutive integers, each appearing twice except for the last term, T(n,2n-1) = n(n+1)/2.

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A360665 Square array T(n, k) = k*((2*n-1)*k+1)/2 read by rising antidiagonals.

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

A360665 Square array T(n, k) = k((2n-1)*k+1)/2 read by rising antidiagonals.