A126224 -id:A126224 - cp's OEIS frontend

A343558 Irregular triangle read by rows: the n-th row gives the row indices of the consecutive elements of the spiral of the n X n matrix defined in A126224.

1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 3, 2, 2, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 3, 2, 2, 2, 3, 3, 1, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 5, 5, 4, 3, 2, 2, 2, 2, 3, 4, 4, 4, 3, 3, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 2, 2, 2, 2, 3, 4, 5, 5, 5, 5, 4, 3, 3, 3, 4, 4

Offset: 1

Stefano Spezia, Apr 19 2021

			The triangle begins
1
1   1   2   2
1   1   1   2   3   3   3   2   2
1   1   1   1   2   3   4   4   4   4   3   2   2   2   3   3
...

Stefano Spezia, First 30 rows of the triangle, flattened

Cf. A000290 (row length), A002265, A002411 (row sums), A010873, A060747, A126224, A343559 (column indices).

a:={};nmax:=6;For[n=1,n<=nmax,n++,For[s=1,s<=2n-1,s++,If[OddQ[s] &&Mod[s,4]==1,k=Ceiling[s/4];For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k]],If[EvenQ[s]&&Mod[s,4]==2,For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k+i]];k+=Ceiling[n-s/2],If[EvenQ[s]&&Mod[s,4]==0,For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k-i]];k=k-i+1,For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k]]]]]]];a

A343559 Irregular triangle read by rows: the n-th row gives the column indices of the consecutive elements of the spiral of the n X n matrix defined in A126224.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 3, 3, 2, 1, 1, 2, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 1, 1, 2, 3, 3, 2, 1, 2, 3, 4, 5, 5, 5, 5, 5, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 4, 4, 3, 2, 2, 3, 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 5, 4, 3, 2, 2, 2, 3, 4, 4, 3

Offset: 1

Stefano Spezia, Apr 19 2021

			The triangle begins
1
1   2   2   1
1   2   3   3   3   2   1   1   2
1   2   3   4   4   4   4   3   2   1   1   1   2   3   3   2
...

Stefano Spezia, First 30 rows of the triangle, flattened

Cf. A000290 (row length), A002265, A002411 (row sums), A010873, A060747, A126224, A343558 (row indices).

a:={};nmax:=6;For[n=1,n<=nmax,n++,For[s=1,s<=2n-1,s++,If[OddQ[s]&&Mod[s,4]==1 ,k=Floor[s/4];For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k+i]];k=+Floor[s/4]+Ceiling[n-s/2],If[EvenQ[s],For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k]],For[i=1,i<=Ceiling[n-s/2],i++,AppendTo[a,k-i]];k=k-i+1]]]]; a

A343155 Irregular triangle T read by rows: T(n, k) is the sum of the consecutive integers placed along the k-th turn of the spiral of the n X n matrix defined in A126224.

Original entry on oeis.org

1, 10, 36, 9, 78, 58, 136, 164, 25, 210, 318, 138, 300, 520, 356, 49, 406, 770, 654, 250, 528, 1068, 1032, 612, 81, 666, 1414, 1490, 1086, 394, 820, 1808, 2028, 1672, 932, 121, 990, 2250, 2646, 2370, 1614, 570, 1176, 2740, 3344, 3180, 2440, 1316, 169, 1378, 3278, 4122, 4102, 3410, 2238, 778

Offset: 1

Stefano Spezia, Apr 07 2021

			The triangle T(n, k) begins:
n\k|   1    2    3    4
---+-------------------
1  |   1
2  |  10
3  |  36    9
4  |  78   58
5  | 136  164   25
6  | 210  318  138
7  | 300  520  356   49
...
For n = 1 the matrix is
      1
and T(1, 1) = 1.
For n = 2 the matrix is
      1, 2
      4, 3
and T(2, 1) = 1 + 2 + 3 + 4 = 4*5/2 = 10.
For n = 3 the matrix is
      1, 2, 3
      8, 9, 4
      7, 6, 5
and T(3, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 8*9/2 = 36; T(3, 2) = 9.
For n = 4 the matrix is
      1,  2,  3,  4
     12, 13, 14,  5
     11, 16, 15,  6
     10,  9,  8,  7
and T(4, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 12*13/2 = 78; T(4, 2) = 13 + 14 + 15 + 16 = (13 + 16)*4/2 = 58.
...

Cf. A000007, A000290, A033585, A037270 (row sums), A110654, A126224.

Table[2(2k-n-1)(3+8k(k-n-1)+4n)+n^2KroneckerDelta[n,2k-1],{n,14},{k,Ceiling[n/2]}]//Flatten

T(n, k) = 2*(2*k - n - 1)*(3 + 8*k*(k - n - 1) + 4*n) + n^2*0^(n+1-2*k) with 0 < k <= ceiling(n/2).

T(n, 1) = A033585(n-1) for n > 1.

A023999 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling inward, starting in a corner.

Original entry on oeis.org

1, 5, 48, 660, 11760, 257040, 6652800, 198918720, 6745939200, 255826771200, 10727081164800, 492775291008000, 24610605962342400, 1327677426915840000, 76940526008586240000, 4766815315895592960000, 314406967644177408000000, 21995911456386651463680000

Offset: 1

Charles Diminnie (charles.diminnie(AT)rampo.angelo.edu)

nonn

Starting in the NW or SE corner, the signs are cyclic (+,-,-,+), starting in the NE or SW corner, the signs are always positive.

			n=4: det of
.1..2..3.4
12.13.14.5
11.16.15.6
10..9..8.7

Alois P. Heinz, Table of n, a(n) for n = 1..200
Gaurav Bhatnagar, Christian Krattenthaler, Spiral determinants, arXiv:1704.02859 [math.CO], 2017.
Charles Vanden Eynden, Problem 1517, Mathematics Magazine, Vol. 70, No. 1, Feb., 1997 p. 65.

Cf. A079340, A078475, A067276, A052182.

Main diagonal of A226167, A126224 (signed version). - Alois P. Heinz, Jan 21 2014

a:= proc(n) option remember; `if`(n<2, (3*n+1)/4,
      4*(3*n-1)*(2*n-5)*(2*n-3) *a(n-2) /(3*n-7))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 21 2014

M[0, 0] = 1;
M[i_, j_] := If[i <= j,
  If[i + j >= 0, If[i != j, M[i + 1, j] + 1, M[i, j - 1] + 1],
   M[i, j + 1] + 1],
  If[i + j > 1, M[i, j - 1] + 1, M[i - 1, j] + 1]
  ]
M[n_] := n^2 + 1 - If[EvenQ[n],
  Table[M[i, j], {j, n/2, -n/2 + 1, -1}, {i, -n/2 + 1, n/2}],
  Table[M[i, j], {j, (n - 1)/2, -(n - 1)/2, -1}, {i, -(n - 1)/2, (n - 1)/2}]]
a[n_]:=Det[M[n]] (* Christian Krattenthaler, Apr 19 2017 *)

A023999(n):=if n=1 then 1 else 2*((-1)^((n+4)*(n-1))/2 *(3*n-1) * (2*n-3)!/(n-2)!)$
makelist(A023999(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n) = (3n-1) * (2n-3)!/(n-2)! for n >= 2. [corrected by Robert Israel, Apr 20 2017]

E.g.f.: ((-2*x-1)*sqrt(1-4*x)+1-4*x)/(16*x-4). - Robert Israel, Apr 20 2017

Edited and extended by Robert G. Wilson v, May 07 2003

A304487 a(n) = (3 + 2n - 3n^2 + 4n^3 - 3((-1 + n) mod 2))/6.

Original entry on oeis.org

1, 4, 15, 36, 73, 128, 207, 312, 449, 620, 831, 1084, 1385, 1736, 2143, 2608, 3137, 3732, 4399, 5140, 5961, 6864, 7855, 8936, 10113, 11388, 12767, 14252, 15849, 17560, 19391, 21344, 23425, 25636, 27983, 30468, 33097, 35872, 38799, 41880, 45121, 48524, 52095

Offset: 1

Stefano Spezia, Aug 17 2018

a(n) is the trace of an n X n matrix A in which the entries are 1 through n^2, spiraling inward starting with 1 in the (1,1)-entry (proved).

The first three terms of a(n) coincide with those of A317614.

			For n = 1 the matrix A is
   1
with trace Tr(A) = a(1) = 1.
For n = 2 the matrix A is
   1, 2
   4, 3
with Tr(A) = a(2) = 4.
For n = 3 the matrix A is
   1, 2, 3
   8, 9, 4
   7, 6, 5
with Tr(A) = a(3) = 15.
For n = 4 the matrix A is
   1,  2,  3, 4
  12, 13, 14, 5
  11, 16, 15, 6
  10,  9,  8, 7
with Tr(A) = a(4) = 36.

Stefano Spezia, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A317614, A228958, A045991, A085046.

Cf. A126224 (determinant of the matrix A), A317298 (first differences).

a_n:=List([1..43], n->(3 + 2*n - 3*n^2 + 4*n^3 - 3*RemInt(-1 + n, 2))/6);

List([1..43],n->(3+2*n-3*n^2+4*n^3-3*((-1+n) mod 2))/6); # Muniru A Asiru, Sep 17 2018

I:=[1,4,15,36,73]; [n le 5 select I[n] else 3*Self(n-1)-2*Self(n-2)-2*Self(n-3)+3*Self(n-4)-Self(n-5): n in [1..43]]; // Vincenzo Librandi, Aug 26 2018

seq((3+2*n-3*n^2+4*n^3-3*modp((-1+n),2))/6,n=1..43); # Muniru A Asiru, Sep 17 2018

Table[1/6 (3 + 2 n - 3 n^2 + 4 n^3 - 3 Mod[-1 + n, 2]), {n, 1, 43}] (* or *)
CoefficientList[ Series[x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)), {x, 0, 43}], x] (* or *)
LinearRecurrence[{3, -2, -2, 3, -1}, {1, 4, 15, 36, 73}, 43]

a(n):=(3 + 2*n - 3*n^2 + 4*n^3 - 3*mod(-1 + n, 2))/6$ makelist(a(n), n, 1, 43);

Vec(x*(1 + x + 5*x^2 + x^3)/((-1 + x)^4*(1 + x)) + O(x^44))

a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n)%2))/6

a(n) = A045991(n) - Sum_{k=2..n-1} A085046(k) for n > 2 (proved).
G.f.: x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)).
a(n) + a(n + 1) = A228958(2*n + 1).
From Colin Barker, Aug 17 2018: (Start)
a(n) = (2*n - 3*n^2 + 4*n^3) / 6 for n even.
a(n) = (3 + 2*n - 3*n^2 + 4*n^3) / 6 for n odd.
a(n) = 3*a(n - 1) - 2*a(n - 2) - 2*a(n - 3) + 3*a(n - 4) - a(n - 5) for n > 5.
(End)
E.g.f.: (1/12)*exp(-x)*(-3 + exp(2*x)*(3 + 6*x + 18*x^2 + 8*x^3)). - Stefano Spezia, Feb 10 2019

cp's OEIS Frontend

A343558 Irregular triangle read by rows: the n-th row gives the row indices of the consecutive elements of the spiral of the n X n matrix defined in A126224.

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A343559 Irregular triangle read by rows: the n-th row gives the column indices of the consecutive elements of the spiral of the n X n matrix defined in A126224.

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A343155 Irregular triangle T read by rows: T(n, k) is the sum of the consecutive integers placed along the k-th turn of the spiral of the n X n matrix defined in A126224.

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A023999 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling inward, starting in a corner.

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A304487 a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n) mod 2))/6.

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A304487 a(n) = (3 + 2n - 3n^2 + 4n^3 - 3((-1 + n) mod 2))/6.