A003983 -id:A003983 - cp's OEIS frontend

A115217 Diagonal sums of "correlation triangle" for 2^n.

1, 2, 6, 13, 30, 62, 133, 270, 558, 1125, 2286, 4590, 9253, 18542, 37230, 74533, 149358, 298862, 598309, 1196910, 2394990, 4790565, 9583470, 19168110, 38340901, 76684142, 153377646, 306759973, 613538670, 1227086702, 2454210853

Offset: 0

Paul Barry, Jan 16 2006

Diagonal sums of number triangle A003983.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-2,-2,4).

Cf. A003983.

LinearRecurrence[{2,2,-3,-2,-2,4},{1,2,6,13,30,62},40] (* Harvey P. Dale, Oct 18 2021 *)

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} [j<=k]*2^(k-j)*[j<=n-2k]*2^(n-2k-j).
From Paul Barry, Jan 18 2006: (Start)
G.f.: 1/((1-2*x)*(1-2*x^2)*(1-x^3)).
a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - 2*a(n-4) - 2*a(n-5) + 4*a(n-6). (End)
E.g.f.: (exp(x)*(7 + 48*exp(x)) + 2*exp(-x/2)*cos(sqrt(3)*x/2) - 36*cosh(sqrt(2)*x) - 30*sqrt(2)*sinh(sqrt(2)*x))/21. - Stefano Spezia, Aug 28 2025

A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2k, 1 + 2(n-k), n).

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Mar 01 2009

The general form of this, and related triangular sequences, takes the form A(n, k, m) = (m*(n-k) + 1)*A(n-1, k-1, m) + (m*k + 1)*A(n-1, k, m) + m*f(n, k)* A(n-2, k-1, m), where f(n,k) is a polynomial in n and k.

Row sums are: 0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, ... = A007590(n+1). - N. J. A. Sloane, Aug 27 2009

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

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A003983, A157457.

T := proc(m,n) return min(1+2*m, 1+2*(n-m), n): end: seq(seq(T(m,n),m=0..n),n=0..14); # Nathaniel Johnston, Apr 29 2011

T[n_, k_]:= Min[1+2*k, 1+2*(n-k), n]; Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten

T(n, k) = min(1 + 2*k, 1 + 2*(n - k), n).
From Yu-Sheng Chang, May 19 2020: (Start)
O.g.f.: F(z,v) = (1+v)*z/((1-v*z-1)*(1-z)*(1-v*z^2)).
T(n,k) = [v^k] (1+v)*(2*v^(n+1)+2-((sqrt(v)-1)^2 * (-1)^n + (sqrt(v)+1)^2) * v^((1/2)*n))/(2*(v-1)^2). (End)

Edited by N. J. A. Sloane, Aug 27 2009

More terms from and partially edited by G. C. Greubel, May 21 2020

A321126 T(n,k) = max(n + k - 1, n + 1, k + 1), square array read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 4, 5, 6, 7, 6, 5, 5, 5, 6, 7, 8, 7, 6, 6, 6, 6, 7, 8, 9, 8, 7, 7, 7, 7, 7, 8, 9, 10, 9, 8, 8, 8, 8, 8, 8, 9, 10, 11, 10, 9, 9, 9, 9, 9, 9, 9, 10, 11, 12, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13

Offset: 0

Franck Maminirina Ramaharo, Nov 21 2018

T(n,k) - 1 is the maximum degree of d in the three-variable bracket polynomial (A,B,d) for the two-bridge knot with Conway's notation C(n,k). Hence, T(n,k) is the maximum number of Jordan curves that are obtained by splitting the crossings of such knot diagram.

			Square array begins:
    1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
    2,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
    3,  3,  3,  4,  5,  6,  7,  8,  9, 10, ...
    4,  4,  4,  5,  6,  7,  8,  9, 10, 11, ...
    5,  5,  5,  6,  7,  8,  9, 10, 11, 12, ...
    6,  6,  6,  7,  8,  9, 10, 11, 12, 13, ...
    7,  7,  7,  8,  9, 10, 11, 12, 13, 14, ...
    8,  8,  8,  9, 10, 11, 12, 13, 14, 15, ...
    9,  9,  9, 10, 11, 12, 13, 14, 15, 16, ...
   10, 10, 10, 11, 12, 13, 14, 15, 16, 17, ...
  ...

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.
Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.
Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013.
Franck Maminirina Ramaharo, Illustration of T(2,2)
Franck Maminirina Ramaharo, Note on sequence A321125 and related ones
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial

T(n,1) = degree of the (n+1)-th row polynomial in A300453.
T(n,k) = degree of the n-th row polynomials in A300454 and A321127, k = 2,n, respectively.
Cf. A321125, A316989.
Cf. A003056, A003983, A051125.

Table[Max[k + 1, n - 1, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten

create_list(max(k + 1, n - 1, n - k + 1), n, 0, 10, k, 0, n);

T(n,k) = T(k,n).
T(n,k) = A051125(n+1,k+1) for 0 <= k <= 2, n >= 0, and T(n,k) =  A051125(n+1,k+1) + A003983(n-2,k-2) for k >= 3, n >= 3.
T(n,n) = A004280(n+1).
G.f.: (1 - (2*x - x^2)*y + (x - 2*x^2 + x^3)*y^2 + (x^2 - x^3)*y^3)/(((1 - x)*(1 - y))^2).

A107044 A symmetric factorial triangle, read by rows: T(n,k) = min(n,k)!.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 6, 2, 1, 1, 2, 6, 6, 2, 1, 1, 2, 6, 24, 6, 2, 1, 1, 2, 6, 24, 24, 6, 2, 1, 1, 2, 6, 24, 120, 24, 6, 2, 1, 1, 2, 6, 24, 120, 120, 24, 6, 2, 1, 1, 2, 6, 24, 120, 720, 120, 24, 6, 2, 1, 1, 2, 6, 24, 120, 720, 720, 120, 24, 6, 2, 1

Offset: 1

Gary W. Adamson, May 09 2005

			First few rows of the triangle are:
1
1, 1
1, 2, 1
1, 2, 2, 1
1, 2, 6, 2, 1
1, 2, 6, 6, 2, 1
1, 2, 6, 24, 6, 2, 1 ...
Row having (1, 2, 3, 2, 1) in A003983 is replaced by (1, 2, 6, 2, 1) in A107044.

Cf. A003983.

Given A003983, replace each term with its factorial.

Edited by Michael Kleber, Apr 06 2009

Corrected by Philippe Deléham, Nov 07 2011

A144480 T(n,k) = binomial(n, k)*min(k + 1, n - k + 1), triangle read by rows (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 18, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 80, 45, 12, 1, 1, 14, 63, 140, 140, 63, 14, 1, 1, 16, 84, 224, 350, 224, 84, 16, 1, 1, 18, 108, 336, 630, 630, 336, 108, 18, 1, 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1

Offset: 0

Roger L. Bagula, Oct 11 2008

			Triangle begins:
  1;
  1,  1;
  1,  4,   1;
  1,  6,   6,   1;
  1,  8,  18,   8,    1;
  1, 10,  30,  30,   10,    1;
  1, 12,  45,  80,   45,   12,    1;
  1, 14,  63, 140,  140,   63,   14,   1;
  1, 16,  84, 224,  350,  224,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  336, 108,  18,  1;
  1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
  ...

Row sums are in A245560.

Cf. A003983, A007318, A168643.

Table[Table[Binomial[n, m]*If[m <= Floor[n/2], 1 + m, 1 + n - m], {m, 0, n}], {n, 0, 10}] // Flatten

create_list(binomial(n, k)*min(k + 1, n - k + 1), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Dec 10 2018 */

If k <= floor(n/2), then T(n,k) = binomial(n, k)*(k + 1), otherwise T(n,k) = binomial(n, k)*(n - k - 1).

T(n,k) = A007318(n,k)*A003983(k+1,n-k+1), i.e., term-by term product of Pascal's triangle A007318 and A003983 as a triangle.

Entry revised by N. J. A. Sloane, Aug 07 2014

Edited by Franck Maminirina Ramaharo, Dec 10 2018

A157457 Read n-th row of triangle in A157458 and regard it as the expansion of a number in base n+1.

Original entry on oeis.org

0, 3, 16, 125, 1116, 12943, 182400, 3080025, 60524200, 1357997531, 34237168560, 957927505717, 29446184348868, 986272776455415, 35746439807927296, 1393753996031259953, 58165330905054360720, 2586788074128361802419

Offset: 0

Roger L. Bagula, Mar 01 2009

nonn

Cf. A139038, A003983, A157458.

t[n_, m_] =Min[1 + 2*m, 1 + 2*(n - m), n];
Table[FromDigits[{Table[t[n, m], {m, 0, n}], n + 1}, n + 1], {n, 0, 20}]

Edited by N. J. A. Sloane, Aug 27 2009

A351154 a(n) is the permanent of the n X n matrix M(n) that is defined as M[i,j,n] = A351153(n, min(i, j)) + abs(i - j).

Original entry on oeis.org

1, 1, 7, 169, 10388, 1324344, 305668180, 116145817656, 67770421715800, 57594670663866124, 68393751368082128320, 109765035421144948709232, 231657098706747226470685920, 628412716450312334529486247152, 2149132484027947970192241804640128, 9113755489596517688997731211571700256

Offset: 0

Stefano Spezia, Feb 02 2022

nonn

Conjectures: (Start)
det(M(0)) = det(M(1)) = 1 and det(M(n)) = -(n - 2)! for n > 1.
abs(det(M(n))) = abs(A159333(n-2)). (End)

			a(3) = 169:
    1    2    3
    2    4    5
    3    5    6
a(4) = 10388:
    1    2    3    4
    2    5    6    7
    3    6    8    9
    4    7    9   10

Cf. A000142, A003983, A049581, A159333, A351153.

A351153[n_,k_]:=n(k-1)-k(k-3)/2; M[i_,j_,n_]:=A351153[n,Min[i,j]]+Abs[i-j]; a[n_]:=Permanent[Table[M[i,j,n],{i,n},{j,n}]]; Join[{1},Array[a,15]]

t(n, k) = n*(k-1) - k*(k-3)/2; \\ A351153
a(n) = matpermanent(matrix(n, n, i, j, t(n, min(i, j)) + abs(i - j))); \\ Michel Marcus, Feb 03 2022

A115296 Skew version of correlation triangle for constant sequence 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 2, 3, 2, 1, 0, 0, 0, 0, 1, 3, 3, 2, 1, 0, 0, 0, 0, 0, 2, 4, 3, 2, 1, 0, 0, 0, 0, 0, 1, 3, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 2, 4, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 4, 5, 4, 3, 2, 1

Offset: 0

Paul Barry, Jan 19 2006

Row sums are A001399. Diagonal sums are A025795.

			Triangle begins
1;
0,1;
0,1,1;
0,0,2,1;
0,0,1,2,1;
0,0,0,2,2,1;
0,0,0,1,3,2,1;
0,0,0,0,2,3,2,1;

G.f.: 1/((1-x*y)*(1-x^2*y)*(1-x^3*y^2)); Number triangle T(n, k)=sum{j=0..k, [j<=n-k]*[j<=2k-n]}; T(n, k)=A003983(k, n-k).

A129804 a(0) = 1, a(1) = 2; for n>0, a(2n) = 3a(2n-1) - a(2n-2), a(2n+1) = 3a(2n) - a(2n-1) - a(n-1).

Original entry on oeis.org

1, 2, 5, 12, 31, 79, 206, 534, 1396, 3642, 9530, 24917, 65221, 170667, 446780, 1169467, 3061621, 8014862, 20982965, 54932637, 143814946, 376508559, 985710731, 2580614104, 6756131581, 17687755722, 46307135585, 121233585812

Offset: 0

Paul Curtz, May 19 2007

Equals the eigensequence of the correlation triangle, A003983. - Gary W. Adamson, Mar 14 2011

Cf. A003983.

a:=proc(n) if n=0 or n=1 then n+1 elif n mod 2 = 0 then 3*a(n-1)-a(n-2) else 3*a(n-1)-a(n-2)-a((n-3)/2) fi end: seq(a(n),n=0..30); - Emeric Deutsch, May 20 2007

{m=27; v=vector(m+1); v[1]=1; v[2]=2; for(n=2, m, k=3*v[n]-v[n-1]; if(n%2==1, k=k-v[(n-1)/2]); v[n+1]=k); print(v)} /* Klaus Brockhaus, May 20 2007 */

More terms from Emeric Deutsch and Klaus Brockhaus, May 20 2007

A157455 Number generated by regarding the numbers in row n of A157454 as digits of a base n number.

Original entry on oeis.org

1, 3, 19, 125, 1141, 12943, 182743, 3080025, 60530761, 1357997531, 34237329611, 957927505717, 29446189175677, 986272776455415, 35746439978786671, 1393753996031259953, 58165330912030118161, 2586788074128361802419

Offset: 1

Roger L. Bagula, Mar 01 2009

Note that for odd n, the digits will include n itself.

			Row 4 of A157454 is 1,3,3,1. 1331 in base 4 = 4^3 + 3*4^2 + 3*4 + 1 = 125, so a(4) = 125.

A139038, A003983

t[n_, m_] =Min[1 + 2*m, 1 + 2*(n - m)];
Table[FromDigits[{Table[t[n, m], {m, 0, n}], n + 1}, n + 1], {n, 0, 20}]

t(n,m)=Min[2*m - 1, 2*(n - m) + 1]; a(n)=FromDigits[{Table[t[n, m], {m, 1, n}], n}, n].

Edited by Franklin T. Adams-Watters, Sep 25 2011

cp's OEIS Frontend

A115217 Diagonal sums of "correlation triangle" for 2^n.

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A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2*k, 1 + 2*(n-k), n).

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A321126 T(n,k) = max(n + k - 1, n + 1, k + 1), square array read by antidiagonals (n >= 0, k >= 0).

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A107044 A symmetric factorial triangle, read by rows: T(n,k) = min(n,k)!.

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A144480 T(n,k) = binomial(n, k)*min(k + 1, n - k + 1), triangle read by rows (n >= 0, 0 <= k <= n).

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A157457 Read n-th row of triangle in A157458 and regard it as the expansion of a number in base n+1.

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A351154 a(n) is the permanent of the n X n matrix M(n) that is defined as M[i,j,n] = A351153(n, min(i, j)) + abs(i - j).

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A115296 Skew version of correlation triangle for constant sequence 1.

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A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2k, 1 + 2(n-k), n).

A129804 a(0) = 1, a(1) = 2; for n>0, a(2n) = 3a(2n-1) - a(2n-2), a(2n+1) = 3a(2n) - a(2n-1) - a(n-1).