A005993 -id:A005993 - cp's OEIS frontend

A269064 At stage 1, start with a unit equilateral triangle. At each successive stage add 3*(n-1) new triangles around outside with vertex-to-vertex contacts. Sequence gives number of triangles at n-th stage.

0, 1, 4, 10, 26, 48, 87, 135, 208, 293, 410, 542, 714, 904, 1141, 1399, 1712, 2049, 2448, 2874, 3370, 3896, 4499, 5135, 5856, 6613, 7462, 8350, 9338, 10368, 11505, 12687, 13984, 15329, 16796, 18314, 19962, 21664, 23503, 25399, 27440, 29541, 31794, 34110, 36586, 39128, 41837, 44615, 47568

Offset: 0

Luce ETIENNE, Feb 18 2016

At stage n, we count (6*n^2-6*n+5-3*(2*n-1)*(-1)^n)/8 unit up-pointing triangles and 3*(2*n^2-2*n+1+(2*n-1)*(-1)^n)/8 unit down-pointing triangles.
At stage n, the total number of unit triangles is (3*n^2-3*n+2)/2 = A005448(n). It is the same total as for A064412. Note also that A064412 gives number of triangles in a geometrical structure according to expansion side-side (mode S-S).
The edges of several unit triangles can form larger size triangles, and these are also up- or down-pointing. The number of all such larger is given by :(14*n^3-9*n^2+11*n+18-(9*n^2+3*n+14)*(-1)^n-4*((-1)^((2*n+1-(-1)^n)/4)))/64 up-pointing triangles and (14*n^3-15*n^2+35*n-6+(9*n^2+21*n+2)*(-1)^n+4*((-1)^((2*n-1+(-1)^n)/4)))/64 down-pointing triangles.
As for A265282 we observe that starting with n = 4 we can see and count hexagonal and dodecagonal forms for example in a reticular system (incomplete with hexagonal holes) by opposition to a compact shape A064412.

			a(0)= 0, a(1) = 1, a(2) = 4, a(3) = 7+3 = 10, a(4) = 19 + 6 + 1 = 26, a(5) = 31 + 12 + 4 + 1 = 48.

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Illustration of initial terms
Kival Ngaokrajang, Illustration of triangles expansion
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Cf. A003215, A005448, A005993, A033428, A064412, A061600, A006003, A039623, A102214, A265282.

[(14*n^3-12*n^2+23*n+6+3*(3*n-2)*(-1)^n+2*((-1)^((2*n-1+(-1)^n) div 4)-(-1)^((6*n-1+(-1)^n) div 4)))/32: n in [0..50]]; // Vincenzo Librandi, Feb 19 2016

Table[(14 n^3 - 12 n^2 + 23 n + 6 + 3 (3 n - 2) (-1)^n + 2 ((-1)^((2*n - 1 + (-1)^n) / 4) - (-1)^((6 n - 1 + (-1)^n) / 4))) / 32, {n, 0, 45}] (* Vincenzo Librandi, Feb 19 2016 *)

concat(0, Vec(x*(1+2*x+2*x^2+8*x^3+2*x^4+5*x^5+x^6)/((1-x)^4*(1+x)^2*(1+x^2)) + O(x^50))) \\ Colin Barker, Feb 24 2016

a(n) = (7*n^3-3*n^2+4*n)/2 for n even.
a(n) = (28*n^3+30*n^2+16*n+7+(-1)^n)/8 for n odd.
a(n) = (14*n^3-12*n^2+23*n+6+3*(3*n-2)*(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)-(-1)^((6*n-1+(-1)^n)/4)))/32.
G.f.: x*(1+2*x+2*x^2+8*x^3+2*x^4+5*x^5+x^6) / ((1-x)^4*(1+x)^2*(1+x^2)). - Colin Barker, Feb 24 2016

A062135 Odd-numbered columns of Losanitsch triangle A034851 formatted as triangle with an additional first column.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 6, 3, 1, 0, 3, 10, 12, 4, 1, 0, 3, 19, 28, 20, 5, 1, 0, 4, 28, 66, 60, 30, 6, 1, 0, 4, 44, 126, 170, 110, 42, 7, 1, 0, 5, 60, 236, 396, 365, 182, 56, 8, 1, 0, 5, 85, 396, 868, 1001, 693, 280, 72, 9, 1

Offset: 0

Wolfdieter Lang, Jun 19 2001

Because the sequence of column m=2*k, k >= 1, of A034851 is the partial sum sequence of the one of column m=2*k-1 the present triangle is essentially Losanitsch's triangle A034851.

Row sums give A051450 with A051450(0) := 1. Column sequences (without leading zeros) are for m=0..6: A000007, A008619, A005993, A005995, A018211, A018213, A062136.

			Triangle begins:
  {1};
  {0,1};
  {0,1,1};
  {0,2,2,1};
  ...
Pe(4,x^2)=1+6*x^2+x^4.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened)

Cf. A034851, A034839.

t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_, k_] := (Binomial[n, k] + Binomial[Quotient[n, 2], Quotient[k, 2]])/2; Flatten[Table[t[n - 1 + m, n - m], {n, 0, 12}, {m, 0, n}]] (* Michael De Vlieger, Sep 28 2024, after Jean-François Alcover at A034851  *)

T(n, m) = A034851(n-1+m, n-m), n >= m >= 0; A034851(n-1, n) := 0, n >= 1, A034851(-1, 0) := 1.
T(n, m) = 0 if n= 1; T(n, m) = T(n-1, m)+sum(T(k, m-1), k=m-1..n-1) if n+m even and T(n, m) = T(n-1, m)+sum(T(k, m-1), k=m-1..n-1)-binomial((n+m-3)/2, m-1) if n+m odd, n >= m >= 1.
G.f. for column m: x^m*Pe(m, x^2)/(((1-x)^(2*m))*(1+x)^m), m >= 0, with Pe(m, x^2)= sum(A034839(m, k)*x^(2*k), k=0..floor(n/2)), the row polynomial of array A034839 (even-indexed entries of the rows of Pascal's triangle).

More terms from Michael De Vlieger, Sep 28 2024

A137508 Successive structures of alkaline earth metals (periodic table elements from 2nd column).

Original entry on oeis.org

2, 2, 2, 8, 2, 2, 8, 8, 2, 2, 8, 18, 8, 2, 2, 8, 18, 18, 8, 2, 2, 8, 18, 32, 18, 8, 2

Offset: 1

Paul Curtz, Apr 23 2008

nonn

Apparently a(n) = A168281(n+1). - Georg Fischer, Nov 11 2021

			27 terms: 2, 2 for beryllium, ... Every structure is palindromic (even and odd mixed). Also 2*A106314.

Cf. A005993, A099956, A168281. Same numbers as in A093907.

A141783 Number of bracelets (turn over necklaces) with n beads: 1 blue, 12 green, and r = n - 13 red.

Original entry on oeis.org

1, 7, 49, 231, 924, 3108, 9324, 25236, 63090, 147070, 323554, 676270, 1352540, 2600612, 4829708, 8692788, 15212379, 25949469, 43249115, 70562765, 112900424, 177412664, 274183208, 417232088, 625848132, 926250780, 1353751140

Offset: 13

Washington Bomfim, Aug 17 2008

Harold S. Grant, On a Formula for Circular Permutations, Mathematics Magazine, Vol. 23, No. 3 (Jan. - Feb., 1950), pp. 133-136

Cf. A005993, A005994, A005995, A018210, A018211, A018212, A018213, A018214, A002620, A062136.

A141783:=n->(1/2)*(binomial(n - 1, 12) + binomial((n - 2 + (n mod 2))/2, 6)); seq(A141783(n), n=13..50); # Wesley Ivan Hurt, Jan 30 2014

Table[(1/2) (Binomial[n - 1, 12] + Binomial[(n - 2 + Mod[n, 2])/2, 6]), {n, 13, 50}] (* Wesley Ivan Hurt, Jan 30 2014 *)

a(n) = 1/2*(binomial(n-1,12) + binomial((n-2+n mod 2)/2, 6)).

a(n) = (1/(2*12!))*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*(n+12)*((n+1)*(n+3)*(n+5)*(n+7)*(n+9)*(n+11) + 1*3*5*7*9*11) - (1/15)*(1/2^10)*(n^5+(65/2)*n^4+400*n^3+(4615/2)*n^2+6154*n+(11895/2))*(1/2)*(1-(-1)^n) [Yosu Yurramendi, Jun 24 2013]

Revised by Washington Bomfim, Jul 24 2012

A212976 Number of (w,x,y) with all terms in {0,...,n} and odd range.

Original entry on oeis.org

0, 6, 12, 36, 60, 114, 168, 264, 360, 510, 660, 876, 1092, 1386, 1680, 2064, 2448, 2934, 3420, 4020, 4620, 5346, 6072, 6936, 7800, 8814, 9828, 11004, 12180, 13530, 14880, 16416, 17952, 19686, 21420, 23364, 25308, 27474, 29640, 32040

Offset: 0

Clark Kimberling, Jun 03 2012

a(n) + A212975(n) = (n+1)^3. Six divides every term.

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A005993, A212959, A212975.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Mod[Max[w, x, y] - Min[w, x, y], 2] == 1,
   s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212976 *)
m/6  (* A005993 except for initial 0 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,6,12,36,60,114},40] (* Harvey P. Dale, Jan 21 2017 *)

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = 6*x*(1 + x^2) and g(x) = ((1-x)^4)*(1+x)^2.
a(n+1) = 6*A005993(n). [Bruno Berselli, Jun 15 2012]

A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)(n + 1) - ij.

Original entry on oeis.org

1, 1, 5, 72, 2309, 140400, 14495641, 2347782144, 562385930985, 190398813728000, 87889475202276461, 53726132414026874880, 42454821207656237294381, 42495322215073539046387712, 52954624815227996007075890625, 80932107560443542398970529579008, 149736953621087625813286348913927569

Offset: 0

Stefano Spezia, Apr 29 2023

nonn

M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
The elements sum of the matrix M(n) is A002415(n+1).
The antidiagonal sum of the matrix M(n) is A005993(n-1).
The n-th row of A107985 gives the row or column sums of the matrix M(n+1).

			a(3) = 72:
           [3, 2, 1]
    M(3) = [2, 4, 2]
           [1, 2, 3]
a(5) = 140400:
           [5, 4, 3, 2, 1]
           [4, 8, 6, 4, 2]
    M(5) = [3, 6, 9, 6, 3]
           [2, 4, 6, 8, 4]
           [1, 2, 3, 4, 5]

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
Wikipedia, Special unitary group

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
    Matrix(n, (i, j)-> min(i, j)*(n+1)-i*j))):
seq(a(n), n=0..16);  # Alois P. Heinz, Apr 30 2023

M[i_, j_, n_]:=Min[i, j](n+1)-i j; Join[{1}, Table[Permanent[Table[M[i, j, n], {i, n}, {j, n}]], {n, 17}]]

a(n) = matpermanent(matrix(n, n, i, j, min(i, j)*(n + 1) - i*j)); \\ Michel Marcus, Apr 30 2023

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023

A109622 Number of different isotemporal classes of diasters with n peripheral edges.

Original entry on oeis.org

1, 1, 4, 7, 15, 23, 38, 53, 77, 101, 136, 171, 219, 267, 330, 393, 473, 553, 652, 751, 871, 991, 1134, 1277, 1445, 1613, 1808, 2003, 2227, 2451, 2706, 2961, 3249, 3537, 3860, 4183, 4543, 4903, 5302, 5701, 6141, 6581, 7064, 7547, 8075, 8603

Offset: 0

Benjamin de Bivort (bivort(AT)fas.harvard.edu), Aug 02 2005

See A092481 for the definition of isotemporal classes.

			A diaster is defined to be any graph with a central edge with vertices of degree j and k and j+k peripheral edges connected to the central edge each terminating in a vertex of degree 1. a(5)=23 refers to diasters with 5 peripheral edges. These can be uniquely arranged with 0, 1 or 2 peripheral edges on a particular side, yielding 1, 10 and 12 isotemporal classes respectively each.

Benjamin de Bivort, Isotemporal classes of diasters, beachballs and daisies, preprint, 2005.

Cf. A092481, A005993.

a(n=2k) = 1 + (Sum_{i=1..(n/2)-1} n*i-i^2+n+1) + (1/2)*((n/2)^2+3*(n/2)+2). a(n=2k+1) = 1 + (Sum_{i=1..(n-1)/2} n*i-i^2+n+1). [Corrected by Sean A. Irvine after private communication with Benjamin de Bivort, Feb 13 2012]

a(n) = A005993(n) - n. - Enrique Pérez Herrero, Apr 22 2012

More terms from Sean A. Irvine, Feb 12 2012

A115263 Correlation triangle for floor((n+2)/2).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 3, 4, 6, 4, 3, 3, 5, 7, 7, 5, 3, 4, 6, 10, 10, 10, 6, 4, 4, 7, 11, 13, 13, 11, 7, 4, 5, 8, 14, 16, 19, 16, 14, 8, 5, 5, 9, 15, 19, 22, 22, 19, 15, 9, 5, 6, 10, 18, 22, 28, 28, 28, 22, 18, 10, 6

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A096338. Diagonal sums are A115264. T(2n,n) is A005993. T(2n,n)-T(2n,n+1) is floor((n+2)/2)(1+(-1)^n)/2 (aerated n+1).

			Triangle begins
1;
1,1;
2,2,2;
2,3,3,2;
3,4,6,4,3;
3,5,7,7,5,3;

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

A136451 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n matrix: 2 on the main antidiagonal, -1 on the adjacent sub-antidiagonals and 0 otherwise.

Original entry on oeis.org

1, 2, -1, -3, 2, 1, -4, 6, 2, -1, 5, -10, -9, 2, 1, 6, -19, -16, 12, 2, -1, -7, 28, 42, -22, -15, 2, 1, -8, 44, 68, -74, -28, 18, 2, -1, 9, -60, -138, 126, 115, -34, -21, 2, 1, 10, -85, -208, 316, 202, -165, -40, 24, 2, -1, -11, 110, 363, -506, -605, 296, 224, -46, -27, 2, 1

Offset: 0

Roger L. Bagula, Mar 19 2008

We start from tri-antidiagonal variants of the Cartan A-n group matrix. For n=1 this is {2}, for n=2 this is {{-1,2},{2,-1}}, for n=3 {{0,-1,2},{-1,2,-1},{2,-1,0}}, for n =4 {{0,0,-1,2},{0,-1,2,-1},{-1,2,-1,0},{2,-1,0,0}} etc. The n-th row of the triangle are the expansion coefficients of the characteristic polynomial.
For n=0, the empty product of the empty matrix is assigned the value T(0,0)=1.
Row sums (characteristic polynomials evaluated at x=0) are 1, 1, 0, 3, -11, -16, 29, 21, 0, 55, -199, -288, 521, 377, 0, 987, -3571, -5168, 9349, 6765, 0, ... (see A038150).

			1;
2, -1;
-3,2, 1;
-4, 6, 2, -1;
5, -10, -9, 2, 1;
6, -19, -16, 12, 2, -1;
-7,28, 42, -22, -15, 2, 1;
-8, 44, 68, -74, -28,18, 2, -1;
9, -60, -138, 126, 115, -34, -21, 2, 1;
10, -85, -208,316, 202, -165, -40, 24, 2, -1;
-11, 110, 363, -506, -605, 296, 224, -46, -27, 2, 1;

Cf. A124018 (variant), A005993 (column k=1), A061927 (bisection column k=2).

A136451x := proc(n,x)
    local A,r,c ;
    A := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
            A[r,c] :=0 ;
        if r+c = 1+n then
            A[r,c] := A[r,c]+2 ;
        elif abs(r+c-1-n)= 1 then
            A[r,c] :=  A[r,c]-1 ;
        end if;
    end do:
    end do:
    (-1)^n*LinearAlgebra[CharacteristicPolynomial](A,x) ;
end proc;
A136451 := proc(n,k)
    coeftayl( A136451x(n,x),x=0,k) ;
end proc:
seq(seq(A136451(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 04 2011

H[n_] := Table[Table[If[i + j - 1 == n, 2,If[i + j - 1 == n + 1, -1, If[i + j - 1 == n - 1, -1, 0]]], {i, 1, n}], {j, 1, n}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[H[n], x], x], {n, 1, 10}]]; Flatten[a']

A177878 Triangle in which row n is generated from (1,2,3,...,n) dot (n, n-1,...,1) with subtractive carryovers.

Original entry on oeis.org

1, 2, 0, 3, 1, 2, 4, 2, 4, 0, 5, 3, 6, 2, 3, 6, 4, 8, 4, 6, 0, 7, 5, 10, 6, 9, 3, 4, 8, 6, 12, 8, 12, 6, 8, 0, 9, 7, 14, 10, 15, 9, 12, 4, 5, 10, 8, 16, 12, 18, 12, 16, 8, 10, 0, 11, 9, 18, 14, 21, 15, 20, 12, 15, 5, 6, 12, 10, 20, 16, 24, 18, 24, 16, 20, 10, 12, 0

Offset: 0

Gary W. Adamson, Dec 13 2010

The subtractive carryover dot product of two vectors (a(1),a(2),...,a(n)) dot (b(1),b(2),...,b(n)) = (c(1),...,c(n)) is defined by c(1) = a(1)*b(1) and c(i) = a(i)*b(i)-c(i-1), i>1.
A177877 = analogous triangle with additive carryovers.
A160770 = the analogous triangle using the triangular series as the generating vector.

			Row 3 = (4, 2, 4, 0) = (1, 2, 3, 4) dot (4, 3, 2, 1) with subtractive carryovers = (4), then (2*3 - 4 = 2), (3*2 - 2 = 4), and (4*1 - 4 = 0).
First few rows of the triangle:
  1;
  2, 0;
  3, 1, 2;
  4, 2, 4, 0;
  5, 3, 6, 2, 3;
  6, 4, 8, 4, 6, 0;
  7, 5, 10, 6, 9, 3, 4;
  8, 6, 12, 8, 12, 6, 8, 0;
  9, 7, 14, 10, 15, 9, 12, 4, 5;
  10, 8, 16, 12, 18, 12, 16, 8, 10, 0;
  11, 9, 18, 14, 21, 15, 20, 12, 15, 5, 6;
  12, 10, 20, 16, 24, 18, 24, 16, 20, 10, 12, 0;
  ...

Cf. A005993 (row sums), A177877, A160770

By rows, dot product of (1,2,3,...) and (...3,2,1) with subtractive carryovers; such that current row product subtracts previous product.

cp's OEIS Frontend

A269064 At stage 1, start with a unit equilateral triangle. At each successive stage add 3*(n-1) new triangles around outside with vertex-to-vertex contacts. Sequence gives number of triangles at n-th stage.

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A062135 Odd-numbered columns of Losanitsch triangle A034851 formatted as triangle with an additional first column.

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A137508 Successive structures of alkaline earth metals (periodic table elements from 2nd column).

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A141783 Number of bracelets (turn over necklaces) with n beads: 1 blue, 12 green, and r = n - 13 red.

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Maple

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A212976 Number of (w,x,y) with all terms in {0,...,n} and odd range.

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Mathematica

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A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)*(n + 1) - i*j.

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Maple

Mathematica

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A109622 Number of different isotemporal classes of diasters with n peripheral edges.

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Extensions

A115263 Correlation triangle for floor((n+2)/2).

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A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)(n + 1) - ij.