A025192 -id:A025192 - cp's OEIS frontend

A256256 Total number of ON cells after n generations of cellular automaton on triangular grid, starting from a node, in which every 60-degree wedge looks like the Sierpiński's triangle.

0, 6, 18, 30, 54, 66, 90, 114, 162, 174, 198, 222, 270, 294, 342, 390, 486, 498, 522, 546, 594, 618, 666, 714, 810, 834, 882, 930, 1026, 1074, 1170, 1266, 1458, 1470, 1494, 1518, 1566, 1590, 1638, 1686, 1782, 1806, 1854, 1902, 1998, 2046, 2142, 2238, 2430, 2454, 2502, 2550, 2646, 2694, 2790, 2886, 3078, 3126, 3222, 3318, 3510, 3606, 3798, 3990, 4374

Offset: 0

Omar E. Pol, Mar 20 2015

nonn

Analog of A160720, but here we are working on the triangular lattice.
The first differences (A256257) gives the number of triangular cells turned ON at every generation.
Also 6 times the sum of all entries in rows 0 to n of Sierpiński's triangle A047999.
Also 6 times the total number of odd entries in first n rows of Pascal's triangle A007318, see formula section.
The structure contains three distinct kinds of polygons formed by triangular ON cells: the initial hexagon, rhombuses (each one formed by two ON cells) and unit triangles.
Note that if n is a power of 2 greater than 2, the structure looks like concentric hexagons with triangular holes, where some of them form concentric stars.

			On the infinite triangular grid we start with all triangular cells turned OFF, so a(0) = 0.
At stage 1, in the structure there are six triangular cells turned ON forming a regular hexagon, so a(1) = 6.
At stage 2, there are 12 new triangular ON cells forming six rhombuses around the initial hexagon, so a(2) = 6 + 12 = 18.
And so on.

Michael De Vlieger, Table of n, a(n) for n = 0..16386
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 30.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A001316, A006046, A007318, A025192, A047999, A151723, A160120, A160720, A161644, A256257.

Prepend[6*FoldList[Plus, 0, Total /@ CellularAutomaton[90, Join[Table[0, {#}], {1}, Table[0, {#}]], #]][[2 ;; -1]], 0] &[63] (* Michael De Vlieger, Nov 03 2022, after Bradley Klee at A006046 *)

a(n) = 6*sum(j=0, n, sum(k=0, j, binomial(j, k) % 2)); \\ Michel Marcus, Apr 01 2015

a(n) = 6*A006046(n).

A321981 Row n gives the chromatic symmetric function of the n-girder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

Original entry on oeis.org

1, 2, 0, 6, 0, 0, 16, 0, 2, 0, 0, 40, 12, 2, 0, 0, 0, 0, 96, 16, 44, 6, 0, 0, 0, 0, 0, 0, 0, 224, 136, 66, 52, 2, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 512, 384, 208, 96, 30, 178, 0, 18, 30, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1152, 1024, 584, 522, 138, 588, 102

Offset: 1

Gus Wiseman, Nov 23 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895).
The n-girder has n vertices and looks like:
  2-4-6-     -n
  |\|\|\ ... \|
  1-3-5-     n-1
Conjecture: All terms are nonnegative (verified up to n = 10). This is a special case of Stanley and Stembridge's poset-chain conjecture.

			Triangle begins:
    1
    2   0
    6   0   0
   16   0   2   0   0
   40  12   2   0   0   0   0
   96  16  44   6   0   0   0   0   0   0   0
  224 136  66  52   2   4   0   2   0   0   0   0   0   0   0
For example, row 6 gives: X_G6 = 96e(6) + 6e(33) + 16e(42) + 44e(51).

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Row sums are A025192.

Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321750, A321911, A321918, A321914, A321979, A321980, A321982.

A328778 Number of indecomposable closed walks of length 2n along the edges of a cube based at a vertex.

Original entry on oeis.org

1, 3, 12, 84, 588, 4116, 28812, 201684, 1411788, 9882516, 69177612, 484243284, 3389702988, 23727920916, 166095446412, 1162668124884, 8138676874188, 56970738119316, 398795166835212, 2791566167846484, 19540963174925388

Offset: 0

Geoffrey Critzer, Oct 27 2019

An indecomposable closed walk returns to its starting vertex exactly once (on the final step).

For n > 1, a(n) is the number of 4-colorings of the grid graph P_2 X P_(n-1). More generally, for q > 1, the number of q-colorings of the grid graph P_2 X P_n is given by q*(q - 1)*((q - 1)*(q - 2) + 1)^(n - 1). - Sela Fried, Sep 25 2023

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A054879, A025192.

nn = 40; list = Range[0, nn]! CoefficientList[Series[ Cosh[x]^3, {x, 0, nn}], x]; a = Sum[list[[i]] x^(i - 1), {i, 1, nn + 1}]; Select[CoefficientList[Series[ 2 - 1/a, {x, 0, nn}], x], # > 0 &]

Vec((1 - 4*x - 9*x^2) / (1 - 7*x) + O(x^25)) \\ Colin Barker, Oct 28 2019

G.f.: 2 - 1/f(x) where f(x) is the g.f. for A054879.
From Colin Barker, Oct 27 2019: (Start)
G.f.: (1 - 4*x - 9*x^2) / (1 - 7*x).
a(n) = 7*a(n-1) for n>2.
a(n) = 12*7^(n - 2) for n>1.
(End)
E.g.f.: (1/49)*(37 + 12*exp(7*x) + 63*x). - Stefano Spezia, Oct 27 2019

A080643 a(0)=0; for n>0, a(n) = 4^n - 2*3^(n-1).

Original entry on oeis.org

0, 2, 10, 46, 202, 862, 3610, 14926, 61162, 249022, 1009210, 4076206, 16422922, 66045982, 265246810, 1064175886, 4266269482, 17093775742, 68461196410, 274103065966, 1097187104842, 4391072942302, 17571265338010, 70305982058446, 281286690353002, 1125335047769662

Offset: 0

Max Alekseyev, Feb 26 2003

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-12).

Complement of A025192 in 4^n.

Join[{0},Table[4^n-2*3^(n-1),{n,30}]] (* or *) LinearRecurrence[{7,-12},{0,2,10},30] (* Harvey P. Dale, Dec 29 2023 *)

def A080643(n): return (1<<(n<<1))-(3**(n-1)<<1) if n else 0 # Chai Wah Wu, Apr 02 2025

G.f.: 2*x*(1-2*x)/((1-3*x)*(1-4*x)). - Colin Barker, Jan 20 2012

E.g.f.: (3*exp(4*x) - 2*exp(3*x) - 1)/3. - Stefano Spezia, Apr 03 2025

A081955 a(n) = 2^r*3^s where r = n(n+1)/2 and s = n(n-1)/2.

Original entry on oeis.org

1, 2, 24, 1728, 746496, 1934917632, 30091839012864, 2807929681968365568, 1572081206902992767287296, 5280985496827154199640037916672, 106440332834866049138191223105387495424, 12872079797383178927229037635891253693013557248

Offset: 0

Amarnath Murthy, Apr 02 2003

nonn

Sequence contains the product of a row in A081954.

Cf. A025192, A081954, A081956.

Do[Print[2^(n*(n+1)/2)*3^(n*(n-1)/2)], {n, 10}] (* Ryan Propper, Jun 15 2005 *)

{a(n) = 3^(n*(n-1)/2) * 2^(n*(n+1)/2)} /* Michael Somos, Dec 17 2009 */

a(n+1) = 2^(n+1)*3^n*a(n), a(1) = 2. - Ryan Propper, Jun 15 2005

A171795(n) = a(-n). a(n+1) * a(n-1) = 6 * a(n)^2. - Michael Somos, Dec 17 2009

More terms from Ryan Propper, Jun 15 2005

A099485 A Fibonacci convolution.

Original entry on oeis.org

1, 2, 5, 14, 37, 96, 251, 658, 1723, 4510, 11807, 30912, 80929, 211874, 554693, 1452206, 3801925, 9953568, 26058779, 68222770, 178609531, 467605822, 1224207935, 3205017984, 8390846017, 21967520066, 57511714181, 150567622478

Offset: 0

Paul Barry, Oct 18 2004

A Chebyshev transform of A025192 with g.f. (1-x)/(1-3*x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))*G(x/(1+x^2)).

Michael De Vlieger, Table of n, a(n) for n = 0..2392
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics. 2017, Vol. 41 Issue 6, pp. 849-853.
Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).

Cf. A000045, A099483, A099484, A001045, A128834.

LinearRecurrence[{3,-2,3,-1},{1,2,5,14},30] (* Harvey P. Dale, Jul 06 2017 *)

G.f.: (1-x+x^2)/((1+x^2)*(1-3*x+x^2)).
a(n) = 3*a(n-1)-2*a(n-2)+3*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^n*(2*3^(n-2*k)+0^(n-2*k))/3.
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*Fibonacci(2*(n-k)+2).
a(n) = (1/6) * (4*Fibonacci(2*n+2) + I^n + (-I)^n). - Ralf Stephan, Dec 04 2004
Also a transformation of the Jacobsthal numbers A001045(n+1) under the mapping G(x)-> (1/(1-x+x^2))*G(x/(1-x+x^2)). - Paul Barry, Dec 11 2004
G.f.: g(f(x))/x, where g is g.f. of A001045 and f is g.f. of A128834. - Oboifeng Dira, Jun 21 2020

A114283 Sequence array for binomial transform of Jacobsthal numbers A001045(n+1).

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 18, 6, 2, 1, 54, 18, 6, 2, 1, 162, 54, 18, 6, 2, 1, 486, 162, 54, 18, 6, 2, 1, 1458, 486, 162, 54, 18, 6, 2, 1, 4374, 1458, 486, 162, 54, 18, 6, 2, 1, 13122, 4374, 1458, 486, 162, 54, 18, 6, 2, 1, 39366, 13122, 4374, 1458, 486, 162, 54, 18, 6, 2, 1

Offset: 0

Paul Barry, Nov 20 2005

Sequence array for A025192. Row sums are 3^n, A000244. Diagonal sums are A015518(n+1). Inverse is A114284.

			Triangle begins
1;
2,1;
6,2,1;
18,6,2,1;
54,18,6,2,1;
162,54,18,6,2,1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

a114283 n k = a114283_tabl !! n !! k
a114283_row n = a114283_tabl !! n
a114283_tabl = iterate
   (\row -> (sum $ zipWith (+) row $ reverse row) : row) [1]
-- Reinhard Zumkeller, Nov 27 2012

Riordan array ((1-x)/(1-3x), x).

A175732 a(n) = gcd(phi(n), psi(n)).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 4, 6, 2, 2, 4, 2, 6, 8, 8, 2, 6, 2, 4, 4, 2, 2, 8, 10, 6, 18, 12, 2, 8, 2, 16, 4, 2, 24, 12, 2, 6, 8, 8, 2, 12, 2, 4, 24, 2, 2, 16, 14, 10, 8, 12, 2, 18, 8, 24, 4, 2, 2, 16, 2, 6, 12, 32, 12, 4, 2, 4, 4, 24, 2, 24, 2, 6, 40, 12, 12, 24, 2, 16, 54, 2, 2, 24, 4, 6, 8, 8, 2, 24, 8

Offset: 1

Enrique Pérez Herrero, Aug 24 2010

nonn

a(n)^2 divides J_2(n), where J_2 is A007434.

If p > 2 is a prime, a(n) = 2. - Enrique Pérez Herrero, Jan 02 2012

			a(56) = gcd(phi(56), psi(56)) = gcd(24, 96) = 24.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Cf. A000010, A001615, A002110, A007434, A025192, A078558.

JordanTotient[n_, k_:1] := DivisorSum[n, #^k * MoebiusMu[n/#] &] /; (n > 0) && IntegerQ[n]; DedekindPsi[n_] := JordanTotient[n, 2]/EulerPhi[n]; A175732[n_] := GCD[EulerPhi[n], DedekindPsi[n]]; Array[A175732, 100]
f1[p_, e_] := (p - 1)*p^(e - 1); f2[p_, e_] := (p + 1)*p^(e - 1); a[1] = 1; a[n_] := Module[{f = FactorInteger[n]}, GCD[Times @@ f1 @@@ f, Times @@ f2 @@@ f]]; Array[a, 40] (* Amiram Eldar, Feb 20 2023 *)

a(n) = {my(f = factor(n)); gcd(prod(i = 1, #f~, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)), prod(i = 1, #f~, (f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)));} \\ Amiram Eldar, Feb 20 2023

a(n) = gcd(A000010(n), A001615(n)).
a(n) >= (n*2^(omega(n)-1))/rad(n).
a(A002110(n)) = A078558(n). - Enrique Pérez Herrero, Dec 04 2012
For k>=1, a(2^k) = 2^(k-1) and a(p^k) = 2*p^(k-1) if p is an odd prime. - Amiram Eldar, Feb 20 2023
a(3^n) = A025192(n). - Enrique Pérez Herrero, Jun 05 2023

A191487 The row sums of the Sierpinski-Stern triangle A191372.

Original entry on oeis.org

0, 1, 3, 8, 9, 22, 24, 26, 27, 62, 66, 70, 72, 76, 78, 80, 81, 178, 186, 194, 198, 206, 210, 214, 216, 224, 228, 232, 234, 238, 240, 242, 243, 518, 534, 550, 558, 574, 582, 590, 594, 610, 618, 626, 630, 638, 642, 646

Offset: 0

Johannes W. Meijer, Jun 05 2011

nonn

The row sums a(n) of the Sierpinski-Stern triangle A191372 equal this sequence.
The differences diff1(n) = a(2*n+3) - a(2*n+1) and diff2(n) = (a(2*n+2) - a(2*n))/3, give rise to patterns that lead to Gould’s sequence A001316, see the examples.
The diff1(n) sequence as a triangle leads to Gould’s sequence in a peculiar way, see A191488. The leading terms of the diff1(n) rows are given by A001550(p+1), p>=1; for p=0 the leading term is 7. The rows sums of diff1(n) as a triangle equal A025192(p+2), p>=1; for p = 0 the row sum is 7. The row sums of diff1(n) as a triangle minus the first term equal 2*A053152(p+1).
The diff2(n) sequence as a triangle leads to Gould’s sequence A001316 in a simple way; just delete the first term and reverse the order of the rest of the terms; more terms require a higher row number. The leading terms of the diff2(n) rows are given by A085281(p), p>=0. The row sums of diff2(n) as a triangle equal A025192(p) and the row sums minus the first term equal A001047(p-1), p>=1; for p=0 the row sum minus the first term is 0.

			The first few rows of diff1(n) as a triangle, row lengths A000079(p) with p>=0, are:
[7]
[14, 4]
[36, 8, 6, 4]
[98, 16, 12, 8, 10, 8, 6, 4]
[276, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
[794, 64, 48, 32, 40, 32, 24, 16, 36, 32, 24, 16, 20, 16, 12, 8, 34, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
The first few rows of diff2(n) as a triangle, row lengths A011782(p) with p>=0, are:
[1]
[2]
[5, 1]
[13, 2, 2, 1]
[35, 4, 4, 2, 4, 2, 2, 1]
[97, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1]
[275, 16, 16, 8, 16, 8, 8, 4, 16, 8, 8, 4, 8, 4, 4, 2, 16, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1]

Cf. A191372, A191488, A001316, A001550, A025192, A053152, A085281, A001047, A007689.

Add the following lines to the Maple program of A191372.
A191487(0):=0: for d from 1 to 2^pmax do A191487(d):= 0: for Tx from 0 to 2^ceil(log(d)/ log(2))-1 do A191487(d):=A191487(d)+S2(Tx,d) od: od: seq(A191487(d),d=0..2^pmax);

a(2*n) = 3*a(n)
diff(n) = a(n+1) - a(n), diff1(n) = a(2*n+3) - a(2*n+1), diff2(n) = (a(2*n+2) - a(2*n))/3
a(2^n+1) - a(2^n) = A085281(n+1) = A007689(n) for n>=0
a(2^(n+1)+1) - a(2^(n+1)-1) = A001550(n+1) for n>=1.

A207608 Triangle of coefficients of polynomials u(n,x) jointly generated with A207609; see the Formula section.

Original entry on oeis.org

1, 2, 3, 3, 4, 11, 3, 5, 26, 20, 3, 6, 50, 74, 29, 3, 7, 85, 204, 149, 38, 3, 8, 133, 469, 547, 251, 47, 3, 9, 196, 952, 1618, 1160, 380, 56, 3, 10, 276, 1764, 4110, 4234, 2124, 536, 65, 3, 11, 375, 3048, 9318, 13036, 9262, 3520, 719, 74, 3, 12, 495, 4983

Offset: 1

Clark Kimberling, Feb 19 2012

As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

			First five rows:
  1;
  2;
  3,  3;
  4, 11,  3;
  5, 26, 20,  3;
Triangle (2, -1/2, 1/2, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, ...) begins:
  1;
  2,   0;
  3,   3,   0;
  4,  11,   3,   0;
  5,  26,  20,   3,   0;
  6,  50,  74,  29,   3,   0;
  7,  85, 204, 149,  38,   3,   0;
  ... - _Philippe Deléham_, Mar 03 2012

Cf. A207609.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1) v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207608 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207609 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else 2*x*u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

u(n,x) = u(n-1,x) + v(n-1,x),
v(n,x) = 2x*u(n-1,x) + (x+1)v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 03 2012: (Start)
As triangle T(n,k), 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-1,k) + T(n-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-y*x)/(1 - (2+y)*x - (y-1)*x^2).
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A025192(n), A001077(n), A180038(n) for x = 0, 1, 2, 3 respectively. (End)

cp's OEIS Frontend

A256256 Total number of ON cells after n generations of cellular automaton on triangular grid, starting from a node, in which every 60-degree wedge looks like the Sierpiński's triangle.

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A321981 Row n gives the chromatic symmetric function of the n-girder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

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A328778 Number of indecomposable closed walks of length 2n along the edges of a cube based at a vertex.

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A080643 a(0)=0; for n>0, a(n) = 4^n - 2*3^(n-1).

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A081955 a(n) = 2^r*3^s where r = n(n+1)/2 and s = n(n-1)/2.

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A099485 A Fibonacci convolution.

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A114283 Sequence array for binomial transform of Jacobsthal numbers A001045(n+1).

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A175732 a(n) = gcd(phi(n), psi(n)).

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