A154272 -id:A154272 - cp's OEIS frontend

A154271 Dirichlet inverse of A154272; Fully multiplicative with a(3^e) = (-1)^e, a(p^e) = 0 for primes p <> 3.

1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Jan 06 2009

Abs(A154272) is a Fredholm-Rueppel-like sequence.

Sequence equals +1 if n is an even power of 3 (3^0, 3^2, 3^4,...), equals -1 if n is an odd power of 3 (3^1, 3^3, 3^5, 3^7,...) and zero elsewhere. - Comment edited by R. J. Mathar, Jun 24 2013

Antti Karttunen, Table of n, a(n) for n = 1..59049 (terms 1..220 from Mats Granvik)

Cf. A154272, A154269, A014578 (Möbius inverse), A154282, A225569.
Cf. A225569 (gives the absolute values when interpreted as the characteristic function of powers of 3, i.e., with starting offset 1 instead of 0).
Cf. A359377, A008836.

nn = 95;a = PadRight[{1, 0, 1}, nn, 0];Inverse[Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]][[All, 1]] (* Mats Granvik, Jul 24 2017 *)

A154271(n) = { my(k=valuation(n,3)); if((3^k)==n,(-1)^k,0); }; \\ Antti Karttunen, Jul 24 2017

(define (A154271 n) (cond ((= 1 n) 1) ((zero? (modulo n 3)) (* -1 (A154271 (/ n 3)))) (else 0))) ;; Antti Karttunen, Jul 24 2017

Fully multiplicative with a(3) = -1, a(p) = 0 for primes p <> 3. - Antti Karttunen, Jul 24 2017
From Amiram Eldar, Nov 03 2023: (Start)
abs(a(n)) = A225569(n-1).
Dirichlet g.f.: 1/(1+3^(-s)). (End)
a(n) = Sum_{d|n} A359377(d)*A008836(n/d). - Ridouane Oudra, Jul 15 2025

Alternative description added to the name by Antti Karttunen, Jul 24 2017

A320530 T(n,k) = k^n + k^(n - 2)n(n - 1)(k(k - 1) + 1)/2 for 0 < k <= n and T(n,0) = A154272(n+1), square array read by antidiagonals upwards.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 7, 3, 1, 0, 7, 26, 16, 4, 1, 0, 11, 88, 90, 29, 5, 1, 0, 16, 272, 459, 220, 46, 6, 1, 0, 22, 784, 2133, 1504, 440, 67, 7, 1, 0, 29, 2144, 9234, 9344, 3775, 774, 92, 8, 1, 0, 37, 5632, 37908, 54016, 29375, 7992, 1246, 121, 9

Offset: 0

Franck Maminirina Ramaharo, Oct 14 2018

Construct a length n ternary word over the alphabet {a, b, c} as follows: letters from the set {a, b} are only used in pairs of at most one, and consist of either (a,b), (b,a) or (b,b). Next, replace each occurrence of a, b and c with a length k binary word such that 'a' has exactly two letters 1, 'b' contains no 0's and 'c' has exactly one letter 0 (empty words otherwise, respectively). Then T(n,k) gives the number of length n*k binary words resulting from this substitution. First column follows from the next definition.
In Kauffman's language, T(n,k) is the number of ways of splitting the crossings of the Pretzel knot shadow P(k, k, ..., k) having n tangles, of k half-twists respectively, such that the final diagram consists of two Jordan curves. This result can be achieved by assigning each tangle of the Pretzel knot a length k binary words in a way that letters 1 and 0 indicate the adequate choice for splitting the crossings.
Columns are linear recurrence sequences with signature (3*k, -3*k^2, k^3).

			Square array begins:
    1,  1,     1,     1,      1,       1,       1, ...
    0,  1,     2,     3,      4,       5,       6, ...
    1,  2,     7,    16,     29,      46,      67, ...
    0,  4,    26,    90,    220,     440,     774, ...
    0,  7,    88,   459,   1504,    3775,    7992, ...
    0, 11,   272,  2133,   9344,   29375,   74736, ...
    0, 16,   784,  9234,  54016,  212500,  649296, ...
    0, 22,  2144, 37908, 295936, 1456250, 5342112, ...
    ...
T(3,2) = 2^3 + 2^(3 - 2)*3*(3 - 1)*(2*(2 - 1) + 1)/2 = 26. The corresponding ternary words are abc, acb, cab, bac, bca, cba, bbc, bcb, cbb, ccc.  Next, let a = {00}, b = {11} and c = {01, 10}. The resulting binary words are
    abc: 001101, 001110;
    acb: 000111, 001011;
    cab: 010011, 100011;
    bac: 110001, 110010;
    bca: 110100, 111000;
    cba: 011100, 101100;
    bbc: 111101, 111110;
    bcb: 110111, 111011;
    cbb: 011111, 101111;
    ccc: 010101, 101010, 010110, 011001, 100101, 101001, 100110, 011010.

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

Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Alexander Stoimenow, Everywhere Equivalent 2-Component Links, Symmetry Vol. 7 (2015), 365-375.
Wikipedia, Pretzel link

Column 1 is column 2 of A300453.
Column 2 is column 2 of A300184.
Cf. A300401, A303273, A320530.

T[n_, k_] = If[k > 0, k^n + k^(n - 2)*n*(n - 1)*(k*(k - 1) + 1)/2, If[k == 0 && (n == 0 || n == 1), 1, 0]];
Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 10}]//Flatten

t(n, k) := k^n + k^(n - 2)*binomial(n, 2)*(2*binomial(k, 2) + 1)$
u(n) := if n = 0 or n = 1 then 1 else 0$
T(n, k) := if k = 0 then u(n) else t(n,k)$
tabl(nn) := for n:0 thru 10 do print(makelist(T(n, k), k, 0, nn))$

T(n,k) = k^n + k^(n - 2)*binomial(n, 2)*(2*binomial(k, 2) + 1), k > 0.
T(n,k) = (3*k)*T(n-1,k) - (3*k^2)*T(n-2,k) + (k^3)*T(n-3,k), n > 3.
T(n,1) = A152947(n+1).
T(n,2) = A300451(n).
T(2,n) = A130883(n).
G.f. for columns: (1 - 2*k*x + (1 - k + 2*k^2)*x^2 )/(1 - k*x)^3.
E.g.f. for columns: ((1 - k + k^2)*x^2 + 2)*exp(k*x)/2.

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 24 2018

Let (A,B,d) denote the three-variable bracket polynomial for the two-bridge knot with Conway's notation C(n,k). Then T(n,k) is the leading coefficient of the reduced polynomial x*(1,1,x). In Kauffman's language, T(n,k) is the number of states of the two-bridge knot C(n,k) corresponding to the maximum number of Jordan curves.

			Square array begins:
  1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 1, 1, 1, ...
  1, 1, 3, 2, 2, 2, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  ...

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 this sequence 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

Cf. A300453, A300454, A316989, A321126, A321127.

b[n_] = If[n == 0 || n == 2, 1, 0];
T[n_, k_] = b[n + k] - (2*b[n]*b[k] + 1)*b[n*k] + b[n] + b[k] + 1;
Table[T[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten

b(n) := if n = 0 or n = 2 then 1 else 0$ /* A154272(n+1) */
T(n, k) := b(n + k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1$
create_list(T(k, n - k), n, 0, 12, k, 0, n);

T(n,0) = T(0,n) = 1, and T(n,k) = b(n+k) - b(n)*b(k) - b(n*k) + c(n)*c(k) for n >= 1, k >= 1, where b(n) = A154272(n+1) and c(n) = A294619(n).
T(n,1) = A300453(n+1,A321126(n,1)).
T(n,2) = A300454(n,A321126(n,2)).
T(n,n) = A321127(n,A004280(n+1)).
G.f.: (1 + (x - x^2)*y - (x - 3*x^2 + x^3)*y^2 - x^2*y^3)/((1 - x)*(1 - y)).
E.g.f.: ((x^2 + 2*exp(x))*exp(y) - x^2 + (2*x - x^2)*y - (1 + x - exp(x))*y^2)/2.

A291728 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 2, 4, 9, 17, 35, 70, 142, 285, 576, 1160, 2340, 4716, 9510, 19171, 38653, 77926, 157110, 316747, 638599, 1287479, 2595698, 5233196, 10550681, 21271280, 42885152, 86460984, 174314476, 351436368, 708532813, 1428476905, 2879960190, 5806303628, 11706120825

Offset: 0

Clark Kimberling, Sep 08 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,0,1,0,0,0,0,...) = A154272, in some cases t(1,0,1,0,0,0,0,...) is a shifted (or differently indexed) version of the indicated sequence:
***
p(S)             t(1,0,1,0,0,0,0,...)
1 - S                A000930 (Narayana's cows sequence)
1 - S^2              A002478 (except for 0's)
1 - S^3              A291723
1 - S^5              A291724
(1 - S)^2            A291725
(1 - S)^3            A291726
(1 - S)^4            A291727
1 - S - S^2          A291728
1 - 2S - S^2         A291729
1 - 2S - 2S^2        A291730
(1 - 2S)^2           A291732
(1 - S)(1 - 2S)      A291734
1 - S - S^3          A291735
1 - S^2 - S^3        A291736
1 - S - S^2 - S^3    A291737
1 - S - S^4          A291738
1 - S^3 - S^6        A291739
(1 - S)(1 - S^2)     A291740
(1 - S)(1 + S^2)     A291741

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1,2,0,1).

Cf. A154272, A290890, A291000, A291382, A291219, A291382.

z = 60; s = x + x^3; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291728 *)

G.f.: (-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x + x^2 + x^3 + 2 x^4 + x^6).

a(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-6) for n >= 7.

A304182 Number of primitive inequivalent mirror-symmetric sublattices of rectangular lattice of index n.

Original entry on oeis.org

1, 3, 2, 4, 2, 6, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 2, 6, 2, 8, 4, 6, 2, 8, 2, 6, 2, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 8, 4, 6, 2, 8, 2, 6, 4, 8, 2, 6, 4, 8, 4, 6, 2, 16, 2, 6, 4, 4, 4, 12, 2, 8, 4, 12, 2, 8, 2, 6, 4, 8, 4, 12, 2, 8, 2, 6, 2, 16, 4

Offset: 1

Andrey Zabolotskiy, May 07 2018

			There are 6 = A001615(4) lattices in Z^2 whose quotient group is C_4. The reflection through an axis relates <(4,0), (1,1)> and <(4,0), (3,1)>. The remaining 4 = a(4) lattices are fixed.

Álvar Ibeas, Table of n, a(n) for n = 1..10000
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 4. Contains errors for n = 24 and 28.]

Cf. A069735 (not only primitive sublattices), A304183 (primitive oblique sublattices), A069734 (all sublattices).
Cf. other columns of tables 4 and 5 from [Rutherford, 2009]: A001615, A060594, A157223, A000089, A157224, A000086, A157227, A019590, A157228, A157226, A157230, A157231, A154272, A157235.
Cf. A034444, A001221, A001620, A306016.

f[p_, e_] := If[p == 2, If[e == 1, 3, 4], 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

From Álvar Ibeas, Mar 18 2021: (Start)
For n odd, a(n) = A034444(n) = 2^(A001221(n)).
For n even, a(n) = A034444(n) + A034444(n/2). If 4|n, a(n) = 2^(A001221(n) + 1); otherwise, a(n) = 3 * 2^(A001221(n) - 1).
Multiplicative with a(2) = 3, a(2^e) = 4 (for e>1), and a(p^e) = 2 (for p>2).
Dirichlet g.f.: (1+2^(-s)) * zeta(s)^2 / zeta(2s).
(End)
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - log(2)/3 - 2*zeta'(2)/zeta(2) - 1)*9*n/Pi^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 31 2022

A132460 Irregular triangle read by rows of the initial floor(n/2) + 1 coefficients of 1/C(x)^n, where C(x) is the g.f. of the Catalan sequence (A000108).

Original entry on oeis.org

1, 1, 1, -2, 1, -3, 1, -4, 2, 1, -5, 5, 1, -6, 9, -2, 1, -7, 14, -7, 1, -8, 20, -16, 2, 1, -9, 27, -30, 9, 1, -10, 35, -50, 25, -2, 1, -11, 44, -77, 55, -11, 1, -12, 54, -112, 105, -36, 2, 1, -13, 65, -156, 182, -91, 13, 1, -14, 77, -210, 294, -196, 49, -2

Offset: 0

Paul D. Hanna, Aug 21 2007

The length of row n is A008619(n).
Essentially equals a signed version of A034807, the triangle of Lucas polynomials. The initial n coefficients of 1/C(x)^n consist of row n followed by floor((n-1)/2) zeros for n > 0.
For the following formula for 1/C(x)^n see the W. Lang reference, proposition 1 on p. 411:
1/C(x)^n = (sqrt(x))^n*(S(n,1/sqrt(x)) - sqrt(x)*S(n-1,1/sqrt(x))*C(x)), n >= 0, with the Chebyshev polynomials S(n,x) with coefficients given in A049310. See also the coefficient array A115139 for P(n,x) = (sqrt(x)^(n-1))*S(n-1, 1/sqrt(x)). - Wolfdieter Lang, Sep 14 2013
This triangular array is composed of interleaved rows of reversed, A127677 (cf. A156308, A217476, A263916) and reversed, signed A111125. - Tom Copeland, Nov 07 2015
It seems that the n-th row lists the coefficients of the HOMFLYPT (HOMFLY) polynomial reduced to one variable for link family n, see Jablan's slide 38. - Andrey Zabolotskiy, Jan 16 2018
For n >= 1 row n gives the coefficients of the Girard-Waring formula for the sum of x1^n + x2^n in terms of the elementary symmetric functions e_1(x1,x2) = x1 + x2 and e_2(x1,x2) = x1*x2. This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions with parts larger than 2 eliminated. E.g., n = 4: x1^4 + x2^4 = 1*e1^4 - 4*e1^3*e2 + 2*e1*e2^2. See also A115131, row n = 4, with the mentioned partitions omitted. - Wolfdieter Lang, May 03 2019
Row n lists the coefficients of the n-th Faber polynomial for the replicable function given in A154272 with offset -1. - Ben Toomey, May 12 2020

			The irregular triangle T(n,k) begins:
n\k 0    1    2    3    4    5    6   7 ...
-------------------------------------------------
0:  1
1:  1
2:  1   -2
3:  1   -3
4:  1   -4    2
5:  1   -5    5
6:  1   -6    9   -2
7:  1   -7   14   -7
8:  1   -8   20  -16    2
9:  1   -9   27  -30    9
10: 1  -10   35  -50   25   -2
11: 1  -11   44  -77   55  -11
12: 1  -12   54 -112  105  -36    2
13: 1  -13   65 -156  182  -91   13
14: 1  -14   77 -210  294 -196   49  -2
... (reformatted - _Wolfdieter Lang_, May 03 2019)

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened)
Tom Copeland, Addendum to Elliptic Lie Triad
G. Dattoli, E. Di Palma and E. Sabia, Cardan Polynomials, Chebyshev Exponents, Ultra-Radicals and Generalized Imaginary Units, Advances in Applied Clifford Algebras, 2014.
Pentti Haukkanen, Jorma Merikoski and Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
S. Jablan, Knots, computers, conjectures
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq. (23) with n -> -n and eq. (20).
Jorma K. Merikoski, Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials, Notes on Num. Theor. and Disc. Math. (2021) Vol. 27, No. 2, 79-87.

Cf. A000108, A008619, A034807 (Lucas polynomials), A111125, A115131 (Waring numbers), A127677, A132461 (row squared sums), A156308, A217476, A263916.

T[0, 0] = 1; T[n_, k_] := (-1)^k (Binomial[n-k, k] + Binomial[n-k-1, k-1]);
Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Jun 04 2018 *)

{T(n,k)=if(k>n\2,0,(-1)^k*(binomial(n-k, k)+binomial(n-k-1, k-1)))}

T(n,k) = (-1)^k*( C(n-k,k) + C(n-k-1,k-1) ) for n >= 0, 0 <= k <= floor(n/2).

T(0,0) = 1; T(n,k) = (-1)^k*n*binomial(n-k,k)/(n-k), k = 0..floor(n/2). - Wolfdieter Lang, May 03 2019

A003050 Number of primitive sublattices of index n in hexagonal lattice: triples x,y,z from Z/nZ with x+y+z = 0, discarding any triple that can be obtained from another by multiplying by a unit and permuting.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 4, 3, 4, 3, 6, 4, 5, 6, 6, 4, 7, 5, 8, 8, 7, 5, 12, 6, 8, 7, 10, 6, 14, 7, 10, 10, 10, 10, 14, 8, 11, 12, 16, 8, 18, 9, 14, 14, 13, 9, 20, 11, 16, 14, 16, 10, 19, 14, 20, 16, 16, 11, 28, 12, 17, 18, 18, 16, 26, 13, 20, 18, 26, 13, 28

Offset: 1

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Also the number of triangles with vertices on points of the hexagonal lattice that have area equal to n/2. - Amihay Hanany, Oct 15 2009 [Here the area is measured in the units of the lattice unit cell area; since the number of the triangles of different shapes with the same half-integral area is infinite, the triangles are probably counted up to the equivalence relation defined in the Davey, Hanany and Rak-Kyeong Seong paper. Also, this comment probably belongs to A003051, not here. - Andrey Zabolotskiy, Mar 10 2018 and Jul 04 2019]
Also number of 2n-vertex connected cubic vertex-transitive graphs which are Cayley graphs for a dihedral group [Potočnik et al.]. - N. J. A. Sloane, Apr 19 2014

			For n = 6 the 3 primitive triples are 510, 411, 321.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217. [Annotated and corrected scanned copy]
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
J. Davey, A. Hanany and Rak-Kyeong Seong, Counting Orbifolds, arXiv:1002.3609 [hep-th], 2010.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Primož Potočnik, Pablo Spiga and Gabriel Verret, A census of small connected cubic vertex-transitive graphs (See the sub-page Table.html, column headed "Dihedrants"). - _N. J. A. Sloane_, Apr 19 2014
Index entries for sequences related to A2 = hexagonal = triangular lattice
Index entries for sequences related to sublattices

Cf. A003051 (not only primitive sublattices), A001615, A006984, A007997, A048259, A054345, A154272, A157235.

Join[{1}, Table[p=Transpose[FactorInteger[n]][[1]]; If[Mod[n,2]==0 || Mod[n,9]==0, c1=0, c1=Product[(1+JacobiSymbol[p[[i]],3]), {i,Length[p]}]]; c2={2,1,0,1,1,1,0,1}[[1+Mod[n,8]]]; n*Product[(1+1/p[[i]]), {i, Length[p]}]/6+c1/3+2^(Length[p]-2+c2), {n,2,100}]] (* T. D. Noe, Oct 18 2009 *)

Let n = Product_{i=1..w} p_i^e_i. Then a(n) = (1/6)*n*Product_{i=1..w} (1 + 1/p_i) + (C_1)/3 + 2^(w-2+C_2),
where C_1 = 0 if 2|n or 9|n, = Product_{i=1..w, p_i > 3} (1 + Legendre(p_i, 3)) otherwise,
and C_2 = 2 if n == 0 (mod 8), 1 if n == 1, 3, 4, 5, 7 (mod 8), 0 if n == 2, 6 (mod 8).

A100063 A Chebyshev transform of Jacobsthal numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1

Offset: 0

Paul Barry, Nov 02 2004

A Chebyshev transform of A001045(n+1): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))*A(x/(1+x^2)).
Also decimal expansion of 1111/9990. - Elmo R. Oliveira, Feb 18 2024
Also partial quotients of the continued fraction expansion of sqrt(5/2). - Hugo Pfoertner, Jan 10 2025

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + x^5 + 2*x^6 + x^7 + x^8 + 2*x^9 + ... - _Michael Somos_, Feb 20 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000012, A001045, A100051, A061347, A057079, A154272.

Cf. A000032, A000045.

PadRight[{1},120,{2,1,1}] (* or *) LinearRecurrence[{0,0,1},{1,1,1,2},120] (* Harvey P. Dale, Jul 08 2015 *)
a[ n_] := If[n<1, Boole[n==0], {2, 1, 1}[[1+Mod[n, 3]]]]; (* Michael Somos, Feb 20 2024 *)

my(x='x+O('x^50)); Vec((1+x)(1+x^2)/(1-x^3)) \\ G. C. Greubel, May 03 2017

{a(n) = if(n<1, n==0, [2, 1, 1][n%3+1])}; /* Michael Somos, Feb 20 2024 */

contfrac(sqrt(5/2),,80) \\ Hugo Pfoertner, Jan 10 2025

G.f.: (1+x)(1+x^2)/(1-x^3).
a(n) = n*Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*A001045(n-2k+1)/(n-k).
Multiplicative with a(3^e) = 2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 11 2005
Dirichlet g.f.: zeta(s)*(1+1/3^s). Dirichlet convolution of A154272 and A000012. - R. J. Mathar, Feb 07 2011
a(n) = 2 if n == 0 (mod 3) and n > 0, and a(n) = 1 otherwise. - Amiram Eldar, Nov 01 2022
a(n) = gcd(Fibonacci(n), Lucas(n)) = gcd(A000045(n), A000032(n)), for n >= 1. - Amiram Eldar, Jul 10 2023

A145377 a(n) = A002324(n) mod 2.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Mar 12 2009

Antti Karttunen, Table of n, a(n) for n = 1..65537
J. S. Rutherford, Generating functions for the cage isomers of the C_{20n} icosahedral fullerenes, J. Mathematical Chem., 14 (1993), 385-390. See Eq. (3).
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. [See Sect. (7), p. 505.]
Index entries for characteristic functions

Essentially same as A195198.

Cf. A002324, A154272, A003050.

Array[Boole@ OddQ@ If[# < 1, 0, DivisorSum[#, KroneckerSymbol[-3, #] &]] &, 105] (* Michael De Vlieger, Nov 05 2017, after Michael Somos at A002324 *)

A002324(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)));
A145377(n) = (A002324(n)%2); \\ Antti Karttunen, Nov 06 2017

a(n) = A195198(n) for n >= 1.

a(n) = Sum_{ m: m^2|n } A154272(n/m^2). - Andrey Zabolotskiy, May 07 2018

More terms from Antti Karttunen, Nov 05 2017

A154282 Dirichlet inverse of A154281.

Original entry on oeis.org

1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Jan 06 2009

Sequence is positive as often as negative.

Multiplicative because A154281 is. - Andrew Howroyd, Aug 05 2018

Mats Granvik, Table of n, a(n) for n = 1..1500

Cf. A154269, A154271, A154272, A154281.

nn = 95; a = PadRight[{1, 0, 0, 1}, nn, 0]; Inverse[Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]][[All, 1]] (* Mats Granvik, Jul 23 2017 *)

a(n) = {my(e=valuation(n,2)); if(e%2 == 0 && n == 1<Andrew Howroyd, Aug 05 2018

Multiplicative with a(2^e) = (-1)^(e/2) if e is even and 0 is e is odd, and a(p^e) = 0 if p is an odd prime. - Amiram Eldar, Aug 27 2023

cp's OEIS Frontend

A154271 Dirichlet inverse of A154272; Fully multiplicative with a(3^e) = (-1)^e, a(p^e) = 0 for primes p <> 3.

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A320530 T(n,k) = k^n + k^(n - 2)*n*(n - 1)*(k*(k - 1) + 1)/2 for 0 < k <= n and T(n,0) = A154272(n+1), square array read by antidiagonals upwards.

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A321125 T(n,k) = b(n+k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

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A291728 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2.

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A304182 Number of primitive inequivalent mirror-symmetric sublattices of rectangular lattice of index n.

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A132460 Irregular triangle read by rows of the initial floor(n/2) + 1 coefficients of 1/C(x)^n, where C(x) is the g.f. of the Catalan sequence (A000108).

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A003050 Number of primitive sublattices of index n in hexagonal lattice: triples x,y,z from Z/nZ with x+y+z = 0, discarding any triple that can be obtained from another by multiplying by a unit and permuting.

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A320530 T(n,k) = k^n + k^(n - 2)n(n - 1)(k(k - 1) + 1)/2 for 0 < k <= n and T(n,0) = A154272(n+1), square array read by antidiagonals upwards.

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).