A016142 -id:A016142 - cp's OEIS frontend

A004488 Tersum n + n.

0, 2, 1, 6, 8, 7, 3, 5, 4, 18, 20, 19, 24, 26, 25, 21, 23, 22, 9, 11, 10, 15, 17, 16, 12, 14, 13, 54, 56, 55, 60, 62, 61, 57, 59, 58, 72, 74, 73, 78, 80, 79, 75, 77, 76, 63, 65, 64, 69, 71, 70, 66, 68, 67, 27, 29, 28, 33, 35, 34, 30, 32, 31, 45, 47, 46, 51

Offset: 0

N. J. A. Sloane

Could also be described as "Write n in base 3, then replace each digit with its base-3 negative" as with A048647 for base 4. - Henry Bottomley, Apr 19 2000
a(a(n)) = n, a self-inverse permutation of the nonnegative integers. - Reinhard Zumkeller, Dec 19 2003
First 3^n terms of the sequence form a permutation s(n) of 0..3^n-1, n>=1; the number of inversions of s(n) is A016142(n-1). - Gheorghe Coserea, Apr 23 2018

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 6561 terms from Alois P. Heinz)
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Column k=0 of A253586, A253587.
Column k=3 of A248813.
Row / column 2 of A325820.
Main diagonal of A004489.
Cf. A048647, A055115, A055116, A055120, A059249, A117966, A117967, A117968, A225901, A242399, A244042, A263273, A289813, A289814, A289815, A289816, A289831, A289838, A300222, A321464.

a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
            where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014

a:= proc(n) local t, r, i;
      t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 07 2011

a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b);
vector(67, i, a(i-1))  \\ Gheorghe Coserea, Apr 23 2018

from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017

Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g., 5 + 8 = "21" + "22" = "10" = 1.
a(n) = Sum(3-d(i)-3*0^d(i): n=Sum(d(i)*3^d(i): 0<=d(i)<3)). - Reinhard Zumkeller, Dec 19 2003
a(3*n) = 3*a(n), a(3*n+1) = 3*a(n)+2, a(3*n+2) = 3*a(n)+1. - Robert Israel, May 09 2014

A078122 Infinite lower triangular matrix, M, that satisfies [M^3](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 9, 1, 1, 93, 117, 27, 1, 1, 1632, 3033, 1080, 81, 1, 1, 68457, 177507, 86373, 9801, 243, 1, 1, 7112055, 24975171, 15562314, 2371761, 88452, 729, 1, 1, 1879090014, 8786827629, 6734916423, 1291958181, 64392813, 796797, 2187, 1

Offset: 0

Paul D. Hanna, Nov 18 2002

M also satisfies: [M^(3k)](i,j) = [M^k](i+1,j+1) for all i,j,k >=0; thus [M^(3^n)](i,j) = M(i+n,j+n) for all n >= 0.

Conjecture: the sum of the n-th row equals the number of partitions of 3^n into powers of 3 (A078125).

			The cube of the matrix is the same matrix excluding the first row and column:
  [1, 0, 0, 0]^3 = [ 1,  0, 0, 0]
  [1, 1, 0, 0]     [ 3,  1, 0, 0]
  [1, 3, 1, 0]     [12,  9, 1, 0]
  [1,12, 9, 1]     [93,117,27, 1]

Alois P. Heinz, Rows n = 0..60, flattened

Cf. A078121, A078123, A078124, A078125, A016142.

S:= proc(i, j) option remember;
       add(M(i, k)*M(k, j), k=0..i)
    end:
M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
       add(S(i-1, k)*M(k, j-1), k=0..i-1))
    end:
seq(seq(M(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 27 2015

m[i_, j_] := m[i, j]=If[j==0||i==j, 1, m3[i-1, j-1]]; m2[i_, j_] := m2[i, j]=Sum[m[i, k]m[k, j], {k, j, i}]; m3[i_, j_] := m3[i, j]=Sum[m[i, k]m2[k, j], {k, j, i}]; Flatten[Table[m[i, j], {i, 0, 8}, {j, 0, i}]]

M(1, j) = A078124(j), M(j+1, j)=3^j, M(j+2, j) = A016142(j).

M(n, k) = the coefficient of x^(3^n - 3^(n-k)) in the power series expansion of 1/Product_{j=0..n-k}(1-x^(3^j)) whenever 0<=k0 (conjecture).

A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 6, 13, 15, 1, 1, 6, 28, 40, 31, 1, 1, 12, 31, 120, 121, 63, 1, 1, 8, 91, 156, 496, 364, 127, 1, 1, 12, 57, 600, 781, 2016, 1093, 255, 1, 1, 12, 112, 400, 3751, 3906, 8128, 3280, 511, 1, 1, 18, 117, 960, 2801, 22932, 19531, 32640

Offset: 1

Álvar Ibeas, Oct 30 2015

All the enumerated lattices have full rank k, since the quotient group is finite.
For m>=1, T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_nm x (C_m)^(k-1); and also, (C_nm)^(k-1) x C_m.
Also, number of subgroups of (C_n)^k isomorphic to C_n (and also, to (C_n)^{k-1}), cf. [Butler, Lemma 1.4.1].
T(n,k) is the sum of the divisors d of n^(k-1) such that n^(k-1)/d is k-free. Namely, the coefficient in n^(-(k-1)*s) of the Dirichlet series zeta(s) * zeta(s-1) / zeta(ks).
Also, number of isomorphism classes of connected (C_n)-fold coverings of a connected graph with circuit rank k.
Columns are multiplicative functions.

			There are 7 = A160870(4,2) lattices of volume 4 in Z^2. Among them, only one (<(2,0), (0,2)>) gives the quotient group C_2 x C_2, whereas the rest give C_4. Hence, T(4,2) = 6 and T(1,2) = 1.
Array begins:
      k=1    k=2    k=3    k=4    k=5    k=6
n=1     1      1      1      1      1      1
n=2     1      3      7     15     31     63
n=3     1      4     13     40    121    364
n=4     1      6     28    120    496   2016
n=5     1      6     31    156    781   3906
n=6     1     12     91    600   3751  22932

Lynne M. Butler, Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc., Vol. 112, No. 539, 1994.

Álvar Ibeas, First 100 antidiagonals, flattened
Jin Ho Kwak, Jang-Ho Chun, and Jaeun Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285. [Page 277]
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. [Examples 2 and 3]

Rows (1-11): A000012, A000225, A003462, A006516, A003463, A160869, A023000, A016152, A016142(n-1), A046915(n-1), A016123(n-1).
Columns (1-17): A000012, A001615, A160889, A160891, A160893, A160895, A160897, A160908, A160953, A160957, A160960, A160972, A161010, A161025, A161139, A161167, A161213.
Main diagonal gives A344210.
Cf. A000203, A008683, A160870.

f[p_, e_, k_] := p^((k - 1)*(e - 1))*(p^k - 1)/(p - 1); T[n_, 1] = T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 08 2022 *)

T(n,k) = J_k(n) / J_1(n) = (Sum_{d|n} mu(n/d) * d^k) / phi(n).
T(n,k) = n^(k-1) * Product_{p|n, p prime} (p^k - 1) / ((p - 1) * p^(k-1)).
Dirichlet g.f. of k-th column: zeta(s-k+1) * Product_{p prime} (1 + p^(-s) + p^(1-s) + ... + p^(k-2-s)).
If n is squarefree, T(n,k) = A160870(n,k) = A000203(n^(k-1)).
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{i=1..n} T(i, k) ~ c * n^k, where c = (1/k) * Product_{p prime} (1 + (p^(k-1)-1)/((p-1)*p^k)).
Sum_{i>=1} 1/T(i, k) = zeta(k-1)*zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(2*k-1)), for k > 2. (End)
T(n,k) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^k). - Ridouane Oudra, Apr 03 2025

A016163 Expansion of 1/((1-5x)(1-9*x)).

Original entry on oeis.org

1, 14, 151, 1484, 13981, 128954, 1176211, 10664024, 96366841, 869254694, 7833057871, 70546348964, 635161281301, 5717672234834, 51465153629131, 463216900240304, 4169104690053361, 37522705149933374, 337708161046665991

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-45).

Cf. A016142, A016161.

[(9^(n+1) - 5^(n+1))/4: n in [0..30]]; // G. C. Greubel, Nov 09 2024

Table[(9^(n+1) - 5^(n+1))/4, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

def A016163(n): return (9^(n+1) - 5^(n+1))/4
[A016163(n) for n in range(31)] # G. C. Greubel, Nov 09 2024

a(n) = ((7+sqrt4)^n - (7-sqrt4)^n)/4. Offset 1. a(3)=151. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) = 14*a(n-1) - 45*a(n-2). - Philippe Deléham, Jan 01 2009
a(0)=1, a(n) = 9*a(n-1) + 5^n. - Vincenzo Librandi, Feb 09 2011
From G. C. Greubel, Nov 09 2024: (Start)
a(n) = (9^(n+1) - 5^(n+1))/4.
E.g.f.: (1/4)*(9*exp(9*x) - 5*exp(5*x)). (End)

A026121 a(n) = 3^n*(3^n-1)/2.

Original entry on oeis.org

0, 3, 36, 351, 3240, 29403, 265356, 2390391, 21520080, 193700403, 1743362676, 15690441231, 141214502520, 1270932117003, 11438393835996, 102945558872871, 926510072902560, 8338590785263203, 75047317454789316, 675425858255365311

Offset: 0

N. J. A. Sloane

Sum of all base-3 numbers with length at most n. - Alois P. Heinz, Feb 23 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (12,-27).

Cf. A000217, A000244.

[3^n*(3^n-1)/2: n in [0..30]]; // Vincenzo Librandi, May 01 2011

seq(binomial(3^n,2),n=0..19); # Zerinvary Lajos, Jan 07 2008

Binomial[3^Range[0,20],2] (* or *) LinearRecurrence[{12,-27},{0,3},30] (* Harvey P. Dale, Oct 07 2012 *)

a(n)=binomial(3^n,2) \\ Charles R Greathouse IV, Jun 11 2015

a(n) = C(3^n,2), n>=0. - Zerinvary Lajos, Jan 07 2008
a(0) = 0, a(1)=3, a(n)=12*a(n-1)-27*a(n-2). - Harvey P. Dale, Oct 07 2012
G.f.: 3*x / ( (9*x-1)*(3*x-1) ). a(n) = 3*A016142(n-1). - R. J. Mathar, Mar 24 2013

A059410 J_n(9) (see A059379).

Original entry on oeis.org

0, 6, 72, 702, 6480, 58806, 530712, 4780782, 43040160, 387400806, 3486725352, 31380882462, 282429005040, 2541864234006, 22876787671992, 205891117745742, 1853020145805120, 16677181570526406, 150094634909578632, 1350851716510730622, 12157665455570144400, 109418989121052006006

Offset: 0

N. J. A. Sloane, Jan 30 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (12,-27).

Cf. A000244, A016142, A024023, A059379, A059380, A059409.

[9^n-3^n: n in [0..20]]; // Vincenzo Librandi, Jun 03 2011

A059410:=n->9^n-3^n: seq(A059410(n), n=0..30); # Wesley Ivan Hurt, Aug 16 2016

Table[9^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[6 x /((1 - 3 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)

a(n) = 9^n - 3^n; a(n) = 12*a(n-1) - 27*a(n-2) for n > 1. - Vincenzo Librandi, Jun 03 2011
From Vincenzo Librandi, Oct 04 2014: (Start)
a(n) = 3^n*(3^n-1) = A000244(n)*A024023(n).
G.f.: 6*x/((1-3*x)*(1-9*x)). (End)
a(n) = 6*A016142(n). - R. J. Mathar, Nov 23 2018
E.g.f.: 2*exp(6*x)*sinh(3*x). - Elmo R. Oliveira, Mar 31 2025

A226804 Expansion of 1/((1-3x)(1-9x)(1-27x)(1-81x)).

Original entry on oeis.org

1, 120, 10890, 914760, 74987451, 6098153040, 494603769780, 40080553611120, 3247001410058901, 263019982119962760, 21304965990387308670, 1725711626112542281080, 139782894999601166334351, 11322421333652608793333280, 917116312670388314093059560

Offset: 0

Vincenzo Librandi, Jul 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (120,-3510,29160,-59049).

Cf. A000292, A016142, A028258.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-9*x)*(1-27*x)*(1-81*x))));

I:=[1,120,10890,914760]; [n le 4 select I[n] else 120*Self(n-1)-3510*Self(n-2)+29160*Self(n-3)-59049*Self(n-4): n in [1..25]];

CoefficientList[Series[1 / ((1 - 3 x) (1 - 9 x) (1 - 27 x) (1 - 81 x)), {x, 0, 20}], x]
LinearRecurrence[{120,-3510,29160,-59049},{1,120,10890,914760},20] (* Harvey P. Dale, Sep 21 2016 *)

G.f.: 1/((1-3*x)*(1-9*x)*(1-27*x)*(1-81*x)).
a(n) = 3^n*(3^(n+1)-1)*(3^(n+2)-1)*(3^(n+3)-1)/416.
a(0)=1, a(1)=120, a(2)=10890, a(3)=914760; for n>3, a(n) = 120*a(n-1) -3510*a(n-2) +29160*a(n-3) -59049*a(n-4).
a(n) -108*a(n-1) +2187*a(n-2) = A016142(n) with a(-1)=a(-2)=0. [Bruno Berselli, Jul 11 2013]

A219205 a(n) = 3^(n-1)*(3^n - 1), n >= 0.

Original entry on oeis.org

0, 2, 24, 234, 2160, 19602, 176904, 1593594, 14346720, 129133602, 1162241784, 10460294154, 94143001680, 847288078002, 7625595890664, 68630372581914, 617673381935040, 5559060523508802, 50031544969859544, 450283905503576874

Offset: 0

Jon Perry, Nov 14 2012

Last digit has period 4: 0, 2, 4, 4.

Index entries for linear recurrences with constant coefficients, signature (12,-27).

Cf. A006516, A016142.

for (j=0;j<20;j++) document.write((Math.pow(3,j)-1)*Math.pow(3,j-1)+", ");

LinearRecurrence[{12, -27}, {0, 2}, 20] (* Hugo Pfoertner, Feb 14 2024 *)

A219205(n):=3^(n-1)*(3^n - 1)$ makelist(A219205(n),n,0,30); /* Martin Ettl, Nov 15 2012 */

a(n) = 2 * A016142(n-1).

a(n) = 12*a(n-1)-27*a(n-2). G.f.: 2*x/((3*x-1)*(9*x-1)). [Colin Barker, Nov 23 2012]

a(19) corrected by Colin Barker, Nov 23 2012

A017897 Expansion of 1/((1-3x)(1-5x)(1-9*x)).

Original entry on oeis.org

1, 17, 202, 2090, 20251, 189707, 1745332, 15900020, 144066901, 1301455397, 11737424062, 105758621150, 952437144751, 8574983669087, 77190104636392, 694787214149480, 6253466332501801, 56283104147438777, 506557473488982322, 4559064943373269010

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Christian Brouder, William J. Keith, and Ângela Mestre, Closed forms for a multigraph enumeration, arXiv preprint arXiv:1301.0874 [math.CO], 2013-2015.
Index entries for linear recurrences with constant coefficients, signature (17,-87,135).

Cf. A005059, A016142, A016163.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-5*x)*(1-9*x)))); // Vincenzo Librandi, Jul 01 2013

I:=[1, 17, 202]; [n le 3 select I[n] else 17*Self(n-1)-87*Self(n-2)+135*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 01 2013

a:= n -> (Matrix(3, (i,j)-> if (i=j-1) then 1 elif j=1 then [17, -87, 135][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..25); # Alois P. Heinz, Aug 04 2008

CoefficientList[Series[1 / ((1 - 3 x) (1 - 5 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 01 2013 *)
LinearRecurrence[{17,-87,135},{1,17,202},30] (* Harvey P. Dale, Sep 26 2014 *)
a[n_]:=(9^(n+2) - 3*5^(n+2) + 2*3^(n+2))/24; Array[a, 30, 0] (* Stefano Spezia, Oct 04 2018 *)

a(n) = (9^(n+2) - 3*5^(n+2) + 2*3^(n+2))/24; \\ Joerg Arndt, Aug 13 2013

def A017897(n): return (9^(n+2) -3*5^(n+2) +2*3^(n+2))//24
[A017897(n) for n in range(41)] # G. C. Greubel, Nov 09 2024

a(n) = term (1,1) in the 3 X 3 matrix [17,1,0; -87,0,1; 135,0,0]^n. - Alois P. Heinz, Aug 04 2008
From Vincenzo Librandi, Jul 01 2013: (Start)
a(n) = 17*a(n-1) - 87*a(n-2) + 135*a(n-3); a(0)=1, a(1)=17, a(2)=202.
a(n) = 14*a(n-1) - 45*a(n-2) + 3^n. (End)
a(n) = (9^(n+2) - 3*5^(n+2) + 2*3^(n+2))/24. - Yahia Kahloune, Aug 13 2013
E.g.f.: exp(3*x)*(6 - 25*exp(2*x) + 27*exp(6*x))/8. - Stefano Spezia, Nov 09 2024

A227524 Expansion of 1/((1-3x)(1-9x)(1-27x)).

Original entry on oeis.org

1, 39, 1170, 32670, 891891, 24169509, 653373540, 17648258940, 476567558181, 12867905191779, 347438670325110, 9380891170278810, 253284485241566871, 6838684914320250849, 184644527001833063880, 4985402537886183692280, 134605871302457221445961

Offset: 0

Vincenzo Librandi, Jul 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (39,-351,729).

Cf. A016142, A226804.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-9*x)*(1-27*x))));

I:=[1, 39, 1170]; [n le 3 select I[n] else 39*Self(n-1)-351*Self(n-2)+729*Self(n-3): n in [1..25]];

CoefficientList[Series[1 / ((1 - 3 x) (1 - 9 x) (1 - 27 x)), {x, 0, 30}], x]
LinearRecurrence[{39,-351,729},{1,39,1170},20] (* Harvey P. Dale, Jul 04 2022 *)

G.f.: 1/((1-3*x)*(1-9*x)*(1-27*x)).
a(n) = 3^n*(3^(n+1)-1)*(3^(n+2)-1)/16.
a(0)=1, a(1)=39, a(2)=1170; for n>2, a(n) = 39*a(n-1)-351*a(n-2)+729*a(n-3).
a(n)-27*a(n-1) = A016142(n), with a(-1)=0; a(n) = A226804(n)-81*A226804(n-1), with A226804(-1)=0. [Bruno Berselli, Jul 17 2013]

cp's OEIS Frontend

A004488 Tersum n + n.

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A078122 Infinite lower triangular matrix, M, that satisfies [M^3](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.

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A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

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A016163 Expansion of 1/((1-5*x)*(1-9*x)).

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A026121 a(n) = 3^n*(3^n-1)/2.

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A059410 J_n(9) (see A059379).

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A226804 Expansion of 1/((1-3x)(1-9x)(1-27x)(1-81x)).

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A016163 Expansion of 1/((1-5x)(1-9*x)).

A017897 Expansion of 1/((1-3x)(1-5x)(1-9*x)).