A140429 -id:A140429 - cp's OEIS frontend

A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 5, 8, 12, 18, 27, 11, 16, 24, 36, 54, 81, 21, 32, 48, 72, 108, 162, 243, 43, 64, 96, 144, 216, 324, 486, 729, 85, 128, 192, 288, 432, 648, 972, 1458, 2187, 171, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 341, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

Paul Curtz, Jan 20 2009

Deleting column k=0 and reading by antidiagonals yields A036561.

Deleting column k=0 and reading the antidiagonals downwards yields A175840.

			The array starts in row n=0 with columns k>=0 as:
   0   1    3    9    27    81    243    729    2187  ... A140429;
   1   2    6   18    54   162    486   1458    4374  ... A025192;
   1   4   12   36   108   324    972   2916    8748  ... A003946;
   3   8   24   72   216   648   1944   5832   17496  ... A080923;
   5  16   48  144   432  1296   3888  11664   34992  ... A257970;
  11  32   96  288   864  2592   7776  23328   69984  ...
  21  64  192  576  1728  5184  15552  46656  139968  ...
Antidiagonal triangle begins as:
   0;
   1,   1;
   1,   2,   3;
   3,   4,   6,   9;
   5,   8,  12,  18,  27;
  11,  16,  24,  36,  54,  81;
  21,  32,  48,  72, 108, 162, 243;
  43,  64,  96, 144, 216, 324, 486, 729;
  85, 128, 192, 288, 432, 648, 972, 1458, 2187; - _G. C. Greubel_, Mar 25 2021

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

Cf. A001045, A004054, A046936.

t:= func< n,k | k eq 0 select (2^(n-k) -(-1)^(n-k))/3 else 2^(n-k)*3^(k-1) >;
[t(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 25 2021

T:=proc(n,k)if(k>0)then return 2^n*3^(k-1):else return (2^n - (-1)^n)/3:fi:end:
for d from 0 to 8 do for m from 0 to d do print(T(d-m,m)):od:od: # Nathaniel Johnston, Apr 13 2011

t[n_, k_]:= If[k==0, (2^(n-k) -(-1)^(n-k))/3, 2^(n-k)*3^(k-1)];
Table[t[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 25 2021 *)

def A155118(n,k): return (2^(n-k) -(-1)^(n-k))/3 if k==0 else 2^(n-k)*3^(k-1)
flatten([[A155118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 25 2021

For the square array:
T(n,k) = 2^n*3^(k-1), k>0.
T(n,k) = T(n-1,k+1) - T(n-1,k), n>0.
Rows:
T(0,k) = A140429(k) = A000244(k-1).
T(1,k) = A025192(k).
T(2,k) = A003946(k).
T(3,k) = A080923(k+1).
T(4,k) = A257970(k+3).
Columns:
T(n,0) = A001045(n) (Jacobsthal numbers J_{n}).
T(n,1) = A000079(n).
T(n,2) = A007283(n).
T(n,3) = A005010(n).
T(n,4) = A175806(n).
T(0,k) - T(k+1,0) = 4*A094705(k-2).
From G. C. Greubel, Mar 25 2021: (Start)
For the antidiagonal triangle:
t(n, k) = T(n-k, k).
t(n, k) = (2^(n-k) - (-1)^(n-k))/3 (J_{n-k}) if k = 0 else 2^(n-k)*3^(k-1).
Sum_{k=0..n} t(n, k) = 3^n - J_{n+1}, where J_{n} = A001045(n).
Sum_{k=0..n} t(n, k) = A004054(n-1) for n >= 1. (End)

a(22) - a(57) from Nathaniel Johnston, Apr 13 2011

A178501 Zero followed by powers of ten.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000, 100000000000000000000

Offset: 0

Reinhard Zumkeller, May 28 2010

The sequence S consisting of the nonnegative numbers arranged in lexicographic order according to their decimal expansion begins 0, 1, 10, 100, 1000, ..., 2, 20, 200, 2000, ..., 3, 30, ... does not have an OEIS entry, since there are uncountably many terms before 2 appears (or even before 100000010000 appears). However, S does begin with the present sequence. - N. J. A. Sloane, Dec 09 2024

a(n)^k + reverse(a(n))^k is a palindrome for any positive integer k. - Bui Quang Tuan, Mar 31 2015

Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A093136, A131577, A140429, A178500; subsequence of A029793.

The powers of 10, A011557, is a subsequence.

Join[{0}, 10^Range[0, 19]] (* Alonso del Arte, Mar 31 2015 *)

a(n)=10^n\10 \\ Charles R Greathouse IV, Oct 07 2015

concat(0, Vec(-x/(10*x-1) + O(x^22))) \\ Elmo R. Oliveira, Jul 21 2025

a(n+1) = A011557(n).
a(n) = A178500(n)/10.
From Paul Barry, Jul 09 2003: (Start)
a(n) = (10^n - 0^n)/10.
E.g.f.: exp(5*x)*sinh(5*x)/5.
Binomial transform of A015577. (End)
G.f.: x/(1 - 10*x). - Chai Wah Wu, Jun 17 2020
From Elmo R. Oliveira, Jul 21 2025: (Start)
a(n) = 10*a(n-1) for n > 1.
a(n) = A093136(n)/2 for n >= 1. (End)

More terms from Elmo R. Oliveira, Jul 21 2025

A174980 Stern's diatomic series type ([0,1], 1).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Apr 03 2010

A variant of Stern's diatomic series A002487. See the link [Luschny] and the Maple function below for the classification by types which is based on a generalization of Dijkstra's fusc function.
a(n) is also the number of superduperbinary integer partitions of n.
It appears that a(n) is equal to the multiplicative inverse of A002487(n+2) mod A002487(n+1). - Gary W. Adamson, Dec 23 2023

			The sequence splits into rows of length 2^k:
  0,
  0, 1,
  0, 2, 1, 1,
  0, 3, 2, 3, 1, 2, 1, 1,
  0, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1,
  ...
.
The first few partitions counted are:
[ 0], []
[ 1], []
[ 2], [[2]]
[ 3], []
[ 4], [[4], [2, 2]]
[ 5], [[4, 1]]
[ 6], [[4, 1, 1]]
[ 7], []
[ 8], [[8], [4, 4], [2, 2, 2, 2]]
[ 9], [[8, 1], [4, 4, 1]]
[10], [[8, 2], [8, 1, 1], [4, 4, 1, 1]]
[11], [[8, 2, 1]]
[12], [[8, 2, 2], [8, 2, 1, 1]]
[13], [[8, 2, 2, 1]]
[14], [[8, 2, 2, 1, 1]]
[15], []
[16], [[16], [8, 8], [4, 4, 4, 4], [2, 2, 2, 2, 2, 2, 2, 2]]
[17], [[16, 1], [8, 8, 1], [4, 4, 4, 4, 1]]
[18], [[16, 2], [8, 8, 2], [16, 1, 1], [8, 8, 1, 1], [4, 4, 4, 4, 1, 1]]
[19], [[16, 2, 1], [8, 8, 2, 1]]
[20], [[16, 4], [16, 2, 2], [8, 8, 2, 2], [16, 2, 1, 1], [8, 8, 2, 1, 1]]
[21], [[16, 4, 1], [16, 2, 2, 1], [8, 8, 2, 2, 1]]
[22], [[16, 4, 2], [16, 4, 1, 1], [16, 2, 2, 1, 1], [8, 8, 2, 2, 1, 1]]
[23], [[16, 4, 2, 1]]
[24], [[16, 4, 4], [16, 4, 2, 2], [16, 4, 2, 1, 1]]

Peter Luschny, row(n) for n = 0..12
Edsger Dijkstra, EWD 578: More about the function 'fusc', Selected Writings on Computing, Springer, 1982, p. 232.
Peter Luschny, Rational Trees and Binary Partitions.
Moritz A. Stern, Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.

Cf. A002487, A070879, A047679, A007306, A174981, A140429 (row sums), A086449.

SternDijkstra := proc(L, p, n) local k, i, len, M; len := nops(L); M := L; k := n; while k > 0 do M[1+(k mod len)] := add(M[i], i=1..len); k := iquo(k, len); od; op(p, M) end:
a := n -> SternDijkstra([0,1], 1, n);

a[0] = 0; a[n_?OddQ] := a[n] = a[(n-1)/2]; a[n_?EvenQ] := a[n] = a[n/2 - 1] + a[n/2] + Boole[ IntegerQ[ Log[2, n/2]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 26 2013 *)

# Generating the partitions.
def SDBinaryPartition(n):
    def Double(W, T):
        B = []
        for L in W:
            A = [a*2 for a in L]
            if T > 0: A += [1]*T
            B.append(A)
        return B
    if n == 2: return [[2]]
    if n <  4: return []
    h = n // 2
    H = SDBinaryPartition(h)
    B = Double(H, n % 2)
    if n % 2 == 0:
        H = SDBinaryPartition(h - 1)
        if H != []: B += Double(H, 2)
        if (n & (n - 1)) == 0: B.append([2]*h)
    return B
for n in range(25): print([n], SDBinaryPartition(n)) # Peter Luschny, Sep 02 2019

def A174980(n):
    M = [0, 1]
    for b in n.bits():
        M[b] = M[0] + M[1]
    return M[0]
print([A174980(n) for n in (0..100)]) # Peter Luschny, Nov 28 2017

Recursion: a(2n + 1) = a(n) and a(2n) = a(n - 1) + a(n) + [n = 2^k] for n = 1, a(0) = 0. [n = 2^k] is 1 if n is a power of 2, 0 otherwise.

A352502 a(n) is the number of integers k in the interval 0..n such that k and n-k can be added without carries in balanced ternary.

Original entry on oeis.org

1, 2, 2, 4, 4, 2, 4, 4, 6, 10, 8, 6, 10, 8, 2, 4, 4, 6, 10, 8, 6, 10, 8, 10, 16, 12, 18, 28, 20, 14, 22, 16, 10, 16, 12, 18, 28, 20, 14, 22, 16, 2, 4, 4, 6, 10, 8, 6, 10, 8, 10, 16, 12, 18, 28, 20, 14, 22, 16, 10, 16, 12, 18, 28, 20, 14, 22, 16, 18, 28, 20, 30

Offset: 0

Rémy Sigrist, Apr 28 2022

Two integers can be added without carries in balanced ternary if they have no equal nonzero digit at the same position.

This sequence has connections with Gould's sequence (A001316); here we work with balanced ternary, there with binary.

			For n = 8:
- we consider the following cases:
              k|    0    1    2    3    4    5    6    7    8
      ---------+---------------------------------------------
        bter(k)|    0    1   1T   10   11  1TT  1T0  1T1  10T
      bter(8-k)|  10T  1T1  1T0  1TT   11   10   1T    1    0
       carries?|  no   yes  no   no   yes  no   no   yes  no
- so a(8) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Wikipedia, Balanced ternary

Cf. A001316, A059095, A140429, A353174 (corresponding k's).

ok(u,v) = { while (u && v, my (uu=[0,+1,-1][1+u%3], vv=[0,+1,-1][1+v%3]); if (abs(uu+vv)>1, return (0)); u=(u-uu)/3; v=(v-vv)/3); return (1) }
a(n) = sum(k=0, n, ok(n-k, k))

a(n) <= n+1 with equality iff n belongs to A140429.
a(3*n) = 3*a(n) - 2.
a(3*n+1) = a(3*n-1) + 2.

A376478 a(1) = 1, a(2) = 2, and a(n) = 3^(n-2) for n > 2.

Original entry on oeis.org

1, 2, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961

Offset: 1

Stefano Spezia, Sep 24 2024

Graham's conjecture: also numbers k such sigma(k) - k = floor(k/2). See Guy.

Also the domination number of the n-Sierpinski gasket graph. - Eric W. Weisstein, Mar 10 2025

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B2.

Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Sierpinski Gasket Graph.
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A000244, A140429, A001065, A004526.

Mathematica
```
LinearRecurrence[{3},{1,2,3},30]
```

def A376478(n): return n if n<3 else 3**(n-2) # Chai Wah Wu, Nov 13 2024

a(n) = 3*a(n-1) for n > 3.
G.f.: (1 - x - 3*x^2)/(1 - 3*x).
E.g.f.: (2 + exp(3*x) + 3*x)/3.

A126184 Number of hex trees with n edges and having no nonroot nodes of outdegree 2.

Original entry on oeis.org

1, 3, 10, 33, 108, 351, 1134, 3645, 11664, 37179, 118098, 373977, 1180980, 3720087, 11691702, 36669429, 114791256, 358722675, 1119214746, 3486784401, 10847773692, 33705582543, 104603532030, 324270949293, 1004193907488

Offset: 0

Emeric Deutsch, Dec 19 2006

nonn

A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).

F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
Index entries for linear recurrences with constant coefficients, signature (6, -9).

Cf. A126183.

Cf. A045891, A140429, A172481.

Maple
```
1,seq(3^(n-2)*(n+8),n=1..28);
```

a(n) = A126183(n,0).
a(n) = (n+8)*3^(n-2) for n >= 1; a(0)=1.
G.f.: (1-3z+z^2)/(1-3z)^2.
From Paul Curtz, Mar 27 2022: (Start)
a(n+1) = 3*a(n) + A140429(n), for n >= 0; a(0)=1.
Binomial transform of A172481(n) for n >= 0.
Also, with a different offset, the binomial transform of A045891(n+2) for n >= 0. (End)

A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

Original entry on oeis.org

0, 1, 1, -3, -3, 9, 9, -27, -27, 81, 81, -243, -243, 729, 729, -2187, -2187, 6561, 6561, -19683, -19683, 59049, 59049, -177147, -177147, 531441, 531441, -1594323, -1594323, 4782969, 4782969, -14348907, -14348907, 43046721, 43046721, -129140163, -129140163, 387420489

Offset: 0

Jean-François Alcover and Paul Curtz, Jul 19 2017

This is the inverse binomial transform of A157241.
Successive differences of A157241 begin:
0,   1,   3,   3,   -5,  -21,  -21,    43,   171,   171, ... = A157241
1,   2,   0,  -8,  -16,    0,   64,   128,     0,  -512, ... = A088138
1,  -2,  -8,  -8,   16,   64,   64,  -128,  -512,  -512, ... = A138230
-3, -6,   0,  24,   48,    0, -192,  -384,     0,  1536, ...
-3,  6,  24,  24,  -48, -192, -192,   384,  1536,  1536, ...
9,  18,   0, -72, -144,    0,  576,  1152,     0, -4608, ...
9, -18, -72  -72,  144,  576,  576, -1152, -4608, -4608, ...
...
a(n) is the n-th term of the first column.
Successive differences of a(n) begin:
0,     1,    1,   -3,   -3,    9,     9,   -27,   -27,    81, ...
1,     0,   -4,    0,   12,    0,   -36,     0,   108,     0, ...
-1,   -4,    4,   12,  -12,  -36,    36,   108,  -108,  -324, ...
-3,    8,    8,  -24,  -24,   72,    72,  -216,  -216,   648, ...
11,    0,  -32,    0,   96,    0,  -288,     0,   864,     0, ...
-11, -32,   32,   96,  -96, -288,   288,   864,  -864, -2592, ...
-21,  64,   64, -192, -192,  576,   576, -1728, -1728,  5184, ...
85,    0, -256,    0,  768,    0, -2304,     0,  6912,     0, ...
...
First column appears to be a subsequence of Jacobsthal numbers A001045 (the trisection A082311 is missing), second column is A104538, and third column is A137717.
a(n) = A128019(n-2) for n > 2. - Georg Fischer, Oct 23 2018

Index entries for linear recurrences with constant coefficients, signature (0, -3).

Cf. A001045, A082311, A088138, A104538, A108411, A128019, A137717, A138230, A140429, A157241.

Join[{0}, LinearRecurrence[{0, -3}, {1, 1}, 40]]
(* or, computation from b = A157241 : *)
b[n_] := (Switch[Mod[n, 3], 0, (-1)^((n + 3)/3), 1, (-1)^((n + 5)/3), 2, (-1)^((n + 4)/3)*2]*2^n + 1)/3; tb = Table[b[n], {n, 0, 40}]; Table[ Differences[tb, n], {n, 0, 40}][[All, 1]]

concat([0], Vec((x + x^2)/(1 + 3*x^2) + O(x^40))) \\ Felix Fröhlich, Oct 23 2018

a(n) = -3*a(n-2) for n > 2.

E.g.f.: (1 - cos(sqrt(3)*x) + sqrt(3)*sin(sqrt(3)*x))/3. - Stefano Spezia, Jul 15 2024

A155734 Binomial transform of A154879.

Original entry on oeis.org

3, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329

Offset: 0

Paul Curtz, Jan 26 2009

Binomial transform of the third differences of A001045.
The binomial transform of the first differences of A001045 is in A133494.
The binomial transform of the 2nd differences of A001045 is in A133494, with the sign of A133494(0) flipped.
The binomial transform of the p-th differences of A001045 is the number A077925(p-1) followed by A000244.

Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A154879, A078008. Essentially the same as A140429 and A000244.

read("transforms") ; A001045 := proc(n) option remember ; if n <= 1 then n; else procname(n-1)+2*procname(n-2) ; fi; end:
a001045 := [seq(A001045(n),n=0..80) ] ; a154879 := DIFF(DIFF(DIFF(a001045))) ; BINOMIAL(a154879) ; # R. J. Mathar, Jul 23 2009

From Colin Barker, Apr 05 2012: (Start)
a(n) = 3*a(n-1) for n > 1.
G.f.: (3-8*x)/(1-3*x). (End)
G.f.: (1 - 2/G(0))/x where G(k) = 1 + 2^k/(1 - 2*x/(2*x + 2^k/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012

Edited and extended by R. J. Mathar, Jul 23 2009

cp's OEIS Frontend

A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A178501 Zero followed by powers of ten.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A174980 Stern's diatomic series type ([0,1], 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

SageMath

Formula

A352502 a(n) is the number of integers k in the interval 0..n such that k and n-k can be added without carries in balanced ternary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A376478 a(1) = 1, a(2) = 2, and a(n) = 3^(n-2) for n > 2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

A126184 Number of hex trees with n edges and having no nonroot nodes of outdegree 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

Views

Author

Keywords