A092810 -id:A092810 - cp's OEIS frontend

A270369 Expansion of g.f. (1-7x)/(1-9x).

1, 2, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442, 774840978, 6973568802, 62762119218, 564859072962, 5083731656658, 45753584909922, 411782264189298, 3706040377703682, 33354363399333138, 300189270593998242, 2701703435345984178, 24315330918113857602, 218837978263024718418

Offset: 0

Colin Barker, Mar 18 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A054879 (partial sums), A132025.

Cf. similar sequences with g.f. (1-k*x)/(1-9*x) and k=0..8: A001019 (k=0; k=8 gives two initial 1's ), A055275 (k=1), A270472 (k=2), A092810 (k=3), A067403 (k=4), A270473 (k=5), A102518 (k=6), this sequence (k=7).

CoefficientList[Series[(1-7x)/(1-9x),{x,0,20}],x] (* or *) Join[ {1}, NestList[9#&,2,20]] (* Harvey P. Dale, Oct 15 2017 *)

PARI
```
Vec((1-7*x)/(1-9*x) + O(x^30))
```

G.f.: (1-7*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 2*9^(n-1) for n>0.
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 25/16.
Sum_{n>=0} (-1)^n/a(n) = 11/20.
Product_{n>=1} (1 - 1/a(n)) = A132025. (End)
E.g.f.: (2*exp(9*x) + 7)/9. - Elmo R. Oliveira, Mar 25 2025

A270472 Expansion of g.f. (1-2x)/(1-9x).

Original entry on oeis.org

1, 7, 63, 567, 5103, 45927, 413343, 3720087, 33480783, 301327047, 2711943423, 24407490807, 219667417263, 1977006755367, 17793060798303, 160137547184727, 1441237924662543, 12971141321962887, 116740271897665983, 1050662447078993847, 9455962023710944623, 85103658213398501607

Offset: 0

Colin Barker, Mar 17 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A005032, A187709 (partial sums).

Cf. A055275: (1-x)/(1-9*x); A092810: (1-3*x)/(1-9*x).

CoefficientList[Series[(1 - 2 x)/(1 - 9 x), {x, 0, 20}], x] (* Michael De Vlieger, Mar 18 2016 *)

PARI
```
Vec((1-2*x)/(1-9*x) + O(x^30))
```

G.f.: (1-2*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 7*9^(n-1) for n>0.
a(n) = A005032(2*n-2). - R. J. Mathar, Jan 28 2025
E.g.f.: (7*exp(9*x) + 2)/9. - Elmo R. Oliveira, Mar 25 2025

A355581 Exponentially-odd 3-smooth numbers: number of the form 2^i * 3^j where i and j are either 0 or odd.

Original entry on oeis.org

1, 2, 3, 6, 8, 24, 27, 32, 54, 96, 128, 216, 243, 384, 486, 512, 864, 1536, 1944, 2048, 2187, 3456, 4374, 6144, 7776, 8192, 13824, 17496, 19683, 24576, 31104, 32768, 39366, 55296, 69984, 98304, 124416, 131072, 157464, 177147, 221184, 279936, 354294, 393216, 497664

Offset: 1

Amiram Eldar, Jul 08 2022

			6 is a term since 6 = 2^1 * 3^1 and the exponents of 2 and 3 are both odd: 1.
24 is a term since 24 = 2^3 * 3^1 and the exponents of 2 and 3 are both odd: 3 and 1, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A003586 and A268335.
Subsequences: A002023, A013711, A092810.
Cf. A355580.

q[n_] := Module[{e = IntegerExponent[n, {2, 3}]}, (e[[1]] == 0 || OddQ[e[[1]]]) && (e[[2]] == 0 || OddQ[e[[2]]]) && Times@@({2, 3}^e) == n]; Select[Range[500000], q]

is(n) = {my(e2 = valuation(n, 2), e3 = valuation(n, 3)); (e2 == 0 || e2%2) && (e3 == 0 || e3%2) && n == 2^e2 * 3^e3};

from itertools import count, takewhile
def aupto(lim):
    pows2 = list(takewhile(lambda x: xMichael S. Branicky, Jul 08 2022

Sum_{n>=1} 1/a(n) = 55/24.

cp's OEIS Frontend

A270369 Expansion of g.f. (1-7x)/(1-9x).

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A270472 Expansion of g.f. (1-2x)/(1-9x).

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A355581 Exponentially-odd 3-smooth numbers: number of the form 2^i * 3^j where i and j are either 0 or odd.

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cp's OEIS Frontend

A270369 Expansion of g.f. (1-7*x)/(1-9*x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A270472 Expansion of g.f. (1-2*x)/(1-9*x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A355581 Exponentially-odd 3-smooth numbers: number of the form 2^i * 3^j where i and j are either 0 or odd.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A270369 Expansion of g.f. (1-7x)/(1-9x).

A270472 Expansion of g.f. (1-2x)/(1-9x).