A270473 -id:A270473 - 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

A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).

Original entry on oeis.org

4, 9, 16, 36, 64, 144, 256, 576, 1024, 2304, 4096, 9216, 16384, 36864, 65536, 147456, 262144, 589824, 1048576, 2359296, 4194304, 9437184, 16777216, 37748736, 67108864, 150994944, 268435456, 603979776, 1073741824, 2415919104, 4294967296, 9663676416, 17179869184

Offset: 1

Karl-Heinz Hofmann, Sep 02 2022

If x is even, y = x + 3; if x is odd, y = x.
Proof for odd x: (2^odd + 2^odd) = 2^(odd + 1) = 2^even --> must be a square.
Proof for even x: 2^even + 2^(even + 3) = 1*(2^even) + (2^even * 2^3) = 1*(2^even) + (2^even * 8) = 1*(2^even) + 8*(2^even) = 9*(2^even); since 9 is a square and 2^even is a square, the multiplication result must be a square too.
And 9 is the only square that can be written as 1 + a power of 2.
Note that a(n) = A272711(n+1) for n=1..23, but beyond it differs more and more.

			2^4 + 2^7 = 144, a square, thus 144 is a term.

Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A029744, A000302, A002063.
Intersection of A000290 and A048645\{1}.
Cf. A272711, A270473 (squares that can be expressed as 3^x + 3^y).
Cf. A220221.

seq(`if`(n::even, 9*2^(n-2), 2^(n+1)),n=1..50); # Robert Israel, Sep 15 2022

Select[Range[2, 2^17]^2, DigitCount[#, 2, 1] <= 2 &] (* Amiram Eldar, Sep 03 2022 *)

a(n) = if (n%2, 2^(n+1), 9*2^(n-2)); \\ Michel Marcus, Sep 15 2022

def A356880(n):
    if n % 2 == 0: return 9*2**(n-2)
    else: return 2**(n+1)

a(n) = A029744(n+1)^2.
a(n) = 9 * 2^(n-2) if n is even (see A002063).
a(n) = 2^(n+1) if n is odd (see A000302).
From Stefano Spezia, Sep 09 2022: (Start)
G.f.: x*(4 + 9*x)/(1 - 4*x^2).
E.g.f.: (9*(cosh(2*x) - 1) + 8*sinh(2*x))/4. (End)

A356879 Numbers k such that the sum k^x + k^y can be a square with {x, y} >= 0.

Original entry on oeis.org

0, 2, 3, 8, 15, 18, 24, 32, 35, 48, 50, 63, 72, 80, 98, 99, 120, 128, 143, 162, 168, 195, 200, 224, 242, 255, 288, 323, 338, 360, 392, 399, 440, 450, 483, 512, 528, 575, 578, 624, 648, 675, 722, 728, 783, 800, 840, 882, 899, 960, 968, 1023, 1058, 1088, 1152, 1155, 1224

Offset: 0

Karl-Heinz Hofmann, Sep 12 2022

nonn

Characteristics of the terms:
  - Any x combined with any y is a solution.
      This special case is valid only for k = 0 (exception: x = y = 0).
  - Any x is possible and if x is odd: y = x. If x is even: y = x + 3.
      This special case is valid only for k = 2 (see A356880).
  - Only even x combined with y = x + 1 gives a solution.
      Those terms are the terms of A132411.
  - Only odd x combined with y = x gives a solution.
      Those terms are the terms of A001105.
  - Any x is possible and if x is odd: y = x. If x is even: y = x + 1.
      Those terms are the terms of A132592.

			Squares that can be produced with k = 8: 8^0 + 8^1 = 9; 8^1 + 8^1 = 16; 8^2 + 8^3 = 576; 8^3 + 8^3 = 1024; 8^4 + 8^5 = 36864; 8^5 + 8^5 = 65536; 8^6 + 8^7 = 2359296, ....

Cf. A132411 is a subsequence (except A132411(1)), A001105 is a subsequence.
Cf. A132592 is a subsequence.
Cf. A356880 (k = 2), A270473 (k = 3).

Select[Range[0, 1225], IntegerQ[Sqrt[# + 1]] || IntegerQ[Sqrt[#/2]] &] (* Amiram Eldar, Sep 18 2022 *)

from gmpy2 import is_square
print([n for n in range(0,1225) if is_square(n+1) or (n % 2 == 0 and is_square(n//2))])

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

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A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).

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A356879 Numbers k such that the sum k^x + k^y can be a square with {x, y} >= 0.

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cp's OEIS Frontend

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

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Mathematica

PARI

Formula

A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A356879 Numbers k such that the sum k^x + k^y can be a square with {x, y} >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

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