A212191 -id:A212191 - cp's OEIS frontend

A005009 a(n) = 7*2^n.

7, 14, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168, 14336, 28672, 57344, 114688, 229376, 458752, 917504, 1835008, 3670016, 7340032, 14680064, 29360128, 58720256, 117440512, 234881024, 469762048, 939524096, 1879048192, 3758096384

Offset: 0

N. J. A. Sloane

The first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini, Sep 07 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), this sequence (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Row sums of (6, 1)-Pascal triangle A093563 and of (1, 6)-Pascal triangle A096956, n>=1.
Cf. A000120, A014311, A118416, A173787, A212191.

a005009 = (* 7) . (2 ^) -- Reinhard Zumkeller, May 03 2012

[7*2^n:n in [0..50]]; // Vincenzo Librandi, Sep 20 2011

7*2^Range[0,50] (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)
NestList[2#&,7,30] (* Harvey P. Dale, Aug 10 2024 *)

a(n)=7<Charles R Greathouse IV, Dec 22 2011

[7*2^n for n in range(51)] # G. C. Greubel, Jan 05 2023

G.f.: 7/(1-2*x).
a(n) = A118416(n+1,4) for n > 3. - Reinhard Zumkeller, Apr 27 2006
a(n) = 2*a(n-1), for n > 0, with a(0)=7 . - Philippe Deléham, Nov 23 2008
a(n) = 7 * A000079(n). - Omar E. Pol, Dec 16 2008
a(n) = A173787(n+3,n). - Reinhard Zumkeller, Feb 28 2010
Intersection of A014311 and A212191: all terms and their squares are the sum of exactly three distinct powers of 2, A000120(a(n)) = A000120(a(n)^2) = 3. - Reinhard Zumkeller, May 03 2012
G.f.: 2/x/G(0) - 1/x + 9, where G(k)= 1 + 1/(1 - x*(7*k+2)/(x*(7*k+9) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
E.g.f.: 7*exp(2*x). - Stefano Spezia, May 15 2021

A212190 Squares that are the sum of exactly three distinct powers of 2.

Original entry on oeis.org

25, 49, 81, 100, 196, 289, 324, 400, 529, 784, 1089, 1156, 1296, 1600, 2116, 3136, 4225, 4356, 4624, 5184, 6400, 8464, 12544, 16641, 16900, 17424, 18496, 20736, 25600, 33856, 50176, 66049, 66564, 67600, 69696, 73984, 82944, 102400, 135424, 200704, 263169

Offset: 1

Reinhard Zumkeller, May 03 2012

nonn

Squares with exactly three ones in their binary representation: A000120(a(n)) = 3;
squares in A014311;
a(n) = A212191(n)^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Cf. A010052, A212192.

a212190 n = a212190_list !! (n-1)
a212190_list = filter ((== 1) . a010052) a014311_list

Select[Range[1000]^2, DigitCount[#, 2, 1] == 3&] (* Jean-François Alcover, Nov 07 2016 *)

A220221 Odd positive integers k such that k^2 has at most three nonzero binary digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 17, 23, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297

Offset: 1

John W. Layman, Dec 07 2012

It is shown in the Szalay reference that if y is a term of this sequence then y=7, y=23, or y=2^t+1 for some positive t. Also see the Bennett reference.

Colin Barker, Table of n, a(n) for n = 1..1000
Michael A. Bennett, Perfect powers with few ternary digits, INTEGERS 12A (2012), #A3.
László Szalay, The equations 2^n+-2^m+-2^l=z^2, Indag. Math. 13 (2002) 131-142.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A212191 (exactly 3 powers).

I:=[1,3,5,7,9,17,23,33,65,129]; [n le 10 select I[n] else 3*Self(n-1)-2*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Nov 07 2014

Select[Range[1, 1000000, 2], Total[IntegerDigits[#^2, 2]] <= 3 &] (* T. D. Noe, Dec 07 2012 *)
CoefficientList[Series[(12 x^8 - 2 x^7 - 10 x^6 + 4 x^5 - 2 x^4 - 2 x^3 - 2 x^2 + 1) / ((x - 1) (2 x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 07 2014 *)

is(n)=n%2 && hammingweight(n^2)<4 \\ Charles R Greathouse IV, Dec 10 2012

Vec(x*(12*x^8-2*x^7-10*x^6+4*x^5-2*x^4-2*x^3-2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Nov 06 2014

a(n) = 3*a(n-1)-2*a(n-2) for n>9. - Colin Barker, Nov 06 2014

G.f.: x*(12*x^8-2*x^7-10*x^6+4*x^5-2*x^4-2*x^3-2*x^2+1) / ((x-1)*(2*x-1)). - Colin Barker, Nov 06 2014

Extended by T. D. Noe, Dec 07 2012

cp's OEIS Frontend

A005009 a(n) = 7*2^n.

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A212190 Squares that are the sum of exactly three distinct powers of 2.

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Haskell

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A220221 Odd positive integers k such that k^2 has at most three nonzero binary digits.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

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