A086341 -id:A086341 - cp's OEIS frontend

A080075 Proth numbers: of the form k*2^m + 1 for k odd, m >= 1 and 2^m > k.

3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 289, 321, 353, 385, 417, 449, 481, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1025, 1089, 1153, 1217, 1281, 1345, 1409

Offset: 1

Eric W. Weisstein, Jan 24 2003

nonn

A Proth number is a square iff it is of the form (2^(m-1)+-1)*2^(m+1)+1 = 4^m+-2^(m+1)+1 = (2^m+-1)^2 for m > 1. See A086341. - Thomas Ordowski, Apr 22 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Bertalan Borsos, Attila Kovács and Norbert Tihanyi, Tight upper and lower bounds for the reciprocal sum of Proth primes, The Ramanujan Journal (2022).
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 38.
Amelia Carolina Sparavigna, Discussion of the groupoid of Proth numbers (OEIS A080075), Politecnico di Torino, Italy (2019).
Eric Weisstein's World of Mathematics, Proth Number.
Wikipedia, Proth number.

Cf. A080076, A086341, A112714, A116882, A157892, A157893.

Select[Range[3, 1500, 2], And[OddQ[#[[1]] ], #[[-1]] >= 1, 2^#[[-1]] > #[[1]] ] &@ Append[QuotientRemainder[#1, 2^#2], #2] & @@ {#, IntegerExponent[#, 2]} &[# - 1] &] (* Michael De Vlieger, Nov 04 2019 *)

is_A080075 = isproth(x)={!bittest(x--,0) && (x>>valuation(x+!x,2))^2 < x } \\ M. F. Hasler, Aug 16 2010; edited by Michel Marcus, Apr 23 2019, M. F. Hasler, Jul 07 2022

next_A080075(N)=N+2^(exponent(N)\2+1)
A080075_first(N)=vector(N,i,if(i>1,next_A080075(N),3)) \\ M. F. Hasler, Jul 07 2022

from itertools import count, islice
def A080075_gen(startvalue=3): # generator of terms >= startvalue
    return filter(lambda n:(n-1&-n+1)**2+1>=n,count(max(startvalue,3)))
A080075_list = list(islice(A080075_gen(),30)) # Chai Wah Wu, Oct 06 2024

a(n) = A116882(n+1)+1. - Klaus Brockhaus, Georgi Guninski and M. F. Hasler, Aug 16 2010
a(n) = A157892(n)*2^A157893(n) + 1. - M. F. Hasler, Aug 16 2010
a(n) ~ n^2/2. - Thomas Ordowski, Oct 19 2014
Sum_{n>=1} 1/a(n) = 1.09332245643583252894473574405304699874426408312553... (Borsos et al., 2022). - Amiram Eldar, Jan 29 2022
a(n+1) = a(n) + 2^round(L(n)/2), where L(n) is the number of binary digits of a(n); equivalently, floor(log_2(a(n))/2 + 1) in the exponent. [Lemma 2.2 in Borsos et al.] - M. F. Hasler, Jul 07 2022

A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 17, 63, 257, 1023, 4097, 16383, 65537, 262143, 1048577, 4194303, 16777217, 67108863, 268435457, 1073741823, 4294967297, 17179869183, 68719476737, 274877906943, 1099511627777, 4398046511103, 17592186044417, 70368744177663, 281474976710657

Offset: 0

Bruno Berselli, Jan 09 2013

This is the Lucas sequence V(3,-4).

Inverse binomial transform of this sequence is A087451.

Bruno Berselli, Table of n, a(n) for n = 0..200
Weerayuth Nilsrakoo and Achariya Nilsrakoo, On One-Parameter Generalization of Jacobsthal Numbers, WSEAS Trans. Math. (2025) Vol. 24, 51-61. See p. 3.
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. for the same recurrence with initial values (i,i+1): A015521 (Lucas sequence U(3,-4); i=0), A122117 (i=1), A189738 (i=3).
Cf. for similar closed form: A014551 (2^n+(-1)^n), A102345 (3^n+(-1)^n), A087404 (5^n+(-1)^n).
Cf. A052539, A024036.

[n le 1 select n+2 else 3*Self(n)+4*Self(n-1): n in [0..25]];

RecurrenceTable[{a[n] == 3 a[n - 1] + 4 a[n - 2], a[0] == 2, a[1] == 3}, a[n], {n, 25}]

a[0]:2$ a[1]:3$ a[n]:=3*a[n-1]+4*a[n-2]$ makelist(a[n], n, 0, 25);

Vec((2-3*x)/((1+x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Jun 26 2015

G.f.: (2-3*x)/((1+x)*(1-4*x)).
a(n) = 4^n+(-1)^n.
a(n) = A086341(A047524(n)) for n>0, a(0)=2.
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 25*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = (2/4^n) * Sum_{k = 0..n} binomial(4*n+1, 4*k). - Peter Bala, Feb 06 2019

A097581 a(n) = 3*2^floor((n-1)/2) + (-1)^n.

Original entry on oeis.org

2, 4, 5, 7, 11, 13, 23, 25, 47, 49, 95, 97, 191, 193, 383, 385, 767, 769, 1535, 1537, 3071, 3073, 6143, 6145, 12287, 12289, 24575, 24577, 49151, 49153, 98303, 98305, 196607, 196609, 393215, 393217, 786431, 786433, 1572863, 1572865

Offset: 1

Pierre CAMI, Sep 20 2004

nonn

Previous name was: a(1)=2 then if n even a(n)=a(n-1)+2 and if n odd a(n)=a(n-2)+a(n-1)-1.

This sequence a(n)=A016116(n-1)+A086341(n). Generalization: starting with a(1) even then if n even a(n)=a(n-1)+2 and if n odd a(n)=a(n-2)+a(n-1)-1 you get a new sequence as a(1) increases. But if a(1) is odd, you get always the same sequence with only less values as a(1) increases. If a(1) is even, the sequence difference between two sequences with different but consecutive a(1) is the sequence of powers of 2 = 2,2,4,4,8,8,16,16,32,32,......

			Starting with a(1)=4 the new sequence is 4,6,9,11,19,21,39,41,79,81,159,161
The sequence difference between sequence starting with a(1)=4 and the sequence starting with a(1)=2 is 2,2,4,4,8,8,16,16,32,32,64,64,.......

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-1,2,2).

Cf. A016116, A086341.

LinearRecurrence[{-1,2,2},{2,4,5},40] (* Harvey P. Dale, Aug 10 2011 *)
Table[3*2^(Floor[(n - 1)/2]) + (-1)^n, {n, 1,50}] (* G. C. Greubel, Apr 18 2017 *)

PARI
```
a(n)=3*2^floor((n-1)/2)+(-1)^n
```

From R. J. Mathar, Nov 13 2009: (Start)
a(n) = -a(n-1) + 2*a(n-2) + 2*a(n-3).
G.f.: x*(2+6*x+5*x^2)/((1+x)*(1-2*x^2)). (End)

Equation in the comment corrected by R. J. Mathar, Nov 13 2009

Better name from Ralf Stephan, Aug 19 2013

A344851 a(n) = (n^2) mod (2^A070939(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 16, 25, 4, 17, 0, 17, 4, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 0, 17, 36, 57, 16, 41, 4, 33, 0, 33, 4, 41, 16, 57, 36, 17, 0, 49, 36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Offset: 0

Rémy Sigrist, May 30 2021

Informally, if n has w binary digits, a(n) is obtained by keeping the w final binary digits of n^2.
For n > 0, a(n) is the final digit of n^2 in base A062383(n).
This sequence has interesting graphical features (see illustration in Links section).

			For n = 42:
- A070939(42) = 6,
- a(42) = (42^2) mod (2^6) = 1764 mod 64 = 36.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
John S. McCaskill and Peter R.Wills, On permutations derived from integer powers x^n, arXiv:1907.01890 [math.NT], 2019.
Rémy Sigrist, Scatterplot of the sequence for n = 2^18..2^19

Cf. A000290, A048152, A062383, A070939, A086341, A116882, A316347 (decimal analog).

{0}~Join~Table[Mod[n^2,2^(1+Floor@Log2@n)],{n,100}] (* Giorgos Kalogeropoulos, Jun 02 2021 *)

PARI
```
a(n) = (n^2) % 2^#binary(n)
```

def a(n): return (n**2) % (2**n.bit_length())
print([a(n) for n in range(75)]) # Michael S. Branicky, May 30 2021

a(n) = 0 iff n = 0 or n > 1 and n belongs to A116882.
a(n) = 1 iff n belongs to A086341.
a(2^k + m) = a(2^(k+1)-m) for any k > 0 and m = 0..2^k.

cp's OEIS Frontend

A080075 Proth numbers: of the form k*2^m + 1 for k odd, m >= 1 and 2^m > k.

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A201455 a(n) = 3*a(n-1) + 4*a(n-2) for n>1, a(0)=2, a(1)=3.

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A097581 a(n) = 3*2^floor((n-1)/2) + (-1)^n.

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A344851 a(n) = (n^2) mod (2^A070939(n)).

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Author

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Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Python

Formula

A201455 a(n) = 3a(n-1) + 4a(n-2) for n>1, a(0)=2, a(1)=3.