A005165 -id:A005165 - cp's OEIS frontend

A004539 Expansion of sqrt(2) in base 2.

1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1

Offset: 1

N. J. A. Sloane

Bailey, Borwein, Crandall, & Pomerance prove a general result that the first n terms contain >> sqrt(n) 1's. Vandehey improves this to sqrt(2*n)(1 + o(1)). - Charles R Greathouse IV, Nov 07 2017

			1.0110101000001001111001...

T. D. Noe, Table of n, a(n) for n = 1..1000
B. Adamczewski and N. Rampersad, On patterns occurring in binary algebraic numbers, Proc. Amer. Math. Soc. 136 (2008), 3105-3109.
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, On the binary expansions of algebraic numbers, J. Théor. Nombres Bordeaux, 16 (2004), 487-518.
R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related to sqrt(2), Mathematics Magazine, Volume 43, Pages 143-145, 1970. Zbl 201.04705.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Richard Isaac, On the simple normality to base 2 of the square root of s, for s not a perfect square., arXiv:math/0512404 [math.NT], 2005-2006.
Jason Kimberley, Index of expansions of sqrt(d) in base b
Thomas Stoll, On families of nonlinear recurrences related to digits, Journal of Integer Sequences 8 (2005), 05.3.2.
Thomas Stoll, On a problem of Erdős and Graham concerning digits, Acta Arithmetica 125 (2006), pp. 89-100.
Thomas Stoll, A fancy way to obtain the binary digits of 759250125 sqrt{2}, (2009), Amer. Math. Monthly, 117 (2010), 611-617.
Joseph Vandehey, On the binary digits of sqrt(2), arXiv:1711.01722 [math.NT], 2017.
Eric Weisstein's World of Mathematics, Wolfram's Iteration
Eric Weisstein's World of Mathematics, Pythagoras's Constant

Cf. A002193 (decimal version), A233836 (run lengths of 0's and 1's).

a004539 n = a004539_list !! (n-1)
a004539_list = w 2 0 where
   w x r = bit : w (4 * (x - (4 * r + bit) * bit)) (2 * r + bit)
     where bit = head (dropWhile (\b -> (4 * r + b) * b < x) [0..]) - 1
-- Reinhard Zumkeller, Dec 16 2013

N[Sqrt[2], 200]; RealDigits[%, 2]
RealDigits[Sqrt[2],2,120][[1]] (* Harvey P. Dale, Aug 03 2024 *)

binary(sqrt(2)) \\ Michel Marcus, Nov 06 2017

a(n) = floor(quadgen(8)<<(n-1))%2; \\ Chittaranjan Pardeshi, Sep 09 2024

bc
```
obase=2 scale=200 sqrt(2)
```

a(k) = floor(Sum_{n>=1} A005875(n)/exp(Pi*n/(2^((2/3)*k+(1/3))))) mod 2. Will give the k-th binary digit of sqrt(2). A005875 : number of ways to write n as sum of 3 squares. - Simon Plouffe, Dec 30 2023

A007351 Where prime race 4m-1 vs. 4m+1 is tied.

Original entry on oeis.org

2, 5, 17, 41, 461, 26833, 26849, 26863, 26881, 26893, 26921, 616769, 616793, 616829, 616843, 616871, 617027, 617257, 617363, 617387, 617411, 617447, 617467, 617473, 617509, 617531, 617579, 617681, 617707, 617719, 618437, 618521, 618593, 618637

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Primes p such that the number of primes <= p of the form 4m-1 is equal to the number of primes <= p of the form 4m+1.

Starting from a(27410)=9103362505753 the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 419467 terms (see a-file) with a(419467)=9543313015351 as its last term. - Sergei D. Shchebetov, Oct 15 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp.54-76.
M. Deléglise, P. Dusart, and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.1565-1575.
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
Andrey S. Shchebetov and Sergei D. Shchebetov, First 419467 terms (zipped file)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007350, A038691.

Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues, Mar 26 2009]

Prime@ Position[Fold[Append[#1, #1[[-1]] + If[Mod[#2, 4] == 3, {1, 0}, {0, 1}]] &, {{0, 0}}, Prime@ Range[2, 10^5]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, May 27 2018 *)

lista(nn) = {nb = 0; forprime(p=2, nn, m = (p % 4); if (m == 1, nb++, if (m == 3, nb--)); if (!nb, print1(p, ", ")););} \\ Michel Marcus, Oct 05 2017

from sympy import nextprime; a, p = 0, 2; R = [p]
while p < 618637:
    p=nextprime(p); a += p%4-2
    if a == 0: R.append(p)
print(*R, sep = ', ')  # Ya-Ping Lu, Jan 18 2025

Corrected and extended by Enoch Haga, Feb 24 2004

A002255 Numbers k such that 7*4^k + 1 is prime.

Original entry on oeis.org

1, 2, 3, 7, 10, 13, 25, 26, 46, 60, 87, 90, 95, 145, 160, 195, 216, 308, 415, 902, 1128, 3307, 6748, 7747, 8348, 11193, 27243, 44033, 47665, 103542, 141517, 280908, 402267, 405615, 745926, 1069956, 1083900, 1457977, 1507881, 1755887

Offset: 1

N. J. A. Sloane

H. Riesel, Prime numbers and computer methods for factorization, in Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Ballinger, Proth Search Page.
C. K. Caldwell, The Prime Pages.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

See A032353 (which is the main entry for this sequence) for more terms.

is(n)=ispseudoprime(7*4^n+1) \\ Charles R Greathouse IV, Jun 06 2017

More terms (from A032353) added by Joerg Arndt, Apr 07 2013

A051200 Except for initial term, primes of form "n 3's followed by 1".

Original entry on oeis.org

3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331

Offset: 1

N. J. A. Sloane

"A remarkable pattern that is entirely accidental and leads nowhere" - M. Gardner, referring to the first 8 terms.
a(2)*a(3)*a(4) = 34179391, a Zeisel number (A051015) with coefficients (10,21).
a(2)*a(3)*a(4)*a(5) = 1139233281421, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(6) = 379741768929343351, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(7) = 1265805010367017001532181, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(8) = 42193497392022209194699696424911, a Zeisel number with coefficients (10,21).
Besides first 3, primes of the form (10^n-7)/3, n>1. See A123568. - Vincenzo Librandi, Aug 06 2010
The integer lengths of the terms of the sequence are 1, 2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, etc. - Harvey P. Dale, Dec 01 2018

Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 194.
W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb, Warsaw, 1964; Problem 88 [in English: 200 Problems from the Elementary Theory of Numbers]
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, pp. 8, 56-57.
F. Smarandache, Properties of numbers, University of Craiova, 1973

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, 3.

Cf. A055520, A089017, A089018, A093671, A056698, A105427, A105428, A033175, A123568.

Join[{3},Select[Rest[FromDigits/@Table[PadLeft[{1},n,3], {n,50}]], PrimeQ]]  (* Harvey P. Dale, Apr 20 2011 *)

Union of 3 and A123568.

More terms from James Sellers

Cross reference added by Harvey P. Dale, May 21 2014

A058006 Alternating factorials: 0! - 1! + 2! - ... + (-1)^n n!

Original entry on oeis.org

1, 0, 2, -4, 20, -100, 620, -4420, 35900, -326980, 3301820, -36614980, 442386620, -5784634180, 81393657020, -1226280710980, 19696509177020, -335990918918980, 6066382786809020, -115578717622022980, 2317323290554617020, -48773618881154822980

Offset: 0

Henry Bottomley, Nov 13 2000

From Harry Richman, Aug 13 2024: (Start)
Euler argued this sequence converges to 0.596347... (A073003 = Gompertz's constant); see Lagarias Section 2.5.
This sequence converges in the p-adic topology, for every prime number p. (End)

			a(5) = 0!-1!+2!-3!+4!-5! = 1-1+2-6+24-120 = -100.
G.f. = 1 + 2*x^2 - 4*x^3 + 20*x^4 - 100*x^5 + 620*x^6 - 4420*x^7 + 35900*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., Vol. 50, No. 4 (2013), pp. 527-628; preprint, arXiv:1303.1856 [math.NT], 2013.
Eric Weisstein's MathWorld, Incomplete Gamma Function.

Cf. A000142, A003422, A005165, A153229 (absolute values), A136580.

Partial sums of A133942.

a058006 n = a058006_list !! n
a058006_list = scanl1 (+) a133942_list
-- Reinhard Zumkeller, Mar 02 2014

a[ n_] := Sum[ (-1)^k k!, {k, 0, n}]; (* Michael Somos, Jan 28 2014 *)

{a(n) = sum(k=0, n, (-1)^k * k!)}; /* Michael Somos, Jan 28 2014 */

a(n) = (-1)^n n! + a(n-1) = A005165(n)(-1)^n + 1.
a(n) = -(n-1)*a(n-1) + n*a(n-2), n>0.
E.g.f.: d/dx ((GAMMA(0,1)-GAMMA(0,1+x))*exp(1+x)). - Max Alekseyev, Jul 05 2010
G.f.: G(0)/(1-x), where G(k)= 1 - (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
0 = a(n)*(-a(n+1) + a(n+3)) + a(n+1)*(2*a(n+1) - 2*a(n+2) -a(n+3)) + a(n+2)*(a(n+2)) if n>=-1. - Michael Somos, Jan 28 2014
a(n) = exp(1)*Gamma(0,1) + (-1)^n*exp(1)*(n+1)!*Gamma(-n-1,1), where Gamma(a,x) is the upper incomplete Gamma function. - Vladimir Reshetnikov, Oct 29 2015

Corrections and more information from Michael Somos, Feb 19 2003

A013594 Smallest order of cyclotomic polynomial containing n or -n as a coefficient.

Original entry on oeis.org

0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305, 11305, 15015, 11305, 17255, 17255, 20615, 20615, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565, 26565

Offset: 1

N. J. A. Sloane

This sequence is infinite - see the Lang reference.

An alternative version would start with 1 rather than 0.

			a(2)=105 because cyclotomic(105) contains "-2" as coefficient, but for n < 105 cyclotomic(n) does not contain 2 or -2.
x^105 - 1 = ( - 1 + x)(1 + x + x^2)(1 + x + x^2 + x^3 + x^4)(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)(1 - x + x^3 - x^4 + x^5 - x^7 + x^8)(1 - x + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12)(1 - x + x^5 - x^6 + x^7 - x^8 + x^10 - x^11 + x^12 - x^13 + x^14 - x^16 + x^17 - x^18 + x^19 - x^23 + x^24)(1 + x + x^2 - x^5 - x^6 - 2x^7 - x^8 - x^9 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 - x^20 - x^22 - x^24 - x^26 - x^28 + x^31 + x^32 + x^33 + x^34 + x^35 + x^36 - x^39 - x^40 - 2x^41 - x^42 - x^43 + x^46 + x^47 + x^48)

Bateman, C. Pomerance and R. C. Vaughan, Colloq. Math. Soc. Janos Bolyai, 34 (1984), 171-202.
S. Lang, Algebra: 3rd edition, Addison-Wesley, 1993, p. 281.
Maier, Prog. Math. 85 (Birkhaueser), 1990, 349-366.
Maier, Prog. Math. 139 (Birkhaueser) 1996, 633-638.

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Arnold and M. Monagan, Calculating cyclotomic polynomials of very large height.
P. Erdős and R. C. Vaughan, Bounds for the r-th coefficients of cyclotomic polynomials, J. London Math. Soc. (2) 8 (1974), 393-400 (MR50 #9835; Zentralblatt 295.10014).
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
H. Maier, Cyclotomic polynomials with large coefficients, Acta Arith. 64 (1993), 227-235.
H. Maier, Cyclotomic polynomials whose orders contain many prime factors, Period. Math. Hungar. 43 (2001), 155-169.
H. L. Montgomery and R. C. Vaughan, The order of magnitude of the mth coefficients of cyclotomic polynomials, Glasgow Math. J. 27 (1985), 143-159.
R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1974), 289-295 (1975).
M. Wallner, Lattice Path Combinatorics, Diplomarbeit, Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien, 2013.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A046887, A013595, A013596, A063696, A063698, A134518, A137979.

Table[Position[Table[Max[Abs[Flatten[CoefficientList[Transpose[FactorList[x^i - 1]][[1]], x]]]], {i, 1, 10000}], j][[1]], {j, 1, 10}] (* Ian Miller, Feb 25 2008 *)

nm=6545; m=0; forstep(n=1, nm, 2, if(issquarefree(n), p=polcyclo(n); o=poldegree(p); for(k=0, o, a=abs(polcoeff(p, k)); if(a>m, m=a; print([m, n, factor(n)])))))

More terms from Eric W. Weisstein

Further terms from T. D. Noe, Oct 29 2007

A003504 a(0)=a(1)=1; thereafter a(n+1) = (1/n)*Sum_{k=0..n} a(k)^2 (a(n) is not always integral!).

Original entry on oeis.org

1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856

Offset: 0

N. J. A. Sloane, R. K. Guy

The sequence appears with a different offset in some other sources. - Michael Somos, Apr 02 2006
Also known as Göbel's (or Goebel's) Sequence. Asymptotically, a(n) ~ n*C^(2^n) where C=1.0478... (A115632). A more precise asymptotic formula is given in A116603. - M. F. Hasler, Dec 12 2007
Let s(n) = (n-1)*a(n). By considering the p-adic representation of s(n) for primes p=2,3,...,43, one finds that a(44) is the first nonintegral value in this sequence. Furthermore, for n>44, the valuation of s(n) w.r.t. 43 is -2^(n-44), implying that both s(n) and a(n) are nonintegral. - M. F. Hasler and Max Alekseyev, Mar 03 2009
a(44) is approximately 5.4093*10^178485291567. - Hans Havermann, Nov 14 2017.
The fractional part is simply 24/43 (see page 709 of Guy (1988)).
The more precise asymptotic formula is a(n+1) ~ C^(2^n) * (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...). - Michael Somos, Mar 17 2012

			a(3) = (1 * 2 + 2^2) / 2 = 3 given a(2) = 2.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Sect. E15.
Clifford Pickover, A Passion for Mathematics, 2005.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..16
Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 1.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
H. Ibstedt, Some sequences of large integers, Fibonacci Quart. 28 (1990), 200-203.
Yuh Kobayashi and Shin-ichiro Seki, On the length over which k-Göbel sequences remain integers, arXiv:2502.17448 [math.CO], 2025.
H. W. Lenstra, Jr., R. K. Guy, and N. J. A. Sloane, Correspondence, 1975-1978
N. Lygeros and M. Mizony, Study of primality of terms of a_k(n)=(1+(sum from 1 to n-1)(a_k(i)^k))/(n-1)
Rinnosuke Matsuhira, Toshiki Matsusaka, and Koki Tsuchida, How long can k-Göbel sequences remain integers?, arXiv:2307.09741 [math.NT], 2023.
D. Rusin, Law of small numbers [Broken link]
D. Rusin, Law of small numbers [Cached copy]
Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
Eric Weisstein's World of Mathematics, Göbel's Sequence
D. Zagier, Problems posed at the St Andrews Colloquium, 1996
D. Zagier, Solution: Day 5, problem 3

Cf. A005166, A005167, A097398, A108394, A115632, A116603 (asymptotic formula).

a:=2: L:=1,1,a: n:=15: for k to n-2 do a:=a*(a+k)/(k+1): L:=L,a od:L; # Robert FERREOL, Nov 07 2015

a[n_] := a[n] = Sum[a[k]^2, {k, 0, n-1}]/(n-1); a[0] = a[1] = 1; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Feb 06 2013 *)
With[{n = 14}, Nest[Append[#, (#.#)/(Length[#] - 1)] &, {1, 1}, n - 2]] (* Jan Mangaldan, Mar 21 2013 *)

A003504(n,s=2)=if(n-->0,for(k=1,n-1,s+=(s/k)^2);s/n,1) \\ M. F. Hasler, Dec 12 2007

a=2; L=[1,1,a]; n=15
for k in range(1,n-1):
    a=a*(a+k)//(k+1)
    L.append(a)
print(L) # Robert FERREOL, Nov 07 2015

a(n+1) = ((n-1) * a(n) + a(n)^2) / n if n > 1. - Michael Somos, Apr 02 2006

0 = a(n)*(+a(n)*(a(n+1) - a(n+2)) - a(n+1) - a(n+1)^2) +a(n+1)*(a(n+1)^2 - a(n+2)) if n>1. - Michael Somos, Jul 25 2016

a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral - H. W. Lenstra, Jr.
Corrected and extended by M. F. Hasler, Dec 12 2007
Further corrections from Max Alekseyev, Mar 04 2009

A001521 a(1) = 1; thereafter a(n+1) = floor(sqrt(2a(n)(a(n)+1))).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 54, 77, 109, 154, 218, 309, 437, 618, 874, 1236, 1748, 2472, 3496, 4944, 6992, 9888, 13984, 19777, 27969, 39554, 55938, 79108, 111876, 158217, 223753, 316435, 447507, 632871, 895015, 1265743, 1790031, 2531486, 3580062, 5062972

Offset: 1

N. J. A. Sloane

Graham and Pollak give an elementary proof of the following result: For given m, define a(n) by a(1) = m and a(n+1) = floor(sqrt(2*a_n*(a_n + 1))), n >= 1. Then a(n) = tau_m(2^((n-1)/2) + 2^((n-2)/2)) where tau_m is the m-th smallest element of {1, 2, 3, ... } union { sqrt(2), 2*sqrt(2), 3*sqrt(2), ... }. For m=1 it follows as a curious corollary that a(2n+1) - 2*a(2n-1) is exactly the n-th bit in the binary expansion of sqrt(2) (A004539).

a(n) is also the curvature (rounded down) of the circle inscribed in the n-th 45-45-90 triangle arranged in a spiral as shown in the illustration in the links section. - Kival Ngaokrajang, Aug 21 2013

R. L. Graham, D. E. Knuth and O. Pataschnic, Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed., Ex. 3.46.
F. K. Hwang, and Shen Lin. "An analysis of Ford and Johnson's sorting algorithm." In Proc. Third Annual Princeton Conf. on Inform. Sci. and Systems, pp. 292-296. 1969.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harvey P. Dale, Table of n, a(n) for n = 1..5000 (first 200 terms from T. D. Noe)
R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related to sqrt(2), Mathematics Magazine, Volume 43, Pages 143-145, 1970. Zbl 201.04705.
R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related to sqrt(2) (annotated and scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Kival Ngaokrajang, Illustration for some initial terms
S. Rabinowitz and P. Gilbert, A nonlinear recurrence yielding binary digits, Math. Mag. 64 (1991), no. 3, 168-171.
Th. Stoll, On Families of Nonlinear Recurrences Related to Digits, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.2.

Cf. A000196.

First, second, and third differences give A017911, A190660, A241576.

a001521 n = a001521_list !! (n-1)
a001521_list = 1 : (map a000196 $ zipWith (*)
                    (map (* 2) a001521_list) (map (+ 1) a001521_list))
-- Reinhard Zumkeller, Dec 16 2013

[Floor(Sqrt(2)^(n-1)+Sqrt(2)^(n-2)): n in [1..45]]; // Vincenzo Librandi, May 24 2015

Digits:=200;
f:=proc(n) option remember;
if n=1 then 1 else floor(sqrt(2*f(n-1)*(f(n-1)+1))); fi; end;
[seq(f(n),n=1..200)];

With[{c=Sqrt[2]},Table[Floor[c^(n-1)+c^(n-2)],{n,1,50}]] (* Harvey P. Dale, May 11 2011 *)
NestList[Floor[Sqrt[2#(#+1)]]&,1,50] (* Harvey P. Dale, Aug 28 2013 *)

a(n)=if(n>1, sqrtint(2^(n-1)) + sqrtint(2^(n-2)), 1) \\ Charles R Greathouse IV, Nov 27 2016

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, v[k]=sqrtint(2*(v[k-1]+1)*v[k-1])); v \\ Charles R Greathouse IV, Jan 23 2020

[floor(sqrt(2)^(n-1))+ floor(sqrt(2)^(n-2)) for n in (1..50)] # Bruno Berselli, May 25 2015

a(n) = floor( sqrt(2)^(n-1) ) + floor( sqrt(2)^(n-2) ), n>1. - Ralf Stephan, Sep 18 2004

k * sqrt(2)^n - 2 < a(n) < k * sqrt(2)^n, where k = (1 + sqrt(2))/2 = A174968 = 1.2071.... Probably the first inequality can be improved (!). - Charles R Greathouse IV, Jan 23 2020

Additional comments from Torsten Sillke, Apr 06 2001

A001272 Numbers k such that k! - (k-1)! + (k-2)! - (k-3)! + ... - (-1)^k*1! is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961

Offset: 1

N. J. A. Sloane

At present the terms greater than or equal to 2653 are only probable primes.
Živković shows that all terms must be less than p = 3612703, which divides the alternating factorial af(k) for k >= p. - T. D. Noe, Jan 25 2008
Notwithstanding Živković's wording, p = 3612703 also divides the alternating factorial for k = 3612702. [Guy: If there is a value of k such that k + 1 divides af(k), then k + 1 will divide af(m) for all m > k.] Therefore af(3612701), approximately 7.3 * 10^22122513, is the final primality candidate. - Hans Havermann, Jun 17 2013
Next term (if it exists) has k > 100000 (per M. Rodenkirch post). - Eric W. Weisstein, Dec 18 2017

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 160, p. 52, Ellipses, Paris 2008.
Martin Gardner, Strong Laws of Small Primes, in The Last Recreations, p. 198 (1997).
R. K. Guy, Unsolved Problems in Number Theory, B43.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 97.

factordb.com, Primality at factor-database of the alternating factorial for n=661.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
M. Rodenkirch, Alternating Factorials.
Eric Weisstein's World of Mathematics, Alternating Factorial.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Miodrag Živković, The number of primes Sum_{i=1..n} (-1)^(n-i)*i! is finite, Math. Comp. 68 (1999), no. 225, 403-409.
Index entries for sequences related to factorial numbers.

Cf. A005165, A002981, A002982, A100289.

with(numtheory); f := proc(n) local i; add((-1)^(n-i)*i!,i=1..n); end; isprime(f(15));

(* This program is not convenient for more than 15 terms *) Reap[For[n = 1, n <= 1000, n++, If[PrimeQ[Sum[(-1)^(n - k) * k!, {k, n}]], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Sep 05 2013 *)
Position[AlternatingFactorial[Range[200]], ?PrimeQ] // Flatten (* _Eric W. Weisstein, Sep 19 2017 *)

661 found independently by Eric W. Weisstein and Rachel Lewis (racheljlewis(AT)hotmail.com); 2653 and 3069 found independently by Chris Nash (nashc(AT)lexmark.com) and Rachel Lewis (racheljlewis(AT)hotmail.com)
3943, 4053, 4998 found by Paul Jobling (paul.jobling(AT)whitecross.com)
8275, 9158 found by team of Rachel Lewis, Paul Jobling and Chris Nash
661! - 660! + 659! - ... was shown to be prime by team of Giovanni La Barbera and others using the Certifix program developed by Marcel Martin, Jul 15 2000 (see link) - Paul Jobling (Paul.Jobling(AT)WhiteCross.com) and Giovanni La Barbera, Aug 02 2000
a(23) = 11164 found by Paul Jobling, Nov 25 2004
Edited by T. D. Noe, Oct 30 2008
Edited by Hans Havermann, Jun 17 2013
a(24) = 43592 from Serge Batalov, Jul 19 2017
a(25) = 59961 from Mark Rodenkirch, Sep 18 2017

A071828 Primes of the form Sum_{i=1..k} (-1)^(k-i)*i!.

Original entry on oeis.org

5, 19, 101, 619, 4421, 35899, 3301819, 1226280710981, 115578717622022981, 32656499591185747972776747396512425885838364422981

Offset: 1

Benoit Cloitre, Jun 08 2002

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B43.

Charles R Greathouse IV, Table of n, a(n) for n = 1..14
M. Zivkovic, The number of primes sum_(i=1..n)(-1)^(n-i)i! is finite, Math. Comp., vol. 68 (1999), pp. 403-409.

Cf. A001272, A005165.

l = Abs@Accumulate[(-1)^#*#! & /@ Range@99]; l[[Flatten@Position[l, ?PrimeQ]]] (* _Hans Rudolf Widmer, Feb 27 2023 *)

for(n=1,120,if(isprime(abs(sum(i=1,n,(-1)^i*i!)))==1,print1(abs(sum(i=1,n,(-1)^i*i!)),",")))

v=[];for(n=1,1000, if(ispseudoprime(t=abs(sum(i=1,n,(-1)^i*i!))), v=concat(v,t))); v \\ Charles R Greathouse IV, Feb 14 2011

a(n) = A005165(A001272(n)).

Edited by R. J. Mathar, Aug 28 2007

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