A040006 -id:A040006 - cp's OEIS frontend

A040000 a(0)=1; a(n)=2 for n >= 1.

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

N. J. A. Sloane, Dec 11 1999

Continued fraction expansion of sqrt(2) is 1 + 1/(2 + 1/(2 + 1/(2 + ...))).
Inverse binomial transform of Mersenne numbers A000225(n+1) = 2^(n+1) - 1. - Paul Barry, Feb 28 2003
A Chebyshev transform of 2^n: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)). - Paul Barry, Oct 31 2004
An inverse Catalan transform of A068875 under the mapping g(x)->g(x(1-x)). A068875 can be retrieved using the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. A040000 and A068875 may be described as a Catalan pair. - Paul Barry, Nov 14 2004
Sequence of electron arrangement in the 1s 2s and 3s atomic subshells. Cf. A001105, A016825. - Jeremy Gardiner, Dec 19 2004
Binomial transform of A165326. - Philippe Deléham, Sep 16 2009
Let m=2. We observe that a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k). Then there is a link with A113311 and A115291: it is the same formula with respectively m=3 and m=4. We can generalize this result with the sequence whose g.f. is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
With offset 1: number of permutations where |p(i) - p(i+1)| <= 1 for n=1,2,...,n-1. This is the identical permutation and (for n>1) its reversal.
Equals INVERT transform of bar(1, 1, -1, -1, ...).
Eventual period is (2). - Zak Seidov, Mar 05 2011
Also decimal expansion of 11/90. - Vincenzo Librandi, Sep 24 2011
a(n) = 3 - A054977(n); right edge of the triangle in A182579. - Reinhard Zumkeller, May 07 2012
With offset 1: minimum cardinality of the range of a periodic sequence with (least) period n. Of course the range's maximum cardinality for a purely periodic sequence with (least) period n is n. - Rick L. Shepherd, Dec 08 2014
With offset 1: n*a(1) + (n-1)*a(2) + ... + 2*a(n-1) + a(n) = n^2. - Warren Breslow, Dec 12 2014
With offset 1: decimal expansion of gamma(4) = 11/9 where gamma(n) = Cp(n)/Cv(n) is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Sep 11 2016
a(n) equals the number of binary sequences of length n where no two consecutive terms differ. Also equals the number of binary sequences of length n where no two consecutive terms are the same. - David Nacin, May 31 2017
a(n) is the period of the continued fractions for sqrt((n+2)/(n+1)) and sqrt((n+1)/(n+2)). - A.H.M. Smeets, Dec 05 2017
Also, number of self-avoiding walks and coordination sequence for the one-dimensional lattice Z. - Sean A. Irvine, Jul 27 2020

			sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))). - _Harry J. Smith_, Apr 21 2009
G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + ...
11/90 = 0.1222222222222222222... - _Natan Arie Consigli_, Sep 11 2016

A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 276-278.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Bruce Fang, Pamela E. Harris, Brian M. Kamau, and David Wang, Vacillating parking functions, arXiv:2402.02538 [math.CO], 2024.
Kshitij Education, Molar specific heat.
MathPath, Square-roots via Continued Fractions.
Narad Rampersad and Max Wiebe, Sums of products of binomial coefficients mod 2 and 2-regular sequences, arXiv:2309.04012 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Square root.
Eric Weisstein's World of Mathematics, Pythagoras's Constant.
Wikipedia, Poisson's constant.
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for eventually constant sequences.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (1).

Convolution square is A008574.
See A003945 etc. for (1+x)/(1-k*x).
From Jaume Oliver Lafont, Mar 26 2009: (Start)
Sum_{0<=k<=n} a(k) = A005408(n).
Prod_{0<=k<=n} a(k) = A000079(n). (End)
Cf. A113311, A115291, A171418, A171440, A171441, A171442, A171443.
Cf. A000674 (boustrophedon transform).
Cf. A001333/A000129 (continued fraction convergents).
Cf. A000122, A002193 (sqrt(2) decimal expansion), A006487 (Egyptian fraction).
Cf. Other continued fractions for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040002 (contfrac(sqrt(5)) = (2,4,4,...)), A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).

a040000 0 = 1; a040000 n = 2
a040000_list = 1 : repeat 2  -- Reinhard Zumkeller, May 07 2012

Digits := 100: convert(evalf(sqrt(2)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[2],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
a[ n_] := 2 - Boole[n == 0]; (* Michael Somos, Dec 28 2014 *)
PadRight[{1},120,2] (* or *) RealDigits[11/90, 10, 120][[1]] (* Harvey P. Dale, Jul 12 2025 *)

{a(n) = 2-!n}; /* Michael Somos, Apr 16 2007 */

a(n)=1+sign(n)  \\ Jaume Oliver Lafont, Mar 26 2009

allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(2)); for (n=0, 20000, write("b040000.txt", n, " ", x[n+1]));  \\ Harry J. Smith, Apr 21 2009

G.f.: (1+x)/(1-x). - Paul Barry, Feb 28 2003
a(n) = 2 - 0^n; a(n) = Sum_{k=0..n} binomial(1, k). - Paul Barry, Oct 16 2004
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*2^(n-2*k)/(n-k). - Paul Barry, Oct 31 2004
A040000(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A068875(n-k). - Paul Barry, Nov 14 2004
From Michael Somos, Apr 16 2007: (Start)
Euler transform of length 2 sequence [2, -1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-v)*(u+v) - 2*v*(u-w).
E.g.f.: 2*exp(x) - 1.
a(n) = a(-n) for all n in Z (one possible extension to n<0). (End)
G.f.: (1-x^2)/(1-x)^2. - Jaume Oliver Lafont, Mar 26 2009
G.f.: exp(2*atanh(x)). - Jaume Oliver Lafont, Oct 20 2009
a(n) = Sum_{k=0..n} A108561(n,k)*(-1)^k. - Philippe Deléham, Nov 17 2013
a(n) = 1 + sign(n). - Wesley Ivan Hurt, Apr 16 2014
10 * 11/90 = 11/9 = (11/2 R)/(9/2 R) = Cp(4)/Cv(4) = A272005/A272004, with R = A081822 (or A070064). - Natan Arie Consigli, Sep 11 2016
a(n) = A001227(A000040(n+1)). - Omar E. Pol, Feb 28 2018

A010467 Decimal expansion of square root of 10.

Original entry on oeis.org

3, 1, 6, 2, 2, 7, 7, 6, 6, 0, 1, 6, 8, 3, 7, 9, 3, 3, 1, 9, 9, 8, 8, 9, 3, 5, 4, 4, 4, 3, 2, 7, 1, 8, 5, 3, 3, 7, 1, 9, 5, 5, 5, 1, 3, 9, 3, 2, 5, 2, 1, 6, 8, 2, 6, 8, 5, 7, 5, 0, 4, 8, 5, 2, 7, 9, 2, 5, 9, 4, 4, 3, 8, 6, 3, 9, 2, 3, 8, 2, 2, 1, 3, 4, 4, 2, 4, 8, 1, 0, 8, 3, 7, 9, 3, 0, 0, 2, 9

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 3 followed by {6} repeated. - Harry J. Smith, Jun 02 2009
In 1594, Joseph Scaliger claimed Pi = sqrt(10), but Ludolph van Ceulen immediately knew this to be wrong. - Alonso del Arte, Jan 17 2013
The 7th-century Hindu mathematician Brahmagupta used this constant as value of Pi. - Stefano Spezia, Jul 08 2022

			3.162277660168379331998893544432718533719555139325216826857504852792594...

Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 27.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 112.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 238.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.4, p. 201.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, pp. 89, 91.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 55.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Josep M. Brunat and Joan-Carles Lario, A problem on concatenated integers, arXiv:2103.05306 [math.NT], 2021. For 1/sqrt(10).
Jason Kimberley, Index of expansions of sqrt(d) in base b.
Robert Nemiroff and Jerry Bonnell, The first 1 million digits of square root of 10.
Robert Nemiroff and Jerry Bonnell, Plouffe's Inverter, The first 1 million digits of square root of 10.
J. J. O'Connor and E. F. Robertson, Ludolph Van Ceulen.
Index entries for algebraic numbers, degree 2

Cf. A000032, A040006 (continued fraction).

SetDefaultRealField(RealField(100)); Sqrt(10); // Vincenzo Librandi, Feb 15 2020

RealDigits[N[Sqrt[10],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

default(realprecision, 20080); x=sqrt(10); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010467.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

Sqrt(10) = sqrt(1 + i*sqrt(15)) + sqrt(1 - i*sqrt(15)) = sqrt(1/2 + 2*i*sqrt(5)) + sqrt(1/2 - 2*i*sqrt(5)), where i = sqrt(-1). - Bruno Berselli, Nov 20 2012
Equals 1/sqrt(10), with offset 0. - Michel Marcus, Mar 10 2021
Equals 2 + Sum_{k>=1} Lucas(k)*binomial(2*k,k)/8^k. - Amiram Eldar, Jan 17 2022
a(k) = floor(Sum_{n>=1} A005875(n)/exp(Pi*n/(10^((2/3)*k+(1/3))))) mod 10. Will give the k-th decimal digit of sqrt(10). A005875 : number of ways to write n as sum of 3 squares. - Simon Plouffe, Dec 30 2023

A005667 Numerators of continued fraction convergents to sqrt(10).

Original entry on oeis.org

1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, 243289797, 1499219281, 9238605483, 56930852179, 350823718557, 2161873163521, 13322062699683, 82094249361619, 505887558869397, 3117419602578001, 19210405174337403, 118379850648602419

Offset: 0

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

a(2*n+1) with b(2*n+1) := A005668(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = -1, a(2*n) with b(2*n) := A005668(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 10*b^2 = +1 (cf. Emerson reference).
Bisection: a(2*n) = T(n,19) = A078986(n), n >= 0 and a(2*n+1) = 3*S(2*n, 2*sqrt(10)), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. See A053120, resp. A049310.
The initial 1 corresponds to a denominator 0 in A005668. But according to standard conventions, a continued fraction starts with b(0) = integer part of the number, and the sequence of convergents p(n)/q(n) start with (p(0),q(0)) = (b(0),1). A fraction 1/0 has no mathematical meaning, the only justification is that initial terms p(-1) = 1, q(-1) = 0 are consistent with the recurrent relations p(n) = b(n)*p(n-1) + b(n-2) and the same for q(n). - M. F. Hasler, Nov 02 2019

			G.f. = 1 + 3*x + 19*x^2 + 117*x^3 + 721*x^4 + 4443*x^5 + 27379*x^6 + ... - _Michael Somos_, Jul 14 2018

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Thm. 1, p. 233.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Tian-Xiao He and Peter J.-S. Shiue, Identities for linear recursive sequences of order 2, Elect. Res. Archive (2021) Vol. 29, No. 5, 3489-3507.
Tanya Khovanova, Recursive Sequences
Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,1).

Cf. A010467, A040006, A084134, A005668 (denominators).

I:=[1, 3]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 09 2013

A005667:=(-1+3*z)/(-1+6*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

Join[{1},Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[10],n]]],{n,1,30}]] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
CoefficientList[Series[(1-3x)/(1-6x-x^2), {x,0,30}], x] (* Vincenzo Librandi, Jun 09 2013 *)
Join[{1},Numerator[Convergents[Sqrt[10],30]]] (* or *) LinearRecurrence[ {6,1},{1,3},30] (* Harvey P. Dale, Aug 22 2016 *)
a[ n_] := (-I)^n ChebyshevT[ n, 3 I]; (* Michael Somos, Jul 14 2018 *)
LucasL[Range[0,30], 6]/2 (* G. C. Greubel, Jun 06 2019 *)

a(n)=([0,1;1,6]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

((1-3*x)/(1-6*x-x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 06 2019

a(n) = 6*a(n-1) + a(n-2).
G.f.: (1-3*x)/(1-6*x-x^2).
a(n) = ((-i)^n)*T(n, 3*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1.
From Paul Barry, Nov 15 2003: (Start)
Binomial transform of A084132.
E.g.f.: exp(3*x)*cosh(sqrt(10)*x).
a(n) = ((3+sqrt(10))^n + (3-sqrt(10))^n)/2.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k) * 10^k * 3^(n-2*k). (End)
a(n) = (-1)^n * a(-n) for all n in Z. - Michael Somos, Jul 14 2018 [This refers to the sequence extended to negative indices according to the recurrence relation, but not to the sequence as it is currently defined. - M. F. Hasler, Nov 02 2019]
a(n) = Lucas(n,6)/2, Lucas polynomial, L(n,x), evaluated at x = 6. - G. C. Greubel, Jun 06 2019

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

A040002 Continued fraction for sqrt(5).

Original entry on oeis.org

2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

N. J. A. Sloane

Decimal expansion of 11/45. - Natan Arie Consigli, Jan 19 2016

			2.236067977499789696409173668... = 2 + 1/(4 + 1/(4 + 1/(4 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 01 2009

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index of eventually constant sequences
Index of divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A002163 (decimal expansion), A001077/A001076 (convergents), A248235 (Egyptian fraction).
Cf. Continued fraction for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040000 (contfrac(sqrt(2)) = (1,2,2,...)), A040002, A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).
Essentially the same as A010709.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[5],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PadRight[{2},120,{4}] (* Harvey P. Dale, Jul 06 2019 *)

{ allocatemem(932245000); default(realprecision, 26000); x=contfrac(sqrt(5)); for (n=0, 20000, write("b040002.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

a(0) = 2, a(n) = 4 n>0. - Natan Arie Consigli, Jan 19 2016
From Elmo R. Oliveira, Feb 16 2024: (Start)
G.f.: 2*(1+x)/(1-x).
E.g.f.: 4*exp(x) - 2.
a(n) = 2*A040000(n). (End)

A182797 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the k X k X k triangular grid.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 24, 5, 0, 0, 6, 192, 60, 6, 0, 0, 6, 2112, 1620, 120, 7, 0, 0, 6, 32640, 98820, 7680, 210, 8, 0, 0, 6, 718080, 13638780, 1574400, 26250, 336, 9, 0, 0, 6, 22665216, 4260983940, 1034019840, 13676250, 72576, 504, 10

Offset: 1

Alois P. Heinz, Dec 02 2010

The k X k X k triangular grid has k rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(k) vertices and 3*A000217(k-1) edges altogether.

The coefficients of the chromatic polynomials for the column sequences are given by the rows of A193283. - Georg Fischer, Jul 31 2023

			Square array A(n,k) begins:
  1,   0,    0,       0,          0,             0,  ...
  2,   0,    0,       0,          0,             0,  ...
  3,   6,    6,       6,          6,             6,  ...
  4,  24,  192,    2112,      32640,        718080,  ...
  5,  60, 1620,   98820,   13638780,    4260983940,  ...
  6, 120, 7680, 1574400, 1034019840, 2175789895680,  ...

Wikipedia, Triangular grid graph
Wikipedia, Chromatic polynomial

Columns k=1-11 give: A000027, A007531, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796.
Rows n=1-10 give: A000007(k-1), A000038(k-1), A040006(k-1), A182798, A153467*4, A153468*5, A153469*6, A153470*7, A153471*8, A153472*9, A153473*10.
Cf. A000217, A193283.

A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 48, 5, 0, 0, 6, 1056, 180, 6, 0, 0, 6, 45696, 32940, 480, 7, 0, 0, 6, 4034304, 30847500, 393600, 1050, 8, 0, 0, 6, 739642368, 148039757460, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 02 2012

The rhombic hexagonal square grid graph RH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges; see A212162 for example. The chromatic polynomial of RH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212195 first at (n,k) = (4,5): A(4,5) = 4034304, A212195(4,5) = 4038432.

			Square array A(n,k) begins:
  1,    0,       0,            0,                 0, ...
  2,    0,       0,            0,                 0, ...
  3,    6,       6,            6,                 6, ...
  4,   48,    1056,        45696,           4034304, ...
  5,  180,   32940,     30847500,      148039757460, ...
  6,  480,  393600,   3312560640,   286169360240640, ...
  7, 1050, 2735250, 123791435250, 97337270132408250, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068246, 6*A068247.
Rows n=1-15 give: A000007, A000038, A040006, 4*A068271, 5*A068272, 6*A068273, 7*A068274, 8*A068275, 9*A068276, 10*A068277, 11*A068278, 12*A068279, 13*A068280, 14*A068281, 15*A068282.
Cf. A212162, A212195, A208054, A208050.

A212195 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the staggered hexagonal square grid graph SH_(k,k).

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 48, 5, 0, 0, 6, 1056, 180, 6, 0, 0, 6, 45696, 32940, 480, 7, 0, 0, 6, 4038432, 30847500, 393600, 1050, 8, 0, 0, 6, 743601024, 148046704020, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 03 2012

The staggered hexagonal square grid graph SH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges; see A212194 for example. The chromatic polynomial of SH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212163 first at (n,k) = (4,5): A(4,5) = 4038432, A212163(4,5) = 4034304.

			Square array A(n,k) begins:
  1,    0,       0,            0,                 0, ...
  2,    0,       0,            0,                 0, ...
  3,    6,       6,            6,                 6, ...
  4,   48,    1056,        45696,           4038432, ...
  5,  180,   32940,     30847500,      148046704020, ...
  6,  480,  393600,   3312560640,   286170443437440, ...
  7, 1050, 2735250, 123791435250, 97337320223288250, ...

Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068248, 6*A068249.
Rows n=1-10, 16-18 give: A000007, A000038, A040006, 4*A068283, 5*A068284, 6*A068285, 7*A068286, 8*A068287, 9*A068288, 10*A068289, 16*A068290, 17*A068291, 18*A068292.
Cf. A212163, A212194.

A020793 Decimal expansion of 1/6.

Original entry on oeis.org

1, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

N. J. A. Sloane

Except for the first term identical to A010722, A040006 and A021019. Except for the first terms the same as A021028, A021100, A021388, A071279, A101272, A168608, A177057,... - M. F. Hasler, Oct 24 2011

Decimal expansion of gamma(1) = 5/3 (with offset 1), where gamma(n) = Cp(n)/Cv(n) = is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Jul 10 2016

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Springer, 2013, see p. 224.

Wikipedia, Poisson's constant.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010722, A021019, A021028, A021100, A021388, A040006, A070064, A071279, A081822, A101272, A168608, A177057, A272001, A272002.

RealDigits[1/6,10,120][[1]] (* or *) PadRight[{1},120,{6}] (* Harvey P. Dale, Dec 30 2018 *)

a(n)=6-5*!n  \\ M. F. Hasler, Oct 24 2011

a(n) = 6^n mod 10. - Zerinvary Lajos, Nov 26 2009
Equals Sum_{k>=1} 1/7^k. - Bruno Berselli, Jan 03 2014
10 * 1/6 = 5/3 = (5/2 R)/(3/2 R) = Cp(1)/Cv(1) = A272002/A272001, with R = A081822 (or A070064). - Natan Arie Consigli, Jul 10 2016
G.f.: (1 + 5*x)/(1 - x). - Ilya Gutkovskiy, Jul 10 2016
Equals Sum_{k>=1} 1/(k*Pi)^2. - Maciej Kaniewski, Sep 14 2017
Equals Sum_{k>=1} (zeta(2*k)-1)/4^k. - Amiram Eldar, Jun 08 2021
K_{n>=2} 2*n/(2*n - 3) = 5/3. (see Clawson at p. 224). - Stefano Spezia, Jul 01 2024
E.g.f.: 6*exp(x) - 5. - Elmo R. Oliveira, Aug 05 2024

A040072 Continued fraction for sqrt(82).

Original entry on oeis.org

9, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18

Offset: 0

N. J. A. Sloane

			9.05538513813741662657380... = 9 + 1/(18 + 1/(18 + 1/(18 + 1/(18 + ...)))). - _Harry J. Smith_, Jun 10 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010533 (decimal expansion), A041144/A041145 (convergents), A248305 (Egyptian fraction).

Cf. A040000, A040006.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[82],300] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
PadRight[{9},120,{18}] (* Harvey P. Dale, Oct 09 2020 *)

{ allocatemem(932245000); default(realprecision, 51000); x=contfrac(sqrt(82)); for (n=0, 20000, write("b040072.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 10 2009

From Elmo R. Oliveira, Feb 10 2024: (Start)
a(n) = 18 = A010857(n) for n >= 1.
G.f.: 9*(1+x)/(1-x).
E.g.f.: 18*exp(x) - 9.
a(n) = 9*A040000(n) = 3*A040006(n). (End)

A040030 Continued fraction for sqrt(37).

Original entry on oeis.org

6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

N. J. A. Sloane

			6.08276253029821968899968... = 6 + 1/(12 + 1/(12 + 1/(12 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 04 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010491 (decimal expansion), A041060/A041061 (convergents), A248263 (Egyptian fraction).

Cf. A040000, A040002, A040006, A010851

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[37],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{6},120,{12}] (* Harvey P. Dale, Jan 02 2017 *)

{ allocatemem(932245000); default(realprecision, 44000); x=contfrac(sqrt(37)); for (n=0, 20000, write("b040030.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

From Elmo R. Oliveira, Feb 06 2024: (Start)
a(n) = 12 for n >= 1.
G.f.: 6*(1+x)/(1-x).
E.g.f.: 12*exp(x) - 6.
a(n) = 6*A040000(n) = 3*A040002(n) = 2*A040006(n). (End)

cp's OEIS Frontend

A040000 a(0)=1; a(n)=2 for n >= 1.

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A010467 Decimal expansion of square root of 10.

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A005667 Numerators of continued fraction convergents to sqrt(10).

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A040002 Continued fraction for sqrt(5).

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A182797 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the k X k X k triangular grid.

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A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

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A212195 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the staggered hexagonal square grid graph SH_(k,k).