A001570 -id:A001570 - cp's OEIS frontend

A006051 Square hex numbers.

1, 169, 32761, 6355441, 1232922769, 239180661721, 46399815451081, 9001325016847969, 1746210653453054881, 338755865444875798921, 65716891685652451935769, 12748738231151130799740241, 2473189499951633722697670961, 479786014252385791072548426169

Offset: 1

N. J. A. Sloane

Numbers n of the form n = y^2 = 3*x^2 - 3*x + 1.

			G.f. = x + 169*x^2 + 32761*x^3 + 6355441*x^4 + 1232922769*x^5 + ...

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 19.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..435
M. Gardner & N. J. A. Sloane, Correspondence, 1973-74
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55.
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
Sociedad Magic Penny Patagonia, Leonardo en Patagonia
Eric Weisstein's World of Mathematics, Hex Number.
Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Cf. A003500.
Intersection of A000290 and A003215.
Values of x are given by A001922, values of y by A001570.

[(7*Evaluate(ChebyshevSecond(n),97) - 7*Evaluate(ChebyshevU(n-1), 97) + 1)/8: n in [1..30]]; // G. C. Greubel, Nov 04 2017; Oct 07 2022

Rest@ CoefficientList[Series[x(1-26x+x^2)/((1-x)(1-194x+x^2)), {x,0,20}], x] (* Michael De Vlieger, Jan 02 2017 *)
LinearRecurrence[{195,-195,1},{1,169,32761},20] (* Harvey P. Dale, Nov 03 2017 *)

{a(n) = sqr( real( (2 + quadgen( 12)) ^ (2*n - 1)) / 2)} /* Michael Somos, Feb 15 2011 */

def A006051(n): return (7*chebyshev_U(n-1,97) - 7*chebyshev_U(n-2,97) + 1)/8
[A006051(n) for n in range(1,31)] # G. C. Greubel, Oct 07 2022

a(n) = A001570(n)^2.
a(1 - n) = a(n).
G.f.: x * (1 - 26*x + x^2) / ((1 - x) * (1 - 194*x + x^2)). - Simon Plouffe in his 1992 dissertation
a(n) = 194*a(n-1) - a(n-2) - 24, a(1)=1, a(2)=169. - James Sellers, Jul 04 2000
a(n+1) = A003215(A001921(n)). - Joerg Arndt, Jan 02 2017
a(n) = (1/8)*(1 + 7*(ChebyshevU(n-1, 97) - ChebyshevU(n-2, 97))). - G. C. Greubel, Oct 07 2022

A094347 a(n) = 14*a(n-1) - a(n-2); a(0) = a(1) = 2.

Original entry on oeis.org

2, 2, 26, 362, 5042, 70226, 978122, 13623482, 189750626, 2642885282, 36810643322, 512706121226, 7141075053842, 99462344632562, 1385331749802026, 19295182152595802, 268747218386539202, 3743165875258953026

Offset: 0

Lekraj Beedassy, Jun 03 2004

Even x satisfying the Pellian x^2 - 3*y^2 = 1. For corresponding y see A028230.

Michael De Vlieger, Table of n, a(n) for n = 0..874
Christian Aebi and Grant Cairns, Less than Equable Triangles on the Eisenstein lattice, arXiv:2312.10866 [math.CO], 2023.
R. K. Guy, Letter to N. J. A. Sloane concerning A001075, A011943, A094347 [Scanned and annotated letter, included with permission]
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (14,-1).

a(n) = 2*A001570(n).
Bisection of A001075.
Cf. A028230.

LinearRecurrence[{14,-1},{2,2},40] (* or *) CoefficientList[ Series[2(1-13x)/(1-14x+x^2),{x,0,39}],x] (* Harvey P. Dale, Apr 23 2011 *)

(a[0]:2, a[1]:2, a[n] := 14*a[n - 1] - a[n-2], makelist(a[n], n, 0, 50)); /* Franck Maminirina Ramaharo, Nov 12 2018 */

G.f.: 2*(1 - 13*x)/(1 - 14*x + x^2). [Philippe Deléham, Nov 17 2008]
a(n) = ((2 + sqrt(3))^(2*n - 1) + (2 - sqrt(3))^(2*n - 1))/2. - Gerry Martens, Jun 03 2015
a(n) = (1/2)*sqrt(4 + (-2*sqrt(-2 + (7 - 4*sqrt(3))^(2*n) + (7 + 4*sqrt(3))^(2*n)) + sqrt(3)*sqrt(2 + (7 - 4*sqrt(3))^(2*n) + (7 + 4*sqrt(3))^(2*n)))^2). - Gerry Martens, Jun 03 2015
E.g.f.: exp(7*x)*(2*cosh(4*sqrt(3)*x) - sqrt(3)*sinh(4*sqrt(3)*x)). - Franck Maminirina Ramaharo, Nov 12 2018

Corrected by Lekraj Beedassy, Jun 11 2004

A103975 Smaller side in (a,a+1,a+1)-integer triangle with integer area.

Original entry on oeis.org

16, 240, 3360, 46816, 652080, 9082320, 126500416, 1761923520, 24540428880, 341804080816, 4760716702560, 66308229755040, 923554499868016, 12863454768397200, 179164812257692800, 2495443916839302016, 34757050023492535440, 484103256412056194160

Offset: 1

Zak Seidov, Feb 23 2005

Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A102341, A103974, A016064, A011945, A103772.

Corresponding areas are given by A104008.

a[n_] := 1/3 (-4 + (2 - Sqrt[3])^(1 + 2 n) + (2 + Sqrt[3])^(1 + 2 n)); A103975 = Expand[a /@ Range[1, 25]] (* Terentyev Oleg, Nov 12 2009 *)
LinearRecurrence[{15,-15,1},{16,240,3360},30] (* Harvey P. Dale, Apr 25 2012 *)

a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3) - Max Alekseyev, May 31 2007
a(n) = 2*A120892(2*n+1) - Max Alekseyev, May 31 2007
a(n) = (1/3)*((2 - sqrt(3))^(1 + 2*n) + (2 + sqrt(3))^(1 + 2*n) - 4). [Terentyev Oleg, Nov 12 2009]
a(n) = (4/3)*(A001570(n+1)-1).
G.f.: -16*x / ((x-1)*(x^2-14*x+1)). - Colin Barker, Apr 09 2013

More terms from Robert G. Wilson v, Mar 24 2005

More terms from Colin Barker, Apr 09 2013

A103772 Larger of two sides in a (k,k,k-1)-integer-sided triangle with integer area.

Original entry on oeis.org

1, 17, 241, 3361, 46817, 652081, 9082321, 126500417, 1761923521, 24540428881, 341804080817, 4760716702561, 66308229755041, 923554499868017, 12863454768397201, 179164812257692801, 2495443916839302017, 34757050023492535441, 484103256412056194161

Offset: 1

Zak Seidov, Feb 23 2005

Corresponding areas are 0, 120, 25080, 4890480, 949077360, 184120982760, ...
Values of (x^2 + y^2)/2, where the pair (x, y) satisfies x^2 - 3*y^2 = -2, i.e., a(n) = {(A001834(n))^2 + (A001835(n))^2}/2 = {(A001834(n))^2 + A046184(n)}/2. - Lekraj Beedassy, Jul 13 2006
The heights of these triangles are given in A028230. (A028230(n), A045899(n), A103772(n)) forms a primitive Pythagorean triple.
Shortest side of (k,k+2,k+3) triangle such that median to longest side is integral. Sequence of such medians is A028230. - James R. Buddenhagen, Nov 22 2013
Numbers n such that (n+1)*(3n-1) is a square. - James R. Buddenhagen, Nov 22 2013

Colin Barker, Table of n, a(n) for n = 1..850
J. B. Cosgrave, The Gauss-Factorial Motzkin connection (Maple worksheet, change suffix to .mw)
J. B. Cosgrave and K. Dilcher, An Introduction to Gauss Factorials, The American Mathematical Monthly, 118 (Nov. 2011), 812-829.
Project Euler, Problem 94: Almost Equilateral Triangles.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A102341, A103974, A016064, A011945, A028230.

I:=[1,17]; [n le 2 select I[n] else 14*Self(n-1)-Self(n-2)+4: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016

a[1] = 1; a[2] = 17; a[3] = 241; a[n_] := a[n] = 15a[n - 1] - 15a[n - 2] + a[n - 3]; Table[ a[n] - 1, {n, 17}] (* Robert G. Wilson v, Mar 24 2005 *)
LinearRecurrence[{15,-15,1},{1,17,241},20] (* Harvey P. Dale, Jan 02 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 17, a[n] == 14 a[n-1] - a[n-2] + 4}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *)

Vec(x*(1+x)^2/((1-x)*(1-14*x+x^2)) + O(x^25)) \\ Colin Barker, Mar 05 2016

a(n) = (4*A001570(n+1) - 1)/3, n > 0. - Ralf Stephan, May 20 2007
a(n) = A052530(n-1)*A052530(n) + 1. - Johannes Boot, May 21 2011
G.f.: x*(1+x)^2/((1-x)*(1-14*x+x^2)). - Colin Barker, Apr 09 2012
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3); a(1)=1, a(2)=17, a(3)=241. - Harvey P. Dale, Jan 02 2016
a(n) = (-1+(7-4*sqrt(3))^n*(2+sqrt(3))-(-2+sqrt(3))*(7+4*sqrt(3))^n)/3. - Colin Barker, Mar 05 2016
a(n) = 14*a(n-1) - a(n-2) + 4. - Vincenzo Librandi, Mar 05 2016
a(n) = A001353(n)^2 + A001353(n-1)^2. - Antonio Alberto Olivares, Apr 06 2020

More terms from Robert G. Wilson v, Mar 24 2005

A153111 Solutions of the Pell-like equation 1 + 6AA = 7BB, with A, B integers.

Original entry on oeis.org

1, 25, 649, 16849, 437425, 11356201, 294823801, 7654062625, 198710804449, 5158826853049, 133930787374825, 3477041644892401, 90269151979827601, 2343520909830625225, 60841274503616428249, 1579529616184196509249, 41006928746285492812225

Offset: 1

Ctibor O. Zizka, Dec 18 2008

B is of the form B(i) = 26*B(i-1) - B(i-2) for B(0) = 1, B(1) = 25 (this sequence).
A is of the form A(i) = 26*A(i-1) - A(i-2) for A(0) = 1, A(1) = 27.
In general a Pell-like equation of the form 1 + X*A*A = (X + 1)*B*B has the solution A(i) = (4*X + 2)*A(i-1) - A(i-2), for A(0) = 1 and A(1) = (4*X + 3), and B(i) = (4*X + 2)*B(i-1) - B(i-2) for B(0) = 1 and B(1) = (4*X + 1).
Examples in the OEIS:
  X = 1 gives A002315 for A(i) and A001653 for B(i);
  X = 2 gives A054320 for A(i) and A072256 for B(i);
  X = 3 gives A028230 for A(i) and A001570 for B(i);
  X = 4 gives A049629 for A(i) and A007805 for B(i);
  X = 5 gives A133283 for A(i) and A157014 for B(i);
  X = 6 gives A157461 for A(i) and this sequence for B(i).
Positive values of x (or y) satisfying x^2 - 26*x*y + y^2 + 24 = 0. - Colin Barker, Feb 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Luigi Cimmino, Algebraic relations for recursive sequences, arXiv:math/0510417 [math.NT], 2005-2008.
Jeroen Demeyer, Diophantine sets of polynomials over number fields, arXiv:0807.1970 [math.NT], 2008.
Franz Lemmermeyer, Conics - a Poor Man's Elliptic Curves, arXiv:math/0311306 [math.NT], 2003.
Index entries for linear recurrences with constant coefficients, signature (26,-1).

Cf. A002315, A001653, A054320, A072256, A001078, A028230, A001570, A049629, A007805, A133283, A140480.

Cf. similar sequences listed in A238379.

I:=[1,25]; [n le 2 select I[n] else 26*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 22 2014

CoefficientList[Series[(1 - x)/(x^2 - 26 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 22 2014 *)
LinearRecurrence[{26, -1}, {1, 25}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(-x*(x-1)/(x^2-26*x+1) + O(x^100)) \\ Colin Barker, Feb 20 2014

a(n) = 26*a(n-1) - a(n-2). - Colin Barker, Feb 20 2014
G.f.: -x*(x - 1) / (x^2 - 26*x + 1). - Colin Barker, Feb 20 2014
a(n) = (1/14)*(7 - sqrt(42))*(1 + (13 + 2*sqrt(42))^(2*n - 1))/(13 + 2*sqrt(42))^(n - 1). - Bruno Berselli, Feb 25 2014
E.g.f.: (1/7)*(7*cosh(2*sqrt(42)*x) - sqrt(42)*sinh(2*sqrt(42)*x))*exp(13*x) - 1. - Franck Maminirina Ramaharo, Jan 07 2019

More terms from Philippe Deléham, Sep 19 2009; corrected by N. J. A. Sloane, Sep 20 2009

Additional term from Colin Barker, Feb 20 2014

A028231 From hexagons in a circle problem.

Original entry on oeis.org

1, 22, 313, 4366, 60817, 847078, 11798281, 164328862, 2288805793, 31878952246, 444016525657, 6184352406958, 86136917171761, 1199732487997702, 16710117914796073, 232741918319147326, 3241676738553266497, 45150732421426583638, 628868577161418904441

Offset: 0

N. J. A. Sloane

Numbers k such that (k^2 + k + 1)/3 is a square. - Arkadiusz Wesolowski, Feb 10 2012

Given by the numerators of the convergents to the continued fraction [1,(1,2)^i,3,(1,2)^{i-1},1]. - Jeffrey Shallit, Dec 11 2017

J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.
T. Nagell, Des équations indéterminées x^2 + x + 1 = y^n et x^2 + x + 1 = 3*y^n, Norsk Mat. Forenings Skrifter, Ser. I, (1921).

Michel Marcus, Table of n, a(n) for n = 0..100
Kevin A. Broughan, An explicit bound for aliquot cycles of repdigits, #A15 INTEGERS Vol 12 (2012) p. 4.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1)

Cf. A001570, which gives the corresponding values of y in 3y^2 = n^2 + n + 1. - Jeffrey Shallit, Dec 11 2017

f:= gfun:-rectoproc({a(n) = 15*a(n-1)-15*a(n-2)+a(n-3),a(0)=1,a(1)=22,a(2)=313},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 12 2017

With[{k = Sqrt@ 3}, Simplify@ Array[k ((2 + k)^(2 # + 1) - (2 - k)^(2 # + 1))/4 - 1/2 &, 19, 0]] (* Michael De Vlieger, Dec 11 2017 *)

a(n) = {w = quadgen(12);w*((2+w)^(2*n+1) - (2-w)^(2*n+1))/4 - 1/2;} /* Michel Marcus, Jul 28 2012 */

a(n) = sqrt(3)*((2+sqrt(3))^(2*n+1) - (2-sqrt(3))^(2*n+1))/4 - 1/2 (see Kevin A. Broughan paper). - Michel Marcus, Jul 28 2012

a(n) = 15*a(n-1)-15*a(n-2)+a(n-3). G.f.: (1+7*x-2*x^2)/((1-x)*(1-14*x+x^2)). - conjectured by Colin Barker, Apr 10 2012; these follow easily from the formula.

Edited by Robert Israel, Dec 12 2017

A059989 Numbers n such that 3n+1 and 4n+1 are both squares.

Original entry on oeis.org

0, 56, 10920, 2118480, 410974256, 79726887240, 15466605150360, 3000441672282656, 582070217817684960, 112918621814958599640, 21905630561884150645256, 4249579410383710266580080, 824396499983877907565890320, 159928671417461930357516142056

Offset: 1

David Radcliffe, Mar 07 2001

			3*56+1=13^2 and 4*56+1=15^2.

Colin Barker, Table of n, a(n) for n = 1..400
Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Cf. A245031.

f:= proc(n) local u;
  u:= <<7,8>|<6,7>>^n . <1,-1>;
  (u[1]^2-1)/3
end proc:
map(f, [$1..30]); # Robert Israel, Mar 03 2016

CoefficientList[Series[56 x/(1 - 195 x + 195 x^2 - x^3), {x, 0, 13}], x] (* Michael De Vlieger, Mar 03 2016 *)

isok(n) = issquare(3*n+1) && issquare(4*n+1) \\ Michel Marcus, Jun 08 2013

concat(0, Vec(56*x^2/((1-x)*(1-194*x+x^2)) + O(x^20))) \\ Colin Barker, Mar 03 2016

a(n) = (A001570(n)^2 - 1)/3.
G.f.: 56*x^2 / (1-195*x+195*x^2-x^3).
From Colin Barker, Mar 03 2016: (Start)
a(n) = 195*a(n-1)-195*a(n-2)+a(n-3) for n>3.
a(n) = (-1)*((97+56*sqrt(3))^(-n)*(-1+(97+56*sqrt(3))^n)*(7+4*sqrt(3)+(-7+4*sqrt(3))*(97+56*sqrt(3))^n))/48.
(End)

Offset changed to 1 by Joerg Arndt, Mar 03 2016

A122571 a(n) = 14*a(n-1) - a(n-2), with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 13, 181, 2521, 35113, 489061, 6811741, 94875313, 1321442641, 18405321661, 256353060613, 3570537526921, 49731172316281, 692665874901013, 9647591076297901, 134373609193269601, 1871582937629476513, 26067787517619401581, 363077442309042145621, 5057016404808970637113

Offset: 1

Roger L. Bagula, Sep 17 2006

Essentially the same as A001570: 1 followed by A001570.

Each term is a sum of two consecutive squares, or a(n) = k^2 + (k+1)^2 for some k. Squares of each term are the hex numbers, or centered hexagonal numbers: a(n) = A001570(n-1) for n > 1. - Alexander Adamchuk, Apr 14 2008

Henry MacKean and Victor Moll, Elliptic Curves, Cambridge University Press, New York, 1997, page 22.

G. C. Greubel, Table of n, a(n) for n = 1..870
Gareth Jones and David Singerman, Belyi Functions, Hypermaps and Galois Groups, Bull. London Math. Soc., 28 (1996), 561-590.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (14,-1).

Cf. A001570 (essentially the same).

[n le 2 select 1 else 14*Self(n-1) -Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 29 2024

LinearRecurrence[{14, -1}, {1, 1}, 25] (* Paolo Xausa, Jan 29 2024 *)

A122571=BinaryRecurrenceSequence(14,-1,1,1)
[A122571(n-1) for n in range(1,41)] # G. C. Greubel, Oct 29 2024

G.f.: x*(1-13*x)/(1-14*x+x^2). - Philippe Deléham, Nov 17 2008
a(n+1) = A001570(n). - Ctibor O. Zizka, Feb 26 2010
a(n) = (1/4)*sqrt( 2 + (2-sqrt(3))^(4*n-6) + (2+sqrt(3))^(4*n-6) ). - Gerry Martens, Jun 03 2015
From G. C. Greubel, Oct 29 2024: (Start)
a(n) = (1/4)*( (2 + sqrt(3))^(2*n-3) + (2 - sqrt(3))^(2*n-3) ).
E.g.f.: -13 + exp(7*x)*( 13*cosh(4*sqrt(3)*x) - (15*sqrt(3)/2)*sinh(4*sqrt(3)*x) ). (End)

Edited by N. J. A. Sloane, Sep 21 2006 and Dec 04 2006

a(19)-a(21) from Paolo Xausa, Jan 29 2024

A128862 Numbers simultaneously triangular and centered triangular.

Original entry on oeis.org

1, 10, 136, 1891, 26335, 366796, 5108806, 71156485, 991081981, 13803991246, 192264795460, 2677903145191, 37298379237211, 519499406175760, 7235693307223426, 100780206894952201, 1403687203222107385, 19550840638214551186, 272308081731781609216

Offset: 1

Steven Schlicker, Apr 24 2007

A129803 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008

			a(2)=10 because 10 is the third triangular number and the fourth centered triangular number.

Michael De Vlieger, Table of n, a(n) for n = 1..875
F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.
J. Sadek and R. Euler, A Formula For All K-Gonal Numbers that Are Centered, 2010, arXiv:1003.2375 [math.NT], 2010.
S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Intersection of A000217 and A005448.

Cf. A001570, A129803.

CP := n -> 1+1/2*3*(n^2-n): N:=10: u:=2: v:=1: x:=3: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+3*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp),CP(s)]: end do: k_pcp;

Rest@ CoefficientList[Series[x (1 - 5 x + x^2)/((1 - x) (1 - 14 x + x^2)), {x, 0, 19}], x] (* Michael De Vlieger, Jul 19 2023 *)

Define x(n) and y(n) by (3+sqrt(3))*(2+sqrt(3))^n = x(n) + y(n)*sqrt(3); let s(n) = (y(n)+1)/2; then a(n) = (1/2)*(2+3*(s(n)^2-s(n))).
a(n) = (3*A001570(n) + 1)/4. - Ralf Stephan, May 20 2007
From Richard Choulet, Oct 01 2007: (Start)
a(n+2) = 14*a(n+1) - a(n) - 3.
a(n+1) = 7*a(n) - 3/2 + (1/2)*sqrt(192*a(n)^2 - 96*a(n) - 15).
G.f.: x*(1-5*x+x^2)/((1-x)*(1-14*x+x^2)). (End)

Offset set to 1 by R. J. Mathar, Apr 28 2020

More terms from Michel Marcus, Jan 20 2021

A144535 Numerators of continued fraction convergents to sqrt(3)/2.

Original entry on oeis.org

0, 1, 6, 13, 84, 181, 1170, 2521, 16296, 35113, 226974, 489061, 3161340, 6811741, 44031786, 94875313, 613283664, 1321442641, 8541939510, 18405321661, 118973869476, 256353060613, 1657092233154, 3570537526921, 23080317394680, 49731172316281, 321467351292366

Offset: 0

N. J. A. Sloane, Dec 29 2008

			0, 1, 6/7, 13/15, 84/97, 181/209, 1170/1351, 2521/2911, 16296/18817, 35113/40545, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Cf. A126664, A144536, A002531/A002530.

Bisections give A001570, A011945.

I:=[0, 1, 6, 13]; [n le 4 select I[n] else 14*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

with(numtheory); Digits:=200: cf:=convert(evalf(sqrt(3)/2,confrac); [seq(nthconver(cf,i), i=0..100)];

CoefficientList[Series[x (1 + 6 x - x^2)/((1 - 4 x + x^2) (1 + 4 x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *)
Numerator[Convergents[Sqrt[3]/2,30]] (* or *) LinearRecurrence[{0,14,0,-1},{0,1,6,13},30] (* Harvey P. Dale, Feb 10 2014 *)

Vec(x*(1+6*x-x^2)/((1-4*x+x^2)*(1+4*x+x^2)) + O(x^30)) \\ Colin Barker, Mar 27 2016

From Colin Barker, Apr 14 2012: (Start)
a(n) = 14*a(n-2) - a(n-4).
G.f.: x*(1 + 6*x - x^2)/((1 - 4*x + x^2)*(1 + 4*x + x^2)). (End)
a(n) = ((-(-2-sqrt(3))^n*(-3+sqrt(3)) + (2-sqrt(3))^n*(-3+sqrt(3)) - (3+sqrt(3))*((-2+sqrt(3))^n - (2+sqrt(3))^n)))/(8*sqrt(3)). - Colin Barker, Mar 27 2016
a(2*n) = 6*a(2*n-1) + a(2*n-2). a(2*n+1) = A003154(A101265(n+1)). - John Elias, Dec 10 2021

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A006051 Square hex numbers.

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A094347 a(n) = 14*a(n-1) - a(n-2); a(0) = a(1) = 2.

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A103975 Smaller side in (a,a+1,a+1)-integer triangle with integer area.

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A103772 Larger of two sides in a (k,k,k-1)-integer-sided triangle with integer area.

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A153111 Solutions of the Pell-like equation 1 + 6*A*A = 7*B*B, with A, B integers.

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A028231 From hexagons in a circle problem.

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A059989 Numbers n such that 3*n+1 and 4*n+1 are both squares.

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A153111 Solutions of the Pell-like equation 1 + 6AA = 7BB, with A, B integers.

A059989 Numbers n such that 3n+1 and 4n+1 are both squares.