1,2

Chebyshev T-sequence with Diophantine property.

a(n) = L(n,14), where L is defined as in A108299; see also A028230 for L(n,-14). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005

Numbers x satisfying x^2 + y^3 = (y+1)^3. Corresponding y given by A001921(n)={A028230(n)-1}/2. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 21 2006

Mod[ a(n), 12 ] = 1. (a(n) - 1)/12 = A076139(n) = Triangular numbers that are one-third of another triangular number. (a(n) - 1)/4 = A076140(n) = Triangular numbers T(k) that are three times another triangular number. - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 06 2007

Also numbers n such that RootMeanSquare(1,3,...,2*n-1) is an integer. [From Ctibor O. ZIZKA (c.zizka(AT)email.cz), Sep 04 2008]

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).

V. Thebault, Consecutive cubes with difference a square, Amer. Math. Monthly, 56 (1949), 174-175.

T. D. Noe, Table of n, a(n) for n=1..101

Tanya Khovanova, Recursive Sequences

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Sociedad Magic Penny Patagonia, Leonardo en Patagonia

Eric Weisstein's World of Mathematics, Hex Number

Index entries for sequences related to Chebyshev polynomials.

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

a(n) = (1/4)*((2+sqrt(3))^(2*n+1)+(2-sqrt(3))^(2*n+1)).

G.f.: (1-x)/(1-14*x+x^2).

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)) then a(n)=q(n, 12). - Benoit Cloitre, Dec 10, 2002

a(n) = S(n, 14) - S(n-1, 14) = T(2*n+1, 2)/2 with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 14)=A007655(n+1) and T(n, 2)=A001075(n).

4*a(n)^2 - 3*b(n)^2 = 1 with b(n)=A028230(n+1), n>=0.

a(n)a(n+3) = 168 + a(n+1)a(n+2). - R. Stephan, May 29 2004

a(n) = 14*a(n-1) - a(n-2), a(-1)=1, a(0)=1. a(-1-n)=a(n) (compare A122571).

a(n) = 12*A076139(n) + 1 = 4*A076140(n) + 1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 06 2007

a(n)=(1/12)*((7-4*Sqrt[3])^n*(3-2*Sqrt[3])+(3+2*Sqrt[3])*(7+4*Sqrt[3])^n -6). - Zak Seidov (zakseidov(AT)yahoo.com), May 06 2007

a(n)=A102871(n)^2+(A102871(n)-1)^2; sum of consecutive squares. E.g. a(4)=36^2+35^2 - Mason Withers (mwithers(AT)semprautilities.com), Jan 26 2008

A001570:=-(-1+z)/(1-14*z+z**2); [S. Plouffe in his 1992 dissertation.]

NestList[3 + 7*#1 + 4*Sqrt[1 + 3*#1 + 3*#1^2] &, 0, 24] - Zak Seidov (zakseidov(AT)yahoo.com), May 06 2007

q=6; s=0; lst={}; Do[s+=n; If[Sqrt[q*s+1]==Floor[Sqrt[q*s+1]], AppendTo[lst, Sqrt[q*s+1]]], {n, 0, 9!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 02 2009]

(PARI) a(n)=real((2+quadgen(12))^(2*n+1))/2 /* Michael Somos Apr 30 2005 */

(PARI) a(n)= n=abs(1+2*n); round(2^(n-2)*prod(k=1, n, 2-sin(2*Pi*k/n)))

Bisection of A003500/4. Cf. A006051, A001922.

One half of odd part of bisection of A001075.

Cf. A077417 with companion A077416.

a(n) = sqrt((3*A028230(n+1)^2 + 1)/4).

Row 14 of array A094954.

a(n) = A098301(n+1) - A001353(n)*A001835(n).

Cf. A076139, A076140, A102871.

A122571 is another version of the same sequence.

Sequence in context: A142646 A083576 A122571 this_sequence A020544 A009015 A067385

Adjacent sequences: A001567 A001568 A001569 this_sequence A001571 A001572 A001573

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Chebyshev comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002