%I A001570 M4915 N2108
%S A001570 1,13,181,2521,35113,489061,6811741,94875313,1321442641,18405321661,
%T A001570 256353060613,3570537526921,49731172316281,692665874901013,9647591076297901,
%U A001570 134373609193269601,1871582937629476513,26067787517619401581,363077442309042145621
%N A001570 Numbers n such that n^2 is simultaneously square and centered hexagonal.
%C A001570 Chebyshev T-sequence with Diophantine property.
%C A001570 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
%C A001570 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
%C A001570 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
%C A001570 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]
%D A001570 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001570 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001570 V. Thebault, Consecutive cubes with difference a square, Amer. Math. 
               Monthly, 56 (1949), 174-175.
%H A001570 T. D. Noe, <a href="b001570.txt">Table of n, a(n) for n=1..101</a>
%H A001570 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A001570 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001570 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001570 Sociedad Magic Penny Patagonia, <a href="http://www.magicpenny.org/engteorema.htm">
               Leonardo en Patagonia</a>
%H A001570 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HexNumber.html">Hex Number</a>
%H A001570 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%H A001570 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A001570 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A001570 a(n) = (1/4)*((2+sqrt(3))^(2*n+1)+(2-sqrt(3))^(2*n+1)).
%F A001570 G.f.: (1-x)/(1-14*x+x^2).
%F A001570 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
%F A001570 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).
%F A001570 4*a(n)^2 - 3*b(n)^2 = 1 with b(n)=A028230(n+1), n>=0.
%F A001570 a(n)a(n+3) = 168 + a(n+1)a(n+2). - R. Stephan, May 29 2004
%F A001570 a(n) = 14*a(n-1) - a(n-2), a(-1)=1, a(0)=1. a(-1-n)=a(n) (compare A122571).
%F A001570 a(n) = 12*A076139(n) + 1 = 4*A076140(n) + 1. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Apr 06 2007
%F A001570 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
%F A001570 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
%p A001570 A001570:=-(-1+z)/(1-14*z+z**2); [S. Plouffe in his 1992 dissertation.]
%t A001570 NestList[3 + 7*#1 + 4*Sqrt[1 + 3*#1 + 3*#1^2] &, 0, 24] - Zak Seidov 
               (zakseidov(AT)yahoo.com), May 06 2007
%t A001570 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]
%o A001570 (PARI) a(n)=real((2+quadgen(12))^(2*n+1))/2 /* Michael Somos Apr 30 2005 
               */
%o A001570 (PARI) a(n)= n=abs(1+2*n); round(2^(n-2)*prod(k=1,n,2-sin(2*Pi*k/n)))
%Y A001570 Bisection of A003500/4. Cf. A006051, A001922.
%Y A001570 One half of odd part of bisection of A001075.
%Y A001570 Cf. A077417 with companion A077416.
%Y A001570 a(n) = sqrt((3*A028230(n+1)^2 + 1)/4).
%Y A001570 Row 14 of array A094954.
%Y A001570 a(n) = A098301(n+1) - A001353(n)*A001835(n).
%Y A001570 Cf. A076139, A076140, A102871.
%Y A001570 A122571 is another version of the same sequence.
%Y A001570 Sequence in context: A142646 A083576 A122571 this_sequence A020544 A009015 
               A067385
%Y A001570 Adjacent sequences: A001567 A001568 A001569 this_sequence A001571 A001572 
               A001573
%K A001570 nonn,easy,nice
%O A001570 1,2
%A A001570 N. J. A. Sloane (njas(AT)research.att.com).
%E A001570 Chebyshev comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 29 2002