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A001570
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Numbers n such that n^2 is simultaneously square and centered hexagonal.
(Formerly M4915 N2108)
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37
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1, 13, 181, 2521, 35113, 489061, 6811741, 94875313, 1321442641, 18405321661, 256353060613, 3570537526921, 49731172316281, 692665874901013, 9647591076297901, 134373609193269601, 1871582937629476513, 26067787517619401581, 363077442309042145621
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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]
a(n), with n>1, is the length of the cevian of equilateral triangle whose side length is the term b(n) of the sequence A028230. This cevian divides the side (2*x+1) of the triangle in two integer segments x and x+1. [From Giacomo Fecondo (jackfertile(AT)alice.it), Oct 09 2010]
For n>=2, a(n) equals the permanent of the (2n-2)X(2n-2) tridiagonal matrix with sqrt(12)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. [From John M. Campbell, Jul 08 2011]
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REFERENCES
| 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.
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LINKS
| 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
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FORMULA
| a(n) = ((2 + sqrt(3))^(2*n - 1) + (2 - sqrt(3))^(2*n - 1)) / 4. - Michael Somos Feb 15 2011
G.f.: x * (1 - x) / (1 -14*x + x^2). - Michael Somos Feb 15 2011
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(0) = a(1) = 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
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EXAMPLE
| x + 13*x^2 + 181*x^3 + 2521*x^4 + 35113*x^5 + 489061*x^6 + 6811741*x^7 + ...
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MAPLE
| A001570:=-(-1+z)/(1-14*z+z**2); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| 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]
f[n_] := Simplify[(2 + Sqrt@3)^(2 n - 1) + (2 - Sqrt@3)^(2 n - 1)]/4; Array[f, 19] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 28 2010]
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PROG
| (PARI) {a(n) = real( (2 + quadgen( 12)) ^ (2*n - 1)) / 2} /* Michael Somos Feb 15 2011 */
(PARI) {a(n) = n = abs( 2*n - 1); round( 2 ^ (n-2) * prod( k=1, n, 2 - sin(2 * Pi * k / n)))} /* Michael Somos Feb 15 2011 */
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CROSSREFS
| 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.
Each term is equal to the root square of 0.25 + 0.75*[A028230(n)]^2 [From Giacomo Fecondo (jackfertile(AT)alice.it), Oct 09 2010]
Row 2 of array A188646.
Sequence in context: A083576 A189432 A122571 * A020544 A009015 A067385
Adjacent sequences: A001567 A001568 A001569 * A001571 A001572 A001573
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Chebyshev comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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