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A122571
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a(1)=a(2)=1, a(n)=14a(n-1)-a(n-2).
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3
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1, 1, 13, 181, 2521, 35113, 489061, 6811741, 94875313, 1321442641, 18405321661, 256353060613, 3570537526921, 49731172316281, 692665874901013, 9647591076297901, 134373609193269601, 1871582937629476513
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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 (alex(AT)kolmogorov.com), Apr 14 2008
A122571(n+1)=A001570(n). [From Ctibor O. Zizka (c.zizka(AT)email.cz), Feb 26 2010]
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REFERENCES
| Gareth Jones and David Singerman, Belyi Functions, Hypermaps and Galois Groups, Bull. London Math. Soc., 28 (1996), 561-590.
Henry MacKean and Victor Moll, Ellipic Curves, Cambridge University Press, New York, 1997, page 22.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
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FORMULA
| Let M be the 8 X 8 matrix {{0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1}, {-1, 0, 0, 0, 14, 0, 0, 0}}; let v[1] = Table[1, {n, 1, 8}], v[n] = M.v[n - 1]; then a(n) =v[4*n][[1]].
a(n)=(1/4)*sqrt(3)*[7-4*sqrt(3)]^n-(1/4)*sqrt(3)*[7+4*sqrt(3)]^n+(1/2)*[7+4*sqrt(3)]^n+(1/2) *[7-4*sqrt(3)]^n, with n>=0 - Paolo P. Lava (paoloplava(AT)gmail.com), Jun 19 2008
G.f.: x*(1-13x)/(1-14*x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2008]
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MATHEMATICA
| M = {{0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1}, {-1, 0, 0, 0, 14, 0, 0, 0}}; v[1] = Table[1, {n, 1, 8}] v[n_] := v[n] = M.v[n - 1] a = Table[v[4*n][[1]], {n, 1, 25}]
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CROSSREFS
| This is simply a variant of A001570.
Sequence in context: A201607 A083576 A189432 * A001570 A020544 A009015
Adjacent sequences: A122568 A122569 A122570 * A122572 A122573 A122574
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KEYWORD
| nonn
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AUTHOR
| Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 17 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 21 2006 and Dec 04 2006
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