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A156712
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Star numbers (A003154) that are also triangular numbers (A000217).
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2
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1, 7, 91, 1261, 17557, 244531, 3405871, 47437657, 660721321, 9202660831, 128176530307, 1785268763461, 24865586158141, 346332937450507, 4823795538148951, 67186804596634801, 935791468814738257, 13033893758809700791, 181538721154521072811
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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From Colin Barker, Jan 06 2015: (Start)
Also indices of centered square numbers (A001844) which are also centered triangular numbers (A005448).
Also indices of centered octagonal numbers (A016754) which are also centered hexagonal numbers (A003215).
Also positive integers y in the solutions to 3*x^2-4*y^2-3*x+4*y = 0, the corresponding values of x being A001922.
(End)
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REFERENCES
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Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427; http://forumgeom.fau.edu/FG2016volume16/FG2016volume16.pdf#page=423
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LINKS
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Colin Barker, Table of n, a(n) for n = 1..875
Wikipedia, Star Numbers
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
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FORMULA
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a(n+3) = 15*a(n+2)-15*a(n+1)+a(n). If x^2-3*y^2=1 with x even then a() =(y+2)/4 evidently related to A001570 by: add 1 and halve.
G.f.: x*(x^2 - 8*x + 1)/(-x^3 + 15*x^2 - 15*x + 1). - Alexander R. Povolotsky, Feb 15 2009
a(n) = (4+(7-4*sqrt(3))^n*(2+sqrt(3))-(-2+sqrt(3))*(7+4*sqrt(3))^n)/8. - Colin Barker, Mar 05 2016
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MAPLE
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f:= gfun[rectoproc]({a(n+3)=15*a(n+2)-15*a(n+1)+a(n), a(1)=1, a(2)=7, a(3)=91}, a(n), 'remember'):
seq(f(n), n=1..30); # Robert Israel, Jan 01 2015
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MATHEMATICA
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f[n_] := (Simplify[(2 + Sqrt@3)^(2 n - 1) + (2 - Sqrt@3)^(2 n - 1)] + 4)/8; Array[f, 17] (* Robert G. Wilson v, Oct 28 2010 *)
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PROG
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(PARI) Vec(-x*(x^2-8*x+1)/((x-1)*(x^2-14*x+1)) + O(x^100)) \\ Colin Barker, Jan 01 2015
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CROSSREFS
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Cf. A000217, A001570, A001844, A001922, A003154, A003215, A005448, A016754.
Sequence in context: A249640 A248226 A165230 * A004368 A243946 A130978
Adjacent sequences: A156709 A156710 A156711 * A156713 A156714 A156715
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KEYWORD
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easy,nonn
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AUTHOR
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Aaron Meyerowitz, Feb 14 2009
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EXTENSIONS
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a(11) onwards from Robert G. Wilson v, Oct 28 2010
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STATUS
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approved
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