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A154139
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Indices k such that 4 plus the k-th triangular number is a perfect square.
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3
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0, 6, 9, 39, 56, 230, 329, 1343, 1920, 7830, 11193, 45639, 65240, 266006, 380249, 1550399, 2216256, 9036390, 12917289, 52667943, 75287480, 306971270, 438807593, 1789159679, 2557558080, 10427986806, 14906540889, 60778761159, 86881687256
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also numbers n such that (ceiling(sqrt(n*(n+1)/2)))^2 - n*(n+1)/2 = 4. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Nov 10 2009]
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LINKS
| F. T. Adams-Watters, SeqFan Discussion, Oct 2009
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FORMULA
| For n>4, a(n) = 6*a(n-2)-a(n-4)+2. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Nov 10 2009]
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5).
G.f.: x^2*(-6-3*x+6*x^2+x^3)/((x-1)*(x^2-2*x-1)*(x^2+2*x-1)) = 1+1/2*(4+11*x)/(x^2-2*x-1)+1/2/(x-1)+1/2*(-3+2*x)/(x^2+2*x-1).
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EXAMPLE
| 0*(0+1)/2+4 = 2^2. 6*(6+1)/2+4 = 5^2. 9*(9+1)/2+4 = 7^2. 39*(39+1)/2+4 = 28^2.
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MAPLE
| a := proc (n) if type(sqrt(4+(1/2)*n*(n+1)), integer) = true then n else end if end proc: seq(a(n), n = 0 .. 10^7); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 31 2009]
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CROSSREFS
| Cf. A000217, A000290, A006451.
Sequence in context: A038263 A004989 A147355 * A187998 A177181 A126110
Adjacent sequences: A154136 A154137 A154138 * A154140 A154141 A154142
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KEYWORD
| nonn
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AUTHOR
| R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 18 2009
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EXTENSIONS
| a(17),a(18) from Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 31 2009
a(19)-a(25) from Donovan Johnson (donovan.johnson(AT)yahoo.com), Nov 01 2010
More terms from Max Alekseyev (maxale(AT)gmail.com), Jan 24 2012
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