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A072221
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a(n) = 6*a(n-1) - a(n-2) + 2, with a(0)=1,a(1)=4.
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6
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1, 4, 25, 148, 865, 5044, 29401, 171364, 998785, 5821348, 33929305, 197754484, 1152597601, 6717831124, 39154389145, 228208503748, 1330096633345, 7752371296324, 45184131144601, 263352415571284, 1534930362283105
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The product of three consecutive triangular numbers with middle term A000217(m) where m is in this sequence is a square.
n is in this sequence just in case the triangle with sides 3,n,n+1 has integer area. Equivalently, n such that 2*(n+2)*(n-1) is a square. [From James Buddenhagen (jbuddenh(AT)gmail.com), Oct 19 2008]
Triangular numbers that are equal to a square plus one have this sequence as indices. For example, 25th triangular number is 25*26/2 = 325 = 18^2 + 1. [From Tanya Khovanova & Alexey Radul (tanyakh(AT)yahoo.com), Aug 08 2009]
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FORMULA
| a(n)={3*A001541(n)-1}/2.
a(n)=3*A001108(n)+1. - David Scheers, Dec 25 2006
a(n)=-1/2+(3/4)*((3+sqrt(8))^n+(3-sqrt(8))^n) for n>=0. a(n)=floor((3/4)*(3+sqrt(8))^n) for n>0. - Franz Vrabec (franz.vrabec(AT)aon.at), Aug 21 2006
G.f.: (1-3x+4x^2)/((1-x)(1-6x+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 09 2008]
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MATHEMATICA
| a[n_] := a[n] = 6a[n - 1] - a[n - 2] + 2; a[0] = 1; a[1] = 4; Table[ a[n], {n, 0, 20}]
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CROSSREFS
| Cf. A000217, A001108, A001541.
Sequence in context: A015533 A079291 A173612 * A055846 A091634 A010909
Adjacent sequences: A072218 A072219 A072220 * A072222 A072223 A072224
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KEYWORD
| nonn
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AUTHOR
| Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 04 2002
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EXTENSIONS
| Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 08 2002
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