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1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, 1721, 1805, 1891, 1979, 2069, 2161, 2255
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Can be obtained as sum of "Smarandache mirror sequence" terms. a(n+1)=a(n)+2(n+1) where a(1)=1 - Felice Russo (frusso(AT)micron.com)
a(n) = A105728(n+2,n+1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2005
a(n+1) is the least k > a(n)+1 such that A000217(a(n))+A000217(k) is a square. - David Wasserman (wasserma(AT)spawar.navy.mil), Jun 30 2005
Values of Fibonacci polynomial n^2-n-1 for n=2,3,4,5,... - Artur Jasinski (grafix(AT)csl.pl), Nov 19 2006
Row sums of triangle A135223 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007
Equals row sums of triangle A143596 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 26 2008]
a(n-1) gives the number of n X k rectangles on an n X n chessboard (for k = 1,2,3,...,n). [From Aaron Dunigan AtLee (aaron(AT)duniganatlee.com), Feb 13 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 23 2009: a(n) = (n + 2 + 1/phi) * (n + 2 - phi); where phi = 1.618033989... Example: a(3) = 19 =(5 + .6180339...) * (3.381966...). Cf. next to leftmost column in A162997 array.
Contribution from Miklos Kristof (kristmikl(AT)freemail.hu), Dec 24 2009: sqrt(a(0)+sqrt(a(1)+sqrt(a(2)+sqrt(a(3)+...)))) = sqrt(1+sqrt(5+sqrt(11+sqrt(19+...)))) = 2.
When n+1 is prime, a(n) gives the number of irreducible representations of any nonabelian group of order (n+1)^3. [From Andrew Rupinski (rupinski(AT)math.upenn.edu), Mar 17 2010]
a(n) = A176271(n+1,n+1). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 13 2010]
The product of any 4 consecutive integers plus 1 is a square (see A062938); the terms of this sequence are the square roots. [From Harvey P. Dale, Oct 19 2011]
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REFERENCES
| Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..950
P. De Geest, World!Of Numbers
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| a(n) = sqrt(A062938(n)). Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 08 2001; From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Oct 31 2010
a(0) = 1, a(1) = 5, a(n) = (n+1)*a(n-1) - (n+2)*a(n-2) for n > 1 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 24 2004
a(n) = A109128(n+2, 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 20 2005
A127701 * [1, 2, 3,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 24 2007
a(n) = 2*T(n+1) - 1, where T(n) = A000217(n) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 15 2007
a(n) = A005408(n) + A002378(n); A084990(n+1) = Sum(a(k): 0<=k<=n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 20 2007
Binomial transform of [1, 4, 2, 0, 0, 0,...] = (1, 5, 11, 19,...) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2007
G.f.: (1+2*x-x^2)/(1-x)^3. a(n)=3*a(n-1)-3*a(n-2)+a(n-3). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 11 2009]
a(n)=a(n-1)+2*(n+1) (with a(0)=1) [From Vincenzo Librandi, Nov 18 2010]
For k<n, a(n)=(k+1)*a(n-k)-k*a(n-k-1)+k*(k+1); e.g., a(5)=41=4*11-3*5+3*4. - Charlie Marion (charliemath(AT)optonline.net), Jan 13 2011
a(n) = lower right term in M^2, M = the 2x2 matrix [1, n; 1,(n+1)]. - Gary W. Adamson, Jun 29 2011
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MAPLE
| a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+2*n od: seq(a[n], n=1..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008
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MATHEMATICA
| FoldList[## + 2 &, 1, 2 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
s = 1; lst = {s}; Do[s += n + 3; AppendTo[lst, s], {n, 1, 100, 2}]; lst (* From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009 *)
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PROG
| (Sage) [n+(n+1)^2 for n in xrange(0, 48)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 03 2008
(MAGMA) [ n + (n+1)^2: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011
(PARI) a(n)=n^2+3*n+1 \\ Charles R Greathouse IV, Jun 10 2011
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CROSSREFS
| Complement of A028392. Third column of array A094954.
Cf. A000217, A002522, A062392, A127701, A135223.
Cf. A143596 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 26 2008]
Cf. A052905 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
Cf. A162997 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 23 2009]
A110331 and A165900 are signed versions.
Cf. A062938 (for the squares of this sequence). [From Harvey P. Dale, Oct 19 2011]
Sequence in context: A108151 A088059 * A165900 A110331 A106071 A073847
Adjacent sequences: A028384 A028385 A028386 * A028388 A028389 A028390
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KEYWORD
| nonn,easy
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AUTHOR
| Patrick De Geest (pdg(AT)worldofnumbers.com)
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EXTENSIONS
| Minor edits by N. J. A. Soane, Jul 04, 2010, following suggestions from the Sequence Fans Mailing List
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