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%I #289 Nov 15 2023 16:25:25
%S 1,5,11,19,29,41,55,71,89,109,131,155,181,209,239,271,305,341,379,419,
%T 461,505,551,599,649,701,755,811,869,929,991,1055,1121,1189,1259,1331,
%U 1405,1481,1559,1639,1721,1805,1891,1979,2069,2161,2255,2351,2449,2549,2651
%N a(n) = n + (n+1)^2.
%C a(n+1) is the least k > a(n) + 1 such that A000217(a(n)) + A000217(k) is a square. - _David Wasserman_, Jun 30 2005
%C Values of Fibonacci polynomial n^2 - n - 1 for n = 2, 3, 4, 5, ... - _Artur Jasinski_, Nov 19 2006
%C A127701 * [1, 2, 3, ...]. - _Gary W. Adamson_, Jan 24 2007
%C Row sums of triangle A135223. - _Gary W. Adamson_, Nov 23 2007
%C Equals row sums of triangle A143596. - _Gary W. Adamson_, Aug 26 2008
%C a(n-1) gives the number of n X k rectangles on an n X n chessboard (for k = 1, 2, 3, ..., n). - _Aaron Dunigan AtLee_, Feb 13 2009
%C sqrt(a(0) + sqrt(a(1) + sqrt(a(2) + sqrt(a(3) + ...)))) = sqrt(1 + sqrt(5 + sqrt(11 + sqrt(19 + ...)))) = 2. - _Miklos Kristof_, Dec 24 2009
%C When n + 1 is prime, a(n) gives the number of irreducible representations of any nonabelian group of order (n+1)^3. - _Andrew Rupinski_, Mar 17 2010
%C a(n) = A176271(n+1, n+1). - _Reinhard Zumkeller_, Apr 13 2010
%C The product of any 4 consecutive integers plus 1 is a square (see A062938); the terms of this sequence are the square roots. - _Harvey P. Dale_, Oct 19 2011
%C Or numbers not expressed in the form m + floor(sqrt(m)) with integer m. - _Vladimir Shevelev_, Apr 09 2012
%C Left edge of the triangle in A214604: a(n) = A214604(n+1,1). - _Reinhard Zumkeller_, Jul 25 2012
%C Another expression involving phi = (1 + sqrt(5))/2 is a(n) = (n + phi)(n + 1 - phi). Therefore the numbers in this sequence, even if they are prime in Z, are not prime in Z[phi]. - _Alonso del Arte_, Aug 03 2013
%C a(n-1) = n*(n+1) - 1, n>=0, with a(-1) = -1, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 5 for b = 2*n+1. In general D = b^2 - 4ac > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - _Wolfdieter Lang_, Aug 15 2013
%C a(n) has prime factors given by A038872. - _Richard R. Forberg_, Dec 10 2014
%C A253607(a(n)) = -1. - _Reinhard Zumkeller_, Jan 05 2015
%C An example of a quadratic sequence for which the continued square root map (see A257574) produces the number 2. There are infinitely many sequences with this property - another example is A028387. See Popular Computing link. - _N. J. A. Sloane_, May 03 2015
%C Left edge of the triangle in A260910: a(n) = A260910(n+2,1). - _Reinhard Zumkeller_, Aug 04 2015
%C Numbers m such that 4m+5 is a square. - _Bruce J. Nicholson_, Jul 19 2017
%C The numbers represented as 131 in base n: 131_4 = 29, 131_5 = 41, ... . If 'digits' larger than the base are allowed then 131_2 = 11 and 131_1 = 5 also. - _Ron Knott_, Nov 14 2017
%C From _Klaus Purath_, Mar 18 2019: (Start)
%C Let m be a(n) or a prime factor of a(n). Then, except for 1 and 5, there are, if m is a prime, exactly two squares y^2 such that the difference y^2 - m contains exactly one pair of factors {x,z} such that the following applies: x*z = y^2 - m, x + y = z with
%C x < y, where {x,y,z} are relatively prime numbers. {x,y,z} are the initial values of a sequence of the Fibonacci type. Thus each a(n) > 5, if it is a prime, and each prime factor p > 5 of an a(n) can be assigned to exactly two sequences of the Fibonacci type. a(0) = 1 belongs to the original Fibonacci sequence and a(1) = 5 to the Lucas sequence.
%C But also the reverse assignment applies. From any sequence (f(i)) of the Fibonacci type we get from its 3 initial values by f(i)^2 - f(i-1)*f(i+1) with f(i-1) < f(i) a term a(n) or a prime factor p of a(n). This relation is also valid for any i. In this case we get the absolute value |a(n)| or |p|. (End)
%C a(n-1) = 2*T(n) - 1, for n>=1, with T = A000217, is a proper subsequence of A089270, and the terms are 0,-1,+1 (mod 5). - _Wolfdieter Lang_, Jul 05 2019
%C a(n+1) is the number of wedged n-dimensional spheres in the homotopy of the neighborhood complex of Kneser graph KG_{2,n}. Here, KG_{2,n} is a graph whose vertex set is the collection of subsets of cardinality 2 of set {1,2,...,n+3,n+4} and two vertices are adjacent if and only if they are disjoint. - _Anurag Singh_, Mar 22 2021
%C Also the number of squares between (n+2)^2 and (n+2)^4. - _Karl-Heinz Hofmann_, Dec 07 2021
%C (x, y, z) = (A001105(n+1), -a(n-1), -a(n)) are solutions of the Diophantine equation x^3 + 4*y^3 + 4*z^3 = 8. - _XU Pingya_, Apr 25 2022
%H Vincenzo Librandi, <a href="/A028387/b028387.txt">Table of n, a(n) for n = 0..1000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/index.html">World!Of Numbers</a>
%H Adalbert Kerber, <a href="/A004211/a004211.pdf">A matrix of combinatorial numbers related to the symmetric groups<</a>, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H Nandini Nilakantan and Anurag Singh, <a href="https://doi.org/10.1007/s12044-018-0429-9">Homotopy type of neighborhood complexes of Kneser graphs, KG_{2,k}</a>, Proceeding-Mathematical Sciences, 128, Article number: 53(2018).
%H Yanni Pei and Jiang Zeng, <a href="https://arxiv.org/abs/2206.11236">Counting signed derangements with right-to-left minima and excedances</a>, arXiv:2206.11236 [math.CO], 2022.
%H Popular Computing (Calabasas, CA), <a href="/A257352/a257352.pdf">The CSR Function</a>, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy.
%H Zdzislaw Skupień and Andrzej Żak, Pair-sums packing and rainbow cliques, in <a href="https://web.archive.org/web/20170810012713/http://www.math.uiuc.edu/~kostochk/Zykov90-Topics_in_Graph_Theory.pdf">Topics In Graph Theory</a>, A tribute to A. A. and T. E. Zykovs on the occasion of A. A. Zykov's 90th birthday, ed. R. Tyshkevich, Univ. Illinois, 2013, pages 131-144, (in English and Russian).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = sqrt(A062938(n)). - _Floor van Lamoen_, Oct 08 2001
%F a(0) = 1, a(1) = 5, a(n) = (n+1)*a(n-1) - (n+2)*a(n-2) for n > 1. - _Gerald McGarvey_, Sep 24 2004
%F a(n) = A105728(n+2, n+1). - _Reinhard Zumkeller_, Apr 18 2005
%F a(n) = A109128(n+2, 2). - _Reinhard Zumkeller_, Jun 20 2005
%F a(n) = 2*T(n+1) - 1, where T(n) = A000217(n). - _Gary W. Adamson_, Aug 15 2007
%F a(n) = A005408(n) + A002378(n); A084990(n+1) = Sum_{k=0..n} a(k). - _Reinhard Zumkeller_, Aug 20 2007
%F Binomial transform of [1, 4, 2, 0, 0, 0, ...] = (1, 5, 11, 19, ...). - _Gary W. Adamson_, Sep 20 2007
%F G.f.: (1+2*x-x^2)/(1-x)^3. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _R. J. Mathar_, Jul 11 2009
%F 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. - _Gary W. Adamson_, Jul 23 2009
%F a(n) = a(n-1) + 2*(n+1), with n > 0, a(0) = 1. - _Vincenzo Librandi_, Nov 18 2010
%F 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_, Jan 13 2011
%F a(n) = lower right term in M^2, M = the 2 X 2 matrix [1, n; 1, (n+1)]. - _Gary W. Adamson_, Jun 29 2011
%F G.f.: (x^2-2*x-1)/(x-1)^3 = G(0) where G(k) = 1 + x*(k+1)*(k+4)/(1 - 1/(1 + (k+1)*(k+4)/G(k+1)); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Oct 16 2012
%F Sum_{n>0} 1/a(n) = 1 + Pi*tan(sqrt(5)*Pi/2)/sqrt(5). - _Enrique Pérez Herrero_, Oct 11 2013
%F E.g.f.: exp(x) (1+4*x+x^2). - _Tom Copeland_, Dec 02 2013
%F a(n) = A005408(A000217(n)). - _Tony Foster III_, May 31 2016
%F From _Amiram Eldar_, Jan 29 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = -Pi*sec(sqrt(5)*Pi/2).
%F Product_{n>=1} (1 - 1/a(n)) = -Pi*sec(sqrt(5)*Pi/2)/6. (End)
%e From _Ilya Gutkovskiy_, Apr 13 2016: (Start)
%e Illustration of initial terms:
%e o o
%e o o o o o o
%e o o o o o o o o o o o o
%e o o o o o o o o o o o o o o o o o o o o
%e o o o o o o o o o o o o o o o o o o o o o o o o o
%e n=0 n=1 n=2 n=3 n=4
%e (End)
%e From _Klaus Purath_, Mar 18 2019: (Start)
%e Examples:
%e a(0) = 1: 1^1-0*1 = 1, 0+1 = 1 (Fibonacci A000045).
%e a(1) = 5: 3^2-1*4 = 5, 1+3 = 4 (Lucas A000032).
%e a(2) = 11: 4^2-1*5 = 11, 1+4 = 5 (A000285); 5^2-2*7 = 11, 2+5 = 7 (A001060).
%e a(3) = 19: 5^2-1*6 = 19, 1+5 = 6 (A022095); 7^2-3*10 = 19, 3+7 = 10 (A022120).
%e a(4) = 29: 6^2-1*7 = 29, 1+6 = 7 (A022096); 9^2-4*13 = 29, 4+9 = 13 (A022130).
%e a(11)/5 = 31: 7^2-2*9 = 31, 2+7 = 9 (A022113); 8^2-3*11 = 31, 3+8 = 11 (A022121).
%e a(24)/11 = 59: 9^2-2*11 = 59, 2+9 = 11 (A022114); 12^2-5*17 = 59, 5+12 = 17 (A022137).
%e (End)
%t FoldList[## + 2 &, 1, 2 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t Table[n + (n + 1)^2, {n, 0, 100}] (* _Vincenzo Librandi_, Oct 17 2012 *)
%t Table[ FrobeniusNumber[{n, n + 1}], {n, 2, 30}] (* _Zak Seidov_, Jan 14 2015 *)
%o (Sage) [n+(n+1)^2 for n in range(0,48)] # _Zerinvary Lajos_, Jul 03 2008
%o (Magma) [n + (n+1)^2: n in [0..60]]; // _Vincenzo Librandi_, Apr 26 2011
%o (PARI) a(n)=n^2+3*n+1 \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Haskell)
%o a028387 n = n + (n + 1) ^ 2 -- _Reinhard Zumkeller_, Jul 17 2014
%Y Complement of A028392. Third column of array A094954.
%Y Cf. A000217, A002522, A062392, A127701, A135223, A143596, A052905, A162997, A062938 (squares of this sequence).
%Y A110331 and A165900 are signed versions.
%Y Cf. A002327 (primes), A094210.
%Y Frobenius number for k successive numbers: this sequence (k=2), A079326 (k=3), A138984 (k=4), A138985 (k=5), A138986 (k=6), A138987 (k=7), A138988 (k=8).
%Y Cf. also A135223, A176271, A214604, A105728, A005408, A002378, A084990, A253607, A260910, A089270.
%K nonn,easy
%O 0,2
%A _Patrick De Geest_
%E Minor edits by _N. J. A. Sloane_, Jul 04 2010, following suggestions from the Sequence Fans Mailing List
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