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A028387 a(n) = n + (n+1)^2. 150
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; text; internal format)
OFFSET

0,2

COMMENTS

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

Values of Fibonacci polynomial n^2 - n - 1 for n = 2, 3, 4, 5, ... - Artur Jasinski, Nov 19 2006

A127701 * [1, 2, 3, ...]. - Gary W. Adamson, Jan 24 2007

Row sums of triangle A135223. - Gary W. Adamson, Nov 23 2007

Equals row sums of triangle A143596. - Gary W. Adamson, 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). - Aaron Dunigan AtLee, Feb 13 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. - Gary W. Adamson, Jul 23 2009

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

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

a(n) = A176271(n+1, n+1). - Reinhard Zumkeller, 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. - Harvey P. Dale, Oct 19 2011

Or numbers not expressed in the form m + floor(sqrt(m)) with integer m. - Vladimir Shevelev, Apr 09 2012

Left edge of the triangle in A214604: a(n) = A214604(n+1,1). - Reinhard Zumkeller, Jul 25 2012

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

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

a(n) has prime factors given by A038872.  - Richard R. Forberg, Dec 10 2014

A253607(a(n)) = -1. - Reinhard Zumkeller, Jan 05 2015

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

Left edge of the triangle in A260910: a(n) = A260910(n+2,1). - Reinhard Zumkeller, Aug 04 2015

Numbers m such that 4m+5 is a square. - Bruce J. Nicholson, Jul 19 2017

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

P. De Geest, World!Of Numbers

A. Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Popular Computing (Calabasas, CA), The CSR Function, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy.

Z. Skupien, A. Zak, Pair-sums packing and rainbow cliques, in Topics In Graph Theory, 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).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = sqrt(A062938(n)). - Floor van Lamoen, Oct 08 2001

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

a(n) = A105728(n+2, n+1). - Reinhard Zumkeller, Apr 18 2005

a(n) = A109128(n+2, 2). - Reinhard Zumkeller, Jun 20 2005

a(n) = 2*T(n+1) - 1, where T(n) = A000217(n). - Gary W. Adamson, Aug 15 2007

a(n) = A005408(n) + A002378(n); A084990(n+1) = Sum(a(k): 0<=k<=n). - Reinhard Zumkeller, Aug 20 2007

Binomial transform of [1, 4, 2, 0, 0, 0, ...] = (1, 5, 11, 19, ...). - Gary W. Adamson, 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). - R. J. Mathar, Jul 11 2009

a(n) = a(n-1) + 2*(n+1), with n>0, a(0) = 1. - 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, Jan 13 2011

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

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

Sum(n>0, 1/a(n)) = 1 + Pi*tan(sqrt(5)*Pi/2)/sqrt(5). - Enrique Pérez Herrero, Oct 11 2013

E.g.f.: exp(x) (1+4x+x^2). - Tom Copeland, Dec 02 2013

a(n)= A005408(A000217(n)). - Tony Foster III, May 31 2016

EXAMPLE

From Ilya Gutkovskiy, Apr 13 2016: (Start)

Illustration of initial terms:

                                        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 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 o o o o o o   o o o o o o o o o

n=0  n=1       n=2           n=3               n=4

(End)

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, Feb 22 2008

MATHEMATICA

FoldList[## + 2 &, 1, 2 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)

s = 1; A028387 = {s}; Do[s += n + 3; AppendTo[A028387, s], {n, 1, 100, 2}]; A028387 (* Zerinvary Lajos, Jul 11 2009 *)

Table[n + (n + 1)^2, {n, 0, 100}] (* Vincenzo Librandi, Oct 17 2012 *)

Table[ FrobeniusNumber[{n, n + 1}], {n, 2, 30}] (* Zak Seidov, Jan 14 2015 *)

PROG

(Sage) [n+(n+1)^2 for n in xrange(0, 48)] # Zerinvary Lajos, 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

(Haskell)

a028387 n = n + (n + 1) ^ 2  -- Reinhard Zumkeller, Jul 17 2014

CROSSREFS

Complement of A028392. Third column of array A094954.

Cf. A000217, A002522, A062392, A127701, A135223, A143596, A052905, A162997, A062938 (squares of this sequence).

A110331 and A165900 are signed versions.

Cf. A002327 (primes).

Cf. A135223, A176271, A214604, A105728, A005408, A002378, A084990.

Cf. A253607.

Cf. A260910.

Sequence in context: A215886 A088059 * A165900 A110331 A106071 A073847

Adjacent sequences:  A028384 A028385 A028386 * A028388 A028389 A028390

KEYWORD

nonn,easy

AUTHOR

Patrick De Geest

EXTENSIONS

Minor edits by N. J. A. Sloane, Jul 04 2010, following suggestions from the Sequence Fans Mailing List

STATUS

approved

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Last modified August 23 06:11 EDT 2017. Contains 290958 sequences.