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A062938 a(n) = n*(n+1)*(n+2)*(n+3)+1, which equals (n^2 +3*n + 1)^2. 12
1, 25, 121, 361, 841, 1681, 3025, 5041, 7921, 11881, 17161, 24025, 32761, 43681, 57121, 73441, 93025, 116281, 143641, 175561, 212521, 255025, 303601, 358801, 421201, 491401, 570025, 657721, 755161, 863041, 982081, 1113025, 1256641 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = product of first four terms of an arithmetic progression + n^4, where the first term is 1 and the common difference is n. E.g. a(1) = 1*2*3*4 +1^4 =25, a(4) = 1*5*9*13 + 4^4= 841 etc. - Amarnath Murthy, Sep 19 2003

Is it possible for one of the squares to be the sum of two or more lesser squares each used only once? - J. M. Bergot, Feb 17 2011

Yes, in fact a(1)-a(11) are examples. [Charles R Greathouse IV, Jun 28 2011]

This sequence demonstrates that the product of any 4 consecutive integers plus 1 is a square.  The square roots are in A028387. [Harvey P. Dale, Oct 19 2011]

The sum of three consecutive terms of the sequence is divisible by 3. The quotient is a square number: [a(n)+a(n+1)+a(n+2)]/3=(n^2+5*n+7)^2. - Carmine Suriano, Jan 23 2012

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

a(n+1) = Numerator of ((n + 2)! + (n - 2)!)/(n!) n=3,4,5,... - Artur Jasinski, Jan 09 2007

a(n) = A028387(n)^2. - Jaroslav Krizek, Oct 31 2010

a(n) = n*(n+1)*(n+2)*(n+3)+1^4 = 1*(1+n)*(1+2*n)*(1+3*n)+n^4=(n^2+3*n+1)^2;

  in general, n*(n+k)*(n+2*k)*(n+3*k)+k^4 = k*(k+n)*(k+2*n)*(k+3*n)+n^4 =

  (n^2+3*k*n+k^2)^2. - Charlie Marion, Jan 13 2011

G.f.: (1+20*x+6*x^2-4*x^3+x^4)/(1-x)^5. - Colin Barker, Jun 30 2012

a(n) = A052762(n+3) + 1. - Bruce J. Nicholson, Apr 22 2017

EXAMPLE

2*3*4*5 + 1 = 121 = 11^2.

MATHEMATICA

Table[(n^2+3*n+1)^2, {n, 0, 50}]

Times@@#+1&/@Partition[Range[0, 50], 4, 1]  (* Harvey P. Dale, Apr 02 2011 *)

PROG

(PARI) j=[]; for(n=0, 70, j=concat(j, (n^2+3*n+1)^2)); j

(PARI) { for (n=0, 1000, write("b062938.txt", n, " ", (n^2 + 3*n + 1)^2) ) } \\ Harry J. Smith, Aug 14 2009

CROSSREFS

Cf. A028387 (for the square roots of the terms of this sequence). [Harvey P. Dale, Oct 19 2011]

Cf. A052762

Sequence in context: A031151 A016970 A174371 * A190875 A205800 A274783

Adjacent sequences:  A062935 A062936 A062937 * A062939 A062940 A062941

KEYWORD

nonn,easy

AUTHOR

Amarnath Murthy, Jul 05 2001

EXTENSIONS

More terms from Jason Earls, Harvey P. Dale and Dean Hickerson, Jul 06 2001

STATUS

approved

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Last modified May 19 10:36 EDT 2019. Contains 323390 sequences. (Running on oeis4.)