

A227168


Squares of gcd( 2*n, n*(n+1)/2 ).


1



1, 1, 36, 4, 25, 9, 196, 16, 81, 25, 484, 36, 169, 49, 900, 64, 289, 81, 1444, 100, 441, 121, 2116, 144, 625, 169, 2916, 196, 841, 225, 3844, 256, 1089, 289, 4900, 324, 1369, 361, 6084, 400
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OFFSET

1,3


COMMENTS

a(n) is defined as A062828(n)^2 for n >= 1. If we extend the sequence to n=0 and negative n by use of the recurrence that relates a(n) to a(n+12), a(n+8) and a(n+4), we obtain a(0)=0, a(1)=4 and a(n) = A176743(n2)^2 for n >= 2.
Define c(n) = a(n+2)  a(n2) for c >= 0. Because a(n) is a shuffle of three interleaved 2nd order polynomials, c(n) is a shuffle of three interleaved 1st order polynomials: c(n) = 4* A062828(n)*(periodically repeated 1, 8, 1, 1).
The sequence a(n) is case p=0 of the family A062828(n)*A062828(n+p):
0, 1, 1, 36, 4, 25, 9, 196, ... = a(n).
0, 1, 6, 12, 10, 15, 42, 56, ... = A130658(n)*A000217(n) = A177002(n1)*A064038(n+1).
0, 6, 2, 30, 6, 70, 12, 126, ... = 2*A198148(n)
0, 2, 5, 18, 28, 20, 27, 70, ... = A177002(n+2)*A160050(n+1) = A014695(n+2)*A000096(n).


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,3,0,0,0,1).


FORMULA

a(n) = A062828(n)^2.
a(4n) = (4*n+1)^2; a(2n+1) = (n+1)^2; a(4n+2) = 4*(4*n+3)^2.
a(n) = 3*a(n4)  3*a(n8) + a(n12).
a(n) * (period 4: repeat 4, 1, 1, 4) = A061038(n).
A005565(n3) = a(n+1) * A061037(n).  Corrected by R. J. Mathar, Jul 25 2013
a(n) = A130658(n1)^2 * A181318(n).  Corrected by R. J. Mathar, Aug 01 2013
G.f.: x*(1 + x + 36*x^2 + 4*x^3 + 22*x^4 + 6*x^5 + 88*x^6 + 4*x^7 + 9*x^8 + x^9 + 4*x^10) / ( (x1)^3*(1+x)^3*(x^2+1)^3 ).  R. J. Mathar, Jul 20 2013


MAPLE

A227168 := proc(n)
A062828(n)^2 ;
end proc: # R. J. Mathar, Jul 25 2013


MATHEMATICA

a[n_] := GCD[2*n, n*(n + 1)/2]^2; Table[a[n], {n, 1, 40}] (* JeanFrançois Alcover, Jul 03 2013 *)


PROG

(PARI) a(n)=if(n%2, n*if(n%4>2, 2, 1), n/2)^2 \\ Charles R Greathouse IV, Jul 07 2013
(MAGMA) [GCD(2*n, n*(n+1)/2)^2: n in [1..50]]; // G. C. Greubel, Sep 20 2018


CROSSREFS

Cf. A062828, A016814, A017138, A005565, A061037, A061038.
Sequence in context: A037935 A159824 A285575 * A100252 A020340 A255868
Adjacent sequences: A227165 A227166 A227167 * A227169 A227170 A227171


KEYWORD

nonn,easy,less


AUTHOR

Paul Curtz, Jul 03 2013


STATUS

approved



