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A124124 Nonnegative integers n such that 2n^2 + 2n - 3 is square. 11
1, 2, 6, 13, 37, 78, 218, 457, 1273, 2666, 7422, 15541, 43261, 90582, 252146, 527953, 1469617, 3077138, 8565558, 17934877, 49923733, 104532126, 290976842, 609257881, 1695937321, 3551015162, 9884647086, 20696833093, 57611945197, 120629983398, 335787024098 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

First differences are apparently in A143608. [R. J. Mathar, Jul 17 2009]

Alternative definition: T_n and (T_n - 1)/2 are triangular numbers. - Raphie Frank, Sep 06 2012

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..2613

Jeremiah Bartz, Bruce Dearden, Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.

Hermann Stamm-Wilbrandt, 4 interlaced bisections

Index entries for linear recurrences with constant coefficients, signature (1, 6, -6, -1, 1).

FORMULA

It appears that a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) if n is even, a(n) = 5*a(n-1)-5*a(n-2)+a(n-3) if n is odd. Can anyone confirm this?

Corrected and confirmed (using the g.f.) by Robert Israel, Aug 27 2014

2*a(n) = sqrt(7+2*A077442(n-1)^2)-1. - R. J. Mathar, Dec 03 2006

a(n)=a(n-1)+6*a(n-2)-6*a(n-3)-a(n-4)+a(n-5). G.f.: -x*(1+x-2*x^2+x^3+x^4)/((x-1)*(x^2-2*x-1)*(x^2+2*x-1)). [R. J. Mathar, Jul 17 2009]

For n>0, a(2n-1) = 2*A001653(n) - A046090(n-1) and a(2n) = 2*A001653(n) + A001652(n-1). - Charlie Marion, Jan 03 2012

From Raphie Frank, Sep 06 2012: (Start)

If y = A006452(n), then a(n) = 2y + ((sqrt(8y^2 - 7) - 1)/2 - (1 - sgn(n))).

Also see A216134 [a(n) = y + ((sqrt(8y^2 - 7) - 1)/2 - (1 - sgn(n)))].

(End)

a(2*n+2) = A098586(2*n). - Hermann Stamm-Wilbrandt, Aug 27 2014

a(2*n+1) = A098790(2*n). - Hermann Stamm-Wilbrandt, Aug 27 2014

a(n) = 7*a(n-2) - 7*a(n-4) + a(n-6), for n>6. - Hermann Stamm-Wilbrandt, Aug 27 2014

a(2*n+1)^2 + (a(2*n+1)+1)^2 = A038761(n)^2 + 2^2. - Hermann Stamm-Wilbrandt, Aug 31 2014

MAPLE

A124124 := proc(n)

coeftayl(x*(1+x-2*x^2+x^3+x^4)/((1-x)*(x^2-2*x-1)*(x^2+2*x-1)), x=0, n);

end proc:

seq(A124124(n), n=1..20); # Wesley Ivan Hurt, Aug 04 2014

# Alternative:

a[1]:= 1: a[2]:= 2: a[3]:= 6:

for n from 4 to 1000 do

a[n]:= (3 + 2*(n mod 2))*(a[n-1]-a[n-2])+a[n-3]

od:

seq(a[n], n=1..100); # Robert Israel, Aug 13 2014

MATHEMATICA

LinearRecurrence[{1, 6, -6, -1, 1}, {1, 2, 6, 13, 37}, 40] (* Harvey P. Dale, Nov 05 2011 *)

CoefficientList[Series[(1 + x - 2*x^2 + x^3 + x^4)/((1 - x)*(x^2 - 2*x - 1)*(x^2 + 2*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 04 2014 *)

PROG

(PARI)

for(n=1, 10^10, if(issquare(2*n^2+2*n-3), print1(n, ", "))) \\ Derek Orr, Aug 13 2014

CROSSREFS

Cf. A001108, A001652, A001653, A006452, A008844, A046090, A046172, A077442, A098790, A216134.

Sequence in context: A162057 A026550 A319751 * A052450 A231385 A001373

Adjacent sequences:  A124121 A124122 A124123 * A124125 A124126 A124127

KEYWORD

nonn,easy

AUTHOR

John W. Layman, Nov 29 2006

EXTENSIONS

More terms from Harvey P. Dale, Feb 07 2011

More terms from Wesley Ivan Hurt, Aug 04 2014

STATUS

approved

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Last modified April 1 04:15 EDT 2020. Contains 333155 sequences. (Running on oeis4.)