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A285955
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Numbers a(n) = T(b(n))*sqrt(T(b(n))+1), where T(b(n)) is the triangular number of b(n)= A000217(b(n)) and b(n)=A006451(n). Also a(n) = y solutions of the Bachet Mordell equation y^2=x^3+K, where x= T(b(n)) = A006454(n) and K = (T(b(n)))^2= A285985(n).
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7
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0, 6, 60, 1320, 12144, 262080, 2405970, 51894744, 476378760, 10274921850, 94320640056, 2034382775040, 18675010652760, 402797515372356, 3697557790357470, 79751873665825680, 732097767490332144, 15790468188346521390, 144951660405354891060, 3126432949419110989944
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OFFSET
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0,2
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COMMENTS
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Numbers a(n) which are the products of the triangular number T(b(n)) and the square root of this triangular number plus one, sqrt(T(b(n))+1), where b(n) is the sequence A006451(n) of numbers n such that T(n)+1 is a square.
This sequence a(n) gives also the y solutions of the 3rd degree Diophantine Bachet-Mordell equation y^2=x^3+K, with x= T(b(n)) = A006454(n) and K = (T(b(n)))^2 = A285985(n), where T(b(n)) is the triangular number of b(n)= A006451(n).
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REFERENCES
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V. Pletser, On some solutions of the Bachet-Mordell equation for large parameter values, to be submitted, April 2017.
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LINKS
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FORMULA
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Since b(n) = 8*sqrt(T(b(n-2))+1)+ b(n-4) = 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4), with b(-1)=-1, b(0)=0, b(1)=2, b(2)=5 (see A006451) and a(n) = T(b(n))*sqrt(T(b(n))+1) (this sequence), one has :
a(n) = ([8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)]*[ 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)+1]/2)* sqrt(([8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)]*[ 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)+1]/2)+1).
Empirical g.f.: 6*x*(1 - x)*(1 + 11*x + 27*x^2 + 11*x^3 + x^4) / ((1 + 14*x - x^2)*(1 + 2*x - x^2)*(1 - 2*x - x^2)*(1 - 14*x - x^2)). - Colin Barker, Apr 30 2017
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EXAMPLE
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For n=2, b(n)=5, a(n)=60.
For n=5, b(n)=90, a(n)= 262080.
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MAPLE
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restart: bm2:=-1: bm1:=0: bp1:=2: bp2:=5: print (‘0, 0’, ’1, 6’, ’2, 60’); for n from 3 to 1000 do b:= 8*sqrt((bp1^2+bp1)/2+1)+bm2; a:=(b*(b+1)/2)* sqrt((b*(b+1)/2)+1); print(n, a); bm2:=bm1; bm1:=bp1; bp1:=bp2; bp2:=b; end do:
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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