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A190577
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a(n) = n*(n+2)*(n+4)*(n+6).
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6
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105, 384, 945, 1920, 3465, 5760, 9009, 13440, 19305, 26880, 36465, 48384, 62985, 80640, 101745, 126720, 156009, 190080, 229425, 274560, 326025, 384384, 450225, 524160, 606825, 698880, 801009, 913920, 1038345, 1175040, 1324785
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OFFSET
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1,1
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COMMENTS
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Each term is the difference between a square and a fourth power:
n*(n+2)*(n+4)*(n+6) = ((n+2)*(n+4)-2^2)^2-2^4. More generally,
n*(n+k)*(n+2*k)*(n+3*k) = ((n+k)*(n+2*k)-k^2)^2-k^4 for any k; in this case, k=2.
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REFERENCES
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Miguel de Guzmán Ozámiz, Para Pensar Mejor, Editions Pyramid, 2001, p. 294-295.
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LINKS
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FORMULA
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a(n) = ((n+2)*(n+4)-2^2)^2-2^4.
G.f.: 3*x*(5*x^3-25*x^2+47*x-35)/(x-1)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, May 15 2023
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EXAMPLE
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a(3) = 945 = 3*(3+2)*(3+4)*(3+6) = ((3+2)*(3+2*2)-2^2)^2-2^4 = 31^2-2^4.
a(13) = 62985 = 13*(13+2)*(13+4)*(13+6) = ((13+2)*(13+2*2)+2^2)^2-2^4 = 251^2-2^4.
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MATHEMATICA
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Table[n (n + 2) (n + 4) (n + 6), {n, 1, 15}]
Table[((n + 2) (n + 4) - 2^2)^2 - 2^4, {n, 1, 15}]
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PROG
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(Python)
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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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