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5, 7, 17, 23, 37, 47, 65, 79, 101, 119, 145, 167, 197, 223, 257, 287, 325, 359, 401, 439, 485, 527, 577, 623, 677, 727, 785, 839, 901, 959, 1025, 1087, 1157, 1223, 1297, 1367, 1445, 1519, 1601, 1679, 1765, 1847, 1937, 2023, 2117, 2207, 2305, 2399, 2501
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OFFSET
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0,1
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COMMENTS
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The elements of this sequence satisfy the property that for every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2. In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2 : in the case of this sequence 7^2, 17^2, and 23^2 is such a triple (i.e. 15-8 =7, 17, 8+15=23, and 8^2+15^2=17^2) .
The first differences of such a sequence is always an interleaved sequence; in this case the interleaved sequence is 2,10,6,14,10,... (A142954).
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LINKS
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FORMULA
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a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: (x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)).
a(n) = (2*n*(n+4)+3*(-1)^n+7)/2.
2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.
a(n) = 4*(n+1) + a(n-2) for n > 1; a(-n) = a(n-4). - Guenther Schrack, Oct 24 2018
E.g.f.: (5 + 5*x + x^2)*cosh(x) + (2 + 5*x + x^2)*sinh(x). - Stefano Spezia, Feb 22 2024
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EXAMPLE
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For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*65-2*37+23=79
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MAPLE
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seq(coeff(series((x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)), x, n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 26 2018
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MATHEMATICA
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LinearRecurrence[{2, 0, -2, 1}, {5, 7, 17, 23}, 50] (* Harvey P. Dale, Apr 02 2018 *)
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PROG
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(Magma) I:=[5, 7, 17, 23]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];
(Maxima) A214345(n):=(2*n*(n+4)+3*(-1)^n+7)/2$
(GAP) a:=[7, 17];; for n in [3..50] do a[n]:=4*(n+1)+a[n-2]; od; Concatenation([5], a); # Muniru A Asiru, Oct 26 2018
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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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