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A214345 Interleaved reading of A073577 and A053755. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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).
LINKS
FORMULA
a(2n+1) = A073577(n+1); a(2n) = A053755(n+1).
a(n+1)-a(n) = A142954(n+1).
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
EXAMPLE
For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*65-2*37+23=79
MAPLE
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
MATHEMATICA
LinearRecurrence[{2, 0, -2, 1}, {5, 7, 17, 23}, 50] (* Harvey P. Dale, Apr 02 2018 *)
PROG
(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$
makelist(A214345(n), n, 0, 30); /* Martin Ettl, Nov 01 2012 */
(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
CROSSREFS
First differences: A142954; 2-element moving average (a(n-1) + a(n))/2: A002378. - Guenther Schrack, Oct 25 2018
Sequence in context: A283145 A191145 A145354 * A166109 A157755 A265812
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified April 15 12:18 EDT 2024. Contains 371686 sequences. (Running on oeis4.)