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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 Guenther Schrack, Table of n, a(n) for n = 0..10001 Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). 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 Cf. A053755, A073577, A178218. 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 Adjacent sequences: A214342 A214343 A214344 * A214346 A214347 A214348 KEYWORD nonn,easy AUTHOR Yasir Karamelghani Gasmallah, Jul 13 2012 STATUS approved

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