A214345 Interleaved reading of A073577 and A053755.

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

Offset: 0

Author

Yasir Karamelghani Gasmallah, Jul 13 2012

Keywords

• nonn
• easy

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).

Examples

			For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*65-2*37+23=79
		

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).

Crossrefs

Cf. A053755, A073577, A178218.

First differences: A142954; 2-element moving average (a(n-1) + a(n))/2: A002378. - Guenther Schrack, Oct 25 2018

Programs

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

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]];

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 *)

Maxima

A214345(n):=(2*n*(n+4)+3*(-1)^n+7)/2$
makelist(A214345(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

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