A142954 -id:A142954 - cp's OEIS frontend

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

Yasir Karamelghani Gasmallah, Jul 13 2012

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).

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

Guenther Schrack, Table of n, a(n) for n = 0..10001
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A053755, A073577, A178218.

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

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

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]];

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

LinearRecurrence[{2,0,-2,1},{5,7,17,23},50] (* Harvey P. Dale, Apr 02 2018 *)

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

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

A142717 First (leftmost) odd term in the n-th row of triangle A120070.

Original entry on oeis.org

3, 5, 15, 21, 35, 45, 63, 77, 99, 117, 143, 165, 195, 221, 255, 285, 323, 357, 399, 437, 483, 525, 575, 621, 675, 725, 783, 837, 899, 957, 1023, 1085, 1155, 1221, 1295, 1365, 1443, 1517, 1599, 1677, 1763, 1845, 1935, 2021, 2115, 2205, 2303, 2397, 2499, 2597

Offset: 1

Paul Curtz, Sep 26 2008

nonn

Also: Records sequence of A100181.

The last (rightmost) term in the n-th row of triangle A120070 is A005408(n).

			The odd terms of A120070 build the irregular triangle
  3;
  5;
  15,7;
  21,9;
  35,27,11;
  45,33,13;
  63,55,39,15;
The leftmost column defines this sequence.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000466, A078371, A100181.

A142717[n_]:=(n+1)^2-If[OddQ[n],1,4];Array[A142717,100] (* or *)
LinearRecurrence[{2,0,-2,1},{3,5,15,21},100] (* Paolo Xausa, Dec 05 2023 *)

First differences: a(n+1)-a(n) = A142954(n).
From R. J. Mathar, Oct 24 2008: (Start)
a(n) = (n+1)^2-1 = A000466((n+1)/2) if n odd.
a(n) = (n+1)^2-4 = A078371(n/2-1) if n even.
a(n) = 2*a(n-1) -2*a(n-3) +a(n-4).
G.f.: x(3-x+5x^2-3x^3)/((1+x)(1-x)^3). (End)

Edited and extended by R. J. Mathar, Oct 24 2008

cp's OEIS Frontend

A214345 Interleaved reading of A073577 and A053755.

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