A178218 Numbers of the form 2k^2-2k+1 or 2k^2-1.

1, 5, 7, 13, 17, 25, 31, 41, 49, 61, 71, 85, 97, 113, 127, 145, 161, 181, 199, 221, 241, 265, 287, 313, 337, 365, 391, 421, 449, 481, 511, 545, 577, 613, 647, 685, 721, 761, 799, 841, 881, 925, 967, 1013, 1057, 1105, 1151, 1201, 1249

Offset: 1

Author

Eddie Gutierrez, Dec 19 2010

Keywords

• nonn
• easy

Comments

Numbers which when squared are used as entries in magic squares. A sequence of numbers whose difference is an interleaved array consisting of 4,6,8,10,12,... and a second sequence 2,4,6,8,10,... . Each entry when squared produces an entry into a tuple used as the right diagonal in a magic square. The difference between square entries produces a third sequence 24,24,120,120,336,336,720,720,1320,1320,..., numbers divisible by 24 and generating the sequence of natural number squares.

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000
• T. C. Brown, A. R. Freedman, and P. JS. Shiue, Progressions of squares, The Australasian Journal of Combinatorics, Volume 27 (2003), p.187.
• Eddie Gutierrez, New Sequence of Squares
• Eddie Gutierrez, The Generation of New Sequences (Part G)
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A028242, A273182.

Programs

Magma

I:=[1, 5, 7, 13]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..60]]; // Vincenzo Librandi, Jun 09 2012

Mathematica

Join[{1}, Flatten[Table[{(n^2 + 1)/2, (n^2 + 2 n - 1)/2}, {n, 3, 50, 2}]]]
Table[(2 n (n + 2) + 3 (-1)^n + 1)/4, {n, 49}] (* Bruno Berselli, Apr 04 2012 *)
CoefficientList[Series[(1+3*x-3*x^2+x^3)/((1-x)^3*(1+x)),{x,0,60}],x] (* Vincenzo Librandi, Jun 09 2012 *)
LinearRecurrence[{2,0,-2,1},{1,5,7,13},60] (* Harvey P. Dale, Jun 09 2019 *)

Maxima

A178218[1]:1$
A178218[n]:=n*(n+1)-A178218[n-1]$
makelist(A178218[n],n,1,30); /* Martin Ettl, Nov 01 2012 */

Python

a = 1
for n in range(2,77):
    print(a, end=",")
    a = n*(n+1) - a
# Alex Ratushnyak, Aug 03 2012

Formula

From Colin Barker, Apr 04 2012: (Start)

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).

G.f.: x*(1+3*x-3*x^2+x^3)/((1-x)^3*(1+x)). (End)

a(n) = (2n(n+2)+3(-1)^n+1)/4. - Bruno Berselli, Apr 04 2012

From Philippe Deléham, Jun 08 2012: (Start)

2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.

(a(2n)+a(2n-1))*A028242(2n) = (a(2n)+a(2n+1))*A028242(2n+1). (End)

a(1)=1, a(n) = n*(n+1) - a(n-1). - Alex Ratushnyak, Aug 03 2012

E.g.f.: ((x^2 + 3*x + 2)*cosh(x) + (x^2 + 3*x - 1)*sinh(x) - 2)/2. - Stefano Spezia, Feb 22 2024