OFFSET
1,2
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).
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(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
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 *)
PROG
(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
(Python)
a = 1
for n in range(2, 77):
print(a, end=", ")
a = n*(n+1) - a
# Alex Ratushnyak, Aug 03 2012
(Maxima)
A178218[1]:1$
makelist(A178218[n], n, 1, 30); /* Martin Ettl, Nov 01 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eddie Gutierrez, Dec 19 2010
STATUS
approved