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A178218 Numbers of the form 2k^2-2k+1 or 2k^2-1. 18
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 (list; graph; refs; listen; history; text; internal format)
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

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)). [Colin Barker, Apr 04 2012]

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

2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2. [Philippe Deléham, Jun 08 2012]

(a(2n)+a(2n-1))*A028242(2n) = (a(2n)+a(2n+1))*A028242(2n+1). [Philippe Deléham, Jun 08 2012]

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

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,

    a = n*(n+1) - a

# from Alex Ratushnyak, Aug 03 2012

(Maxima)

A178218[1]:1$

A178218[n]:=n*(n+1)-A178218[n-1]$

makelist(A178218[n], n, 1, 30); /* Martin Ettl, Nov 01 2012 */

CROSSREFS

Cf. A273182.

Sequence in context: A253297 A163385 A288449 * A314323 A314324 A247011

Adjacent sequences:  A178215 A178216 A178217 * A178219 A178220 A178221

KEYWORD

nonn,easy

AUTHOR

Eddie Gutierrez, Dec 19 2010

STATUS

approved

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Last modified September 19 12:57 EDT 2019. Contains 327198 sequences. (Running on oeis4.)