A273182 -id:A273182 - cp's OEIS frontend

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

Eddie Gutierrez, Dec 19 2010

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.

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).

Cf. A028242, A273182.

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

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 *)

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

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

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

A273187 a(n) is the third number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of magic square of squares.

Original entry on oeis.org

99, 449, 2499, 14449, 84099, 490049, 2856099, 16646449, 97022499, 565488449, 3295908099, 19209960049, 111963852099, 652573152449, 3803475062499, 22168277222449, 129206188272099, 753068852410049, 4389206926188099, 25582172704718449, 149103829302122499

Offset: 0

Eddie Gutierrez, May 17 2016

			a(2)= 449*6 - (99 + 96)= 2499;
a(3)= 2499*6 - (449 + 96)= 14449.

Colin Barker, Table of n, a(n) for n = 0..1000
E. Gutierrez, Recursion Methods to Generate New Integer Sequences (Part VIF)
E. Gutierrez, Table of Tuples and Use of Magic Ratio for Tuple Conversion (Part IB)
E. Gutierrez, Table of Tuples for Square of Squares (Part IC)
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A273182, A273189.

CoefficientList[Series[(99 - 244 x + 49 x^2)/((1 - x) (1 - 6 x + x^2)), {x, 0, 20}], x] (* Michael De Vlieger, May 20 2016 *)

Vec((99-244*x+49*x^2)/((1-x)*(1-6*x+x^2)) + O(x^50)) \\ Colin Barker, May 18 2016

a(0)= 99, a(1)= 449, a(n+1)= a(n)*6 - a(n-1) - k where k=96.
From Colin Barker, May 18 2016: (Start)
a(n) = (24+25/2*(3-2*sqrt(2))^(1+n)+25/2*(3+2*sqrt(2))^(1+n)).
a(n) = 7*a(n-1)-7*a(n-2)+a(n-3) for n>2.
G.f.: (99-244*x+49*x^2) / ((1-x)*(1-6*x+x^2)).
(End)
a(n) = 24 + 25*A001541(n+1). - R. J. Mathar, Jun 07 2016

A273189 a(n) is the third number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of a magic square of squares.

Original entry on oeis.org

51, 401, 2451, 14401, 84051, 490001, 2856051, 16646401, 97022451, 565488401, 3295908051, 19209960001, 111963852051, 652573152401, 3803475062451, 22168277222401, 129206188272051, 753068852410001, 4389206926188051, 25582172704718401, 149103829302122451

Offset: 0

Eddie Gutierrez, May 17 2016

The multiplying factor 6 (in the recursion formulas below) appears to come from the ratio of b(1)/b(0) of the sequence. Each of the lines of tables (V vs VII) or (VI vs VIII) in oddwheel.com/ImaginaryB.html generates this factor.
k is obtained from the difference of the offsets of two relate sequences. this one, (II), starting at 51 and a second, (I), at 99 (to be submitted separately). Thus, k =[Ic(n)- IIc(n)]*2. When n=0, Ic(0)=99 and IIc(0)=51 giving the value for k of (99-51)*2=96. Furthermore, k is the same constant number for any value of n.
The differences between number in the sequence are identical in both of the related sequences.

			a(2)= 401*6 - (51 - 96)= 2451;
a(3)= 2451*6 - (401 - 96)= 14401;
a(4)= 14401*6 - (2451 - 96)= 84051.

Colin Barker, Table of n, a(n) for n = 0..1000
E. Gutierrez, Recursion Methods to Generate New Integer Sequences (Part VIF)
E. Gutierrez, Table of Tuples and Use of Magic Ratio for Tuple Conversion (Part IB)
E. Gutierrez, Table of Tuples for Square of Squares (Part IC)
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A178218, A273182, A273187.

CoefficientList[Series[(51 + 44 x + x^2)/((1 - x) (1 - 6 x + x^2)), {x, 0, 20}], x] (* Michael De Vlieger, May 18 2016 *)
LinearRecurrence[{7,-7,1},{51,401,2451},30] (* Harvey P. Dale, Feb 21 2020 *)

Vec((51+44*x+x^2)/((1-x)*(1-6*x+x^2)) + O(x^50)) \\ Colin Barker, May 18 2016

a(0)= 51, a(1)= 401, a(n+1)= a(n)*6 - a(n-1) + k where k=96.
From Colin Barker, May 18 2016: (Start)
a(n) = (-24+25/2*(3-2*sqrt(2))^(1+n)+25/2*(3+2*sqrt(2))^(1+n)).
a(n) = 7*a(n-1)-7*a(n-2)+a(n-3) for n>2.
G.f.: (51+44*x+x^2) / ((1-x)*(1-6*x+x^2)).
(End)

More terms from Colin Barker, May 18 2016

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

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A273187 a(n) is the third number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of magic square of squares.

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A273189 a(n) is the third number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of a magic square of squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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