Eddie Gutierrez - cp's OEIS frontend

A351704 Sums of the ascending diagonals of the triangle A162609.

1, 1, 2, 3, 7, 10, 20, 26, 45, 55, 86, 101, 147, 168, 232, 260, 345, 381, 490, 535, 671, 726, 892, 958, 1157, 1235, 1470, 1561, 1835, 1940, 2256, 2376, 2737, 2873, 3282, 3435, 3895, 4066, 4580, 4770, 5341, 5551, 6182, 6413, 7107, 7360, 8120, 8396, 9225, 9525, 10426, 10751, 11727, 12078, 13132, 13510, 14645

Offset: 0

Eddie Gutierrez, May 05 2022

nonn

Each term is the sum of an ascending diagonal of the triangle A162609.

			a(4) = (64 + 8 + 12)/12 = 7
a(5) = (250 - 75 + 50 + 15)/24 = 10.

Eddie Gutierrez, The Wheel Octagon Triangle - Ascending Diagonals (Part VIIIb)

Cf. A162609.

// Calculates and prints out the triangle and terms of ascending diagonals (on first line). To get more terms increment j.
#include 
int main()
{
   int n, j=8, k, C, F1, F2,s;
   F1=1; F2=1;
   printf("%d ", F1);
   printf("%d ", F2);
   for (s=0;s<=j;s++)
   {
      F1=F1 + 2*s*s + 2*s + 1;
      F2=F2 + 2*s*s + 3*s + 2;
      printf("%d ", F1);
      printf("%d ", F2);
   }
   printf("\n");
   return 0;
}

a(n) = (n^3 + 2*n + 12)/12, for even n.

a(n) = (2*n^3 - 3*n^2 + 10*n + 15)/24, for odd n.

A347026 Irregular triangle read by rows in which row n lists the first n odd numbers, followed by the first n odd numbers in decreasing order.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 7, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 17, 15, 13, 11, 9, 7, 5, 3, 1

Offset: 1

Eddie Gutierrez, Aug 11 2021

The terms of this sequence are the numbers in an irregular triangle corresponding to the addition of rows when multiplying two large numbers via a novel method (see Links).
Sums of the rising diagonals yield sequence A007980.
When the 2n terms in row n are used as the coefficients of a (2n-1)st-order polynomial in x, dividing that polynomial by x+1 produces a (2n-2)nd-order polynomial whose coefficients are the n-th row of A004737 (if that sequence is taken as an irregular triangle with 2n-1 terms in its n-th row). E.g., for n=3, (x^5 + 3x^4 + 5x^3 + 5x^2 + 3x + 1)/(x+1) = x^4 + 2x^3 + 3x^2 + 2x + 1.

			Triangle begins:
  1,  1;
  1,  3,  3,  1;
  1,  3,  5,  5,  3,  1;
  1,  3,  5,  7,  7,  5,  3,  1;
  1,  3,  5,  7,  9,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 11,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 13, 13, 11,  9,  7,  5,  3,  1;
  1,  3,  5,  7,  9, 11, 13, 15, 15, 13, 11,  9,  7,  5,  3,  1;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eddie Gutierrez, A Novel Method for Multiplying Large Numbers.
Eddie Gutierrez, The Column Addition Triangle (CAT)-A Pascal Analog (Part I).
Eddie Gutierrez, The Column Addition Triangle (CAT) and Polynomials.

Even-indexed rows of A157454.
Antidiagonal sums give A007980.
Row lengths give nonzero terms of A005843.
Cf. A004737.

#include 
int main()
{
   int n, k;
   for (n=1; n<=13; n++)
   {
      for (k=1; k<=n; k++)
      {
         printf("%d ", 2*k - 1);
      }
      for (k=n+1; k<=2*n; k++)
      {
         printf("%d ", 4*n - 2*k + 1);
      }
      printf("\n");
   }
   return 0;
}

Array[Join[#, Reverse[#]] &@Range[1, 2 # - 1, 2] &, 9] // Flatten (* Michael De Vlieger, Aug 18 2021 *)
Flatten[Table[Join[Range[1,2n+1,2],Range[2n+1,1,-2]],{n,0,10}]] (* Harvey P. Dale, Aug 31 2024 *)

row(n) = n*=2; vector(n, k, min(2*k-1, 2*(n-k)+1)); \\ Michel Marcus, Aug 17 2021

T(n,k) = 2k - 1 for 1 <= k <= n,

4n - 2k + 1 for n+1 <= k <= 2n.

Better definition from Omar E. Pol, Aug 14 2021

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

A273182 a(n) is the second 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

14, 84, 490, 2856, 16646, 97020, 565474, 3295824, 19209470, 111960996, 652556506, 3803378040, 22167711734, 129202892364, 753049642450, 4389094962336, 25581520131566, 149100025827060, 869018634830794, 5065011783157704, 29521052064115430, 172061300601534876

Offset: 0

Eddie Gutierrez, May 17 2016

The multiplying factor 6 appears to come from the ratio of a(1)/a(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.

			a(2) = 84*6 -14 = 490; a(3) = 490*6 - 84 = 2856; a(4) = 2856*6 - 490 = 16646.

Colin Barker, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
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 (6,-1).

Cf. A178218, A273187, A273189.

CoefficientList[Series[14/(1 - 6 x + x^2), {x, 0, 21}], x] (* Michael De Vlieger, May 18 2016 *)

Vec(14/(1-6*x+x^2) + O(x^50)) \\ Colin Barker, May 18 2016

a(0)=14, a(1)= 84, a(n+1)= a(n)*6 - a(n-1).
G.f.: 14 / (1-6*x+x^2). - Colin Barker, May 18 2016
E.g.f.: 7*(3*sqrt(2)*sinh(2*sqrt(2)*x) + 4*cosh(2*sqrt(2)*x))*exp(3*x)/2. - Ilya Gutkovskiy, May 18 2016

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

A217893 50k^2-40k-17 interleaved with 50k^2+10k+13 for k=>0.

Original entry on oeis.org

-17, 13, -7, 73, 103, 233, 313, 493, 623, 853, 1033, 1313, 1543, 1873, 2153, 2533, 2863, 3293, 3673, 4153, 4583, 5113, 5593, 6173, 6703, 7333, 7913, 8593, 9223, 9953, 10633, 11413, 12143, 12973, 13753, 14633, 15463, 16393, 17273, 18253, 19183, 20213, 21193

Offset: 0

Eddie Gutierrez, Oct 14 2012

The sequence (the fourth in the family) is present as a family of interleaved sequences (five in total) which are separated or factored out to give individual sequences. The first sequence is the parent having the formulas: 50*n^2-100*n+25 and 50*n^2-50*n+25 whose entries are all divisible by 25 and is identical to A178218. The fourth sequence has the formulas 50*n^2-40*n-17 and 50*n^2+10*n+13 and is part of a group where each of the sequences are new, except for the parent (in the factored form).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eddie Gutierrez New Interleaved Sequences Part H or Oddwheel.com, Section B1 Line 28 (square_sequencesVIII.html), Part H.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A178218, A214345, A214393, A214405, A216876.

&cat[[50*k^2-40*k-17,50*k^2+10*k+13]: k in [0..23]]; // Bruno Berselli, Oct 23 2012

Flatten[Table[{50 n^2 - 40 n - 17, 50 n^2 + 10 n + 13}, {n, 0, 23}]] (* Bruno Berselli, Oct 23 2012 *)
CoefficientList[Series[(-17 + 47*x - 33*x^2 + 53*x^3)/((1+x)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)

vector(48, n, k=(n-1)\2; if(n%2, 50*k^2-40*k-17, 50*k^2+10*k+13)) \\ Bruno Berselli, Oct 23 2012

G.f.: (-17+47*x-33*x^2+53*x^3)/((1+x)*(1-x)^3).
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = 1+(10*n*(5*n-8)-75*(-1)^n+3)/4. [Bruno Berselli, Oct 15 2012]

Definition rewritten by Bruno Berselli, Nov 09 2012

A217894 50k^2-20k-23 interleaved with 50k^2+30k+17 for k=>0.

Original entry on oeis.org

-23, 17, 7, 97, 137, 277, 367, 557, 697, 937, 1127, 1417, 1657, 1997, 2287, 2677, 3017, 3457, 3847, 4337, 4777, 5317, 5807, 6397, 6937, 7577, 8167, 8857, 9497, 10237, 10927, 11717, 12457, 13297, 14087, 14977, 15817, 16757, 17647, 18637, 19577, 20617, 21607

Offset: 0

Eddie Gutierrez, Oct 14 2012

The sequence (the fifth and last in the family) is present as a family of interleaved sequences (five in total) which are separated or factored out to give individual sequences. The first sequence is the parent having the formulas: 50*n^2-100*n+25 and 50*n^2-50*n+25 whose entries are all divisible by 25 and is identical to A178218. The fifth sequence has the formulas 50*n^2-20*n-23 and 50*n^2+30*n+17 and is part of a group where each of the sequences are new, except for the parent (in the factored form).

			a(9) = 2*a(8) - 2*a(6) + a(5) = 1394 - 734 + 277 = 937.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eddie Gutierrez New Interleaved Sequences Part H or Oddwheel.com, Section B1 Line 28 (square_sequencesVIII.html), Part H.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A178218, A214345, A214393, A214405, A216876.

&cat[[50*k^2-20*k-23, 50*k^2+30*k+17]: k in [0..23]]; // Bruno Berselli, Oct 23 2012

Flatten[Table[{50*n^2 - 20*n - 23, 50*n^2 + 30*n + 17}, {n, 0, 23}]] (* Bruno Berselli, Oct 23 2012 *)
CoefficientList[Series[(-23 + 63*x - 27*x ^ 2 + 37*x^3)/((1+x)*(1-x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 23 2012 *)

vector(48, n, k=(n-1)\2; if(n%2, 50*k^2-20*k-23, 50*k^2+30*k+17)) \\ Bruno Berselli, Oct 23 2012

G.f.: (-23+63*x-27*x^2+37*x^3)/((1+x)*(1-x)^3).
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = (10*n*(5*n-4)-75*(-1)^n-1)/4 -4. [Bruno Berselli, Oct 15 2012]

Definition rewritten by Bruno Berselli, Nov 22 2012

A216853 18k^2-12k-7 interleaved with 18k^2+6k+5 for k>=0.

Original entry on oeis.org

-7, 5, -1, 29, 41, 89, 119, 185, 233, 317, 383, 485, 569, 689, 791, 929, 1049, 1205, 1343, 1517, 1673, 1865, 2039, 2249, 2441, 2669, 2879, 3125, 3353, 3617, 3863, 4145, 4409, 4709, 4991, 5309, 5609, 5945, 6263, 6617, 6953, 7325, 7679, 8069, 8441, 8849

Offset: 0

Eddie Gutierrez, Sep 17 2012

The sequence (the third in the family) is present as a family of single interleaved sequence of which there are many which are separated or factored out to give individual sequences. The larger sequence produces two smaller interleaved sequences where one of them has the formulas above and the other interleaved sequence has the formulas (18n^2-24n-1) and (18n^2-6n+5). The latter interleaved sequence is A214493. There are three sequences in this family.

Eddie Gutierrez New Interleaved Sequences Part B on oddwheel.com, Section B1 Line No. 22 (square_sequencesII.html) Part B
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A178218, A214345, A214393, A214405, A216876.

&cat[[18*k^2-12*k-7, 18*k^2+6*k+5]: k in [0..22]]; // Bruno Berselli, Oct 05 2012

Flatten[Table[{18 n^2 - 12 n - 7, 18 n^2 + 6 n + 5}, {n, 0, 22}]] (* Bruno Berselli, Oct 05 2012 *)

vector(46, n, k=(n-1)\2; if(n%2, 18*k^2-12*k-7, 18*k^2+6*k+5)) \\ Bruno Berselli, Oct 05 2012

G.f.: -(7-19*x+11*x^2-17*x^3)/((1+x)*(1-x)^3). - Bruno Berselli, Oct 05 2012

a(n) = (6*n*(3*n-4)-27*(-1)^n-1)/4. - Bruno Berselli, Oct 05 2012

Definition rewritten by Bruno Berselli, Oct 25 2012

A216852 18k^2-36k+9 interleaved with 18k^2-18k+9 for k>=0.

Original entry on oeis.org

9, 9, -9, 9, 9, 45, 63, 117, 153, 225, 279, 369, 441, 549, 639, 765, 873, 1017, 1143, 1305, 1449, 1629, 1791, 1989, 2169, 2385, 2583, 2817, 3033, 3285, 3519, 3789, 4041, 4329, 4599, 4905, 5193, 5517, 5823, 6165, 6489, 6849, 7191, 7569, 7929, 8325, 8703

Offset: 0

Eddie Gutierrez, Sep 17 2012

The sequence is present as a family of single interleaved sequence of which there are many which are separated or factored out to give individual sequences. The larger sequence produces two smaller interleaved sequences where one of them has the formulas above and the other interleaved sequence has the formulas (18n^2-24n+1) and (18n^2-6n+5). The latter interleaved sequence is A214493. There are three sequences in this family.

Eddie Gutierrez New Interleaved Sequences Part B on oddwheel.com, Section B1 Line No. 22 (square_sequencesII.html) Part B
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Cf. A178218, A214345, A214393, A214405, A216844, A216865, A216875, A216876.

&cat[[18*k^2-36*k+9, 18*k^2-18*k+9]: k in [0..23]]; // Bruno Berselli, Oct 01 2012

Flatten[Table[{18 n^2 - 36 n + 9, 18 n^2 - 18 n + 9}, {n, 0, 23}]] (* Bruno Berselli, Oct 01 2012 *)
Flatten[Table[18n^2+9-{36n,18n},{n,0,50}]] (* or *) LinearRecurrence[ {2,0,-2,1},{9,9,-9,9},100] (* Harvey P. Dale, Apr 26 2014 *)

vector(47, n, k=(n-1)\2; if(n%2, 18*k^2-36*k+9, 18*k^2-18*k+9)) \\ Bruno Berselli, Oct 01 2012

From Bruno Berselli, Oct 01 2012: (Start)
G.f.: 9*(1-x-3*x^2+5*x^3)/((1+x)*(1-x)^3).
a(n) = (9/4)*(2*n*(n-4)-3*(-1)^n+7).
a(n) = 9*A178218(n-3) with A178218(-3)=1, A178218(-2)=1, A178218(-1)=-1, A178218(0)=1. (End)
a(0)=9, a(1)=9, a(2)=-9, a(3)=9, a(n)=2*a(n-1)-2*a(n-3)+a(n-4). - Harvey P. Dale, Apr 26 2014

Definition rewritten by Bruno Berselli, Oct 25 2012

A216865 16k^2-32k+8 interleaved with 16k^2-16k+8 for k>=0.

Original entry on oeis.org

8, 8, -8, 8, 8, 40, 56, 104, 136, 200, 248, 328, 392, 488, 568, 680, 776, 904, 1016, 1160, 1288, 1448, 1592, 1768, 1928, 2120, 2296, 2504, 2696, 2920, 3128, 3368, 3592, 3848, 4088, 4360, 4616, 4904, 5176, 5480, 5768, 6088, 6392, 6728, 7048, 7400, 7736

Offset: 0

Eddie Gutierrez, Sep 18 2012

The sequence (the first in the family) is present as a family of single interleaved sequence of which are separated or factored out of the larger sequence to give individual sequences. The larger sequence produces four smaller interleaved sequences where one of them has the formula above and a second interleaved sequences having the formulas (16n^2-24n+1) and (16n^2-6n+5). This interleaved sequence is A214393. The fourth interleaved sequence in the group has the formulas (16n^2-8n-7) and (16n^2+2n+5) and it is A214405. There are a total of four sequences in this family.

Eddie Gutierrez New Interleaved Sequences Part C on oddwheel.com, Section B1 Line No. 23 (square_sequencesIII.html) Part C
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A178218, A214345, A214393, A214405, A216844, A216875, A216876.

&cat[[16*k^2-32*k+8, 16*k^2-16*k+8]: k in [0..23]]; // Bruno Berselli, Oct 01 2012

Flatten[Table[{16 n^2 - 32 n + 8, 16 n^2 - 16 n + 8}, {n, 0, 23}]] (* Bruno Berselli, Sep 30 2012 *)

vector(47, n, k=(n-1)\2; if(n%2, 16*k^2-32*k+8, 16*k^2-16*k+8)) \\ Bruno Berselli, Oct 01 2012

G.f.: 8*(1-x-3*x^2+5*x^3)/((1+x)*(1-x)^3). [Bruno Berselli, Sep 30 2012]
a(n) = 2*(2*n*(n-4)-3*(-1)^n+7). [Bruno Berselli, Sep 30 2012]
a(n) = 8*A178218(n-3) with A178218(-3)=1, A178218(-2)=1, A178218(-1)=-1, A178218(0)=1. [Bruno Berselli, Oct 01 2012]

Definition rewritten by Bruno Berselli, Oct 25 2012

cp's OEIS Frontend

User: Eddie Gutierrez

A351704 Sums of the ascending diagonals of the triangle A162609.

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A347026 Irregular triangle read by rows in which row n lists the first n odd numbers, followed by the first n odd numbers in decreasing order.

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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.

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A273182 a(n) is the second number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of a magic square of squares.

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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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A217893 50k^2-40k-17 interleaved with 50k^2+10k+13 for k=>0.

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A217894 50k^2-20k-23 interleaved with 50k^2+30k+17 for k=>0.

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