A000466 -id:A000466 - cp's OEIS frontend

A173294 Values of 16n^2+24n+7, n>=0, each duplicated.

7, 7, 47, 47, 119, 119, 223, 223, 359, 359, 527, 527, 727, 727, 959, 959, 1223, 1223, 1519, 1519, 1847, 1847, 2207, 2207, 2599, 2599, 3023, 3023, 3479, 3479, 3967, 3967, 4487, 4487, 5039, 5039, 5623, 5623, 6239, 6239, 6887, 6887, 7567, 7567, 8279, 8279, 9023, 9023, 9799, 9799, 10607, 10607, 11447, 11447, 12319, 12319, 13223, 13223, 14159, 14159, 15127, 15127, 16127

Offset: 0

Paul Curtz, Feb 15 2010

The Leibniz series for Pi/4 involves 1, -1/3, 1/5, -1/7, 1/9, -1/11, .. inverses of the odd numbers. The first differences of this sequence of fractions are -4/3, 8/15, -12/35, 16/63, -20/99, 24/143,... = (-1)^(n+1)*A008586(n+1)/A000466(n+1).
a(n) is the difference of the n-th denominator and numerator, A000466(n+1)+(-1)^n*A008586(n+1). (Note that A000466 is a bisection of A005563, which establishes a very distant relation between this sequence and the Lyman series.)
If one would add the n-th denominator and numerator, -1, 23, 23, 79, 79, 167, 167, 287, 287, 439,...(duplicated values of 16n^2+40n+23 and a -1) would result.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

With[{c=16n^2+24n+7},Table[{c,c},{n,0,40}]]//Flatten (* or *) LinearRecurrence[ {1,2,-2,-1,1},{7,7,47,47,119},80] (* Harvey P. Dale, Jan 16 2019 *)

a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
G..f: ( -7-26*x^2+x^4 ) / ( (1+x)^2*(x-1)^3 ).
a(2n) = a(2n+1) = 16n^2+24n+7.

A191567 Four interlaced 2nd order polynomials: a(4k) = k(1+2k); a(1+2k) = 2(1+2k)(3+2k); a(2+4k) = 4(1+k)(1+2k).

Original entry on oeis.org

0, 6, 4, 30, 3, 70, 24, 126, 10, 198, 60, 286, 21, 390, 112, 510, 36, 646, 180, 798, 55, 966, 264, 1150, 78, 1350, 364, 1566, 105, 1798, 480, 2046, 136, 2310, 612, 2590, 171, 2886, 760, 3198, 210, 3526, 924, 3870, 253, 4230, 1104, 4606, 300, 4998, 1300, 5406, 351

Offset: 0

Paul Curtz, Jun 12 2011

a(n) = T(0,n) and differences T(n,k) = T(n-1,k+1) - T(n-1,k) define the array
0,   6,  4,  30,    3,  70,   24,  126,   10,  198,   60,  286,   21,  390,  ..
6,  -2, 26, -27,   67, -46,  102, -116,  188, -138,  226, -265,  369, -278, ..
-8, 28 -53,  94, -113, 148, -218,  304, -326,  364, -491,  634, -647,  676, ...
T(3,n) mod 9 is the sequence 1, 1, 1, 4, 4, 4, 7, 7, 7, 4, 4, 4 (and periodically repeated with period 12).
A064680(2+n) divides a(n), where b(n) = a(n)/A064680(2+n) = 0, 1, 2, 3, 1, 5, 6, 7, 2,... for n>=0, obeys b(4*k) = k and has recurrence b(n) = 2*b(n-4) - b(n-8).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A014105, A000466, A000384, A177427.

a:=[0,6,4,30,3,70,24,126,10,198,60,286];; for n in [13..60] do a[n]:= 3*a[n-4]-3*a[n-8]+a[n-12]; od; a; # G. C. Greubel, Feb 26 2019

I:=[0,6,4,30,3,70,24,126,10,198,60,286]; [n le 12 select I[n] else 3*Self(n-4)-3*Self(n-8)+Self(n-12): n in [1..60]]; // Vincenzo Librandi, Apr 23 2017

Table[Which[OddQ@ n, 2 (1 + 2 #) (3 + 2 #) &[(n - 1)/2], Mod[n, 4] == 0, # (1 + 2 #) &[n/4], True, 4 (1 + #) (1 + 2 #) &[(n - 2)/4]], {n, 0, 60}] (* or *)
CoefficientList[Series[x(6 +4x +30x^2 +3x^3 +52x^4 +12x^5 +36x^6 +x^7 +6x^8 -2x^10)/((1-x)^3*(1+x)^3*(1+x^2)^3), {x, 0, 60}], x] (* Michael De Vlieger, Apr 22 2017 *)
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1}, {0,6,4,30,3,70,24,126,10,198,60, 286}, 80] (* Vincenzo Librandi, Apr 23 2017 *)

m=60; v=concat([0,6,4,30,3,70,24,126,10,198,60,286], vector(m-12)); for(n=13, m, v[n]=3*v[n-4]-3*v[n-8]+v[n-12]); v \\ G. C. Greubel, Feb 26 2019

(x*(6+4*x+30*x^2+3*x^3+52*x^4+12*x^5+36*x^6+x^7+6*x^8-2*x^10)/((1-x)^3 *(1+x)^3*(1+x^2)^3 )).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n) = A061037(n+2) + A181318(n). - Paul Curtz, Jul 19 2011
a(n) = A060819(n) * A145979(n). - Paul Curtz, Sep 06 2011
G.f.: x*(6+4*x+30*x^2+3*x^3+52*x^4+12*x^5+36*x^6+x^7+6*x^8-2*x^10) /( (1-x)^3 *(1+x)^3 *(1+x^2)^3 ). - R. J. Mathar, Jun 17 2011
Let BEB(n) = a(n)/A061038(n+2) = A060819(n)/A145979(n). Then (BEB(n))^2 = A181318(n)/A061038(n+2) = BEB(n) - A061037(n+2)/A061038(n+2). - Paul Curtz, Jul 19 2011, index corrected by R. J. Mathar, Sep 09 2011
From Luce ETIENNE, Apr 18 2017: (Start)
a(n) = n*(n + 2)*(37 - 27*(-1)^n - 3*((-1)^((2*n + 1 - (-1)^n)/4) + (-1)^((2*n - 1 + (-1)^n)/4)))/32.
a(n) = n*(n+2)*(37-27*cos(n*Pi) - 6*cos(n*Pi/2))/32.
a(n) = n*(n + 2)*(37 - 27*(-1)^n - 3*(i^n + (-i)^n))/32, where i=sqrt(-1). (End)

A208853 Array of hypotenuses of primitive Pythagorean triangles when read by SW-NE diagonals.

Original entry on oeis.org

5, 13, 17, 29, 25, 37, 53, 41, 0, 65, 85, 65, 61, 73, 101, 125, 97, 85, 89, 109, 145, 173, 137, 0, 113, 0, 0, 197, 229, 185, 157, 145, 149, 169, 205, 257, 293, 241, 205, 185, 181, 193, 221, 265, 325, 365, 305, 0, 233, 221, 0

Offset: 1

Wolfdieter Lang, Mar 05 2012

All primitive Pythagorean triples (see the links) (a,b,c), with a odd, b even, hence c odd, are given by c=u^2 + v^2, with u odd, u=2*n+1, n>=1, v even, v=2*m, m>=1, and gcd(u,v)=1. The present array is c=c(n,m) = (2*n-1)^2 + (2*m)^2, if gcd(2*n-1,2*m)=1 and 0 otherwise. The corresponding triangle, read by SW-NE diagonals, is T(n,m):= c(n-m+1,m). The 0 entries indicate that there are only non-primitive triples for these n,m values. See the example section for the scaling factor g=gcd(u,v)^2 for such non-primitive triangles.
For the increasingly ordered c-values see A008846 (with multiplicity see A020882).
All primitive Pythagorean triples are given by
(a(n,m)=A208854(n,m), b(n,m)=A208855(n,m), c(n,m)), n>=1, m>=1. If this is (0,0,0) then no primitive triple exists for these n,m values. See the example section.
In the prime factorization of c(n,m) (which is odd) all prime factors are of the type 4*k+1 (see A002144). See the Niven-Zuckerman-Montgomery reference, Theorem 3.20, p. 164. For the general representation of positive integers as the sum of two squares see Theorem 2.15 by Fermat, p. 55. E.g.: c(5,2) = 85 = 5*17. c = 5*7^2 = 245 has a non-primitive solution 7^2*(1^2 + 2^2) = 7^2*c(1,1), therefore c(4,7)=0 in this array.
The triples with an even cathetus (b) and the hypotenuse (c) differing by 1 unit are (2*k+1, 4*T(k), 4*T(k)+1), k >= 1, with the triangular numbers A000217. The c values are given in A001844. E.g., (n,m)=(1,1), k=1. (3,4,5); (n,m)=(2,1), k=2, (5,12,13); (n,m)=(2,2), k=3, (7,24,25). See the example section for the table.
The triples with an odd cathetus (a) and the hypotenuse differing by 2 units are (4*k^2-1, 4*k, 4*k^2+1), k >= 1. These triples are given in (A000466(k), A008586(k), A053755(k)). E.g., (n,m)=(1,4), k=4, (63,16,65).
The triples with the catheti differing by one length unit are generated by a substitution rule for the (u,v) values starting with (1,1). See a Wolfdieter Lang comment on A001653 for this rule. - Wolfdieter Lang, Mar 08 2012

			Triangle T(n,m):
......m|  1     2     3     4    5    6     7     8    9    10 ...
......v|  2     4     6     8   10   12    14    16   18    20 ...
n,  u
1,  1     5
2,  3    13    17
3,  5    29    25    37
4,  7    53    41     0    65
5,  9    85    65    61    73  101
6, 11   125    97    85    89  109  145
7, 13   173   137     0   113    0    0   197
8, 15   229   185   157   145  149  169   205   257
9, 17   293   241   205   185  181  193   221   265  325
10,19   365   305     0   233  221    0     0   281    0   401
...
Array c(n,m):
......m|  1    2    3     4     5    6     7     8    9    10 ...
......v|  2    4    6     8    10   12    14    16   18    20 ...
n,  u
1,  1     5   17   37    65   101  145   197   257  325   401
2   3    13   25    0    73   109    0   205   265    0   409
3,  5    29   41   61    89     0  169   221   281  349     0
4,  7    53   65   85   113   149  193     0   305  373   449
5,  9    85   97    0   145   181    0   277   337    0   481
6, 11   125  137  157   185   221  265   317   377  445   521
7, 13   173  185  205   233   269  313   365   425  493   569
8, 15   229  241    0   289     0    0   421   481    0     0
9, 17   293  305  325   353   389  433   485   545  613   689
10,19   365  377  397   425   461  505   557   617  685   761
...
------------------------------------------------------------------
Array of triples (a(n,m)=A208854,b(n,m)=A208855,c(n,m)):
......m|    1           2               3             4  ...
......v|    2           4               6             8  ...
n,  u
1,  1    (3,4,5)     (15,8,17)      (35,12,37)     (63,16,65)
2,  3    (5,12,13)   (7,24,25)       (0,0,0)       (55,48,73)
3,  5   (21,20,29)   (9,40,41)      (11,60,61)     (39,80,89)
4,  7   (45,28,53)   (33,56,65)     (13,84,85)    (15,112,113)
5,  9   (77,36,85)   (65,72,97)      (0,0,0)      (17,144,145)
6, 11  (117,44,125) (105,88,137)   (85,132,157)   (57,176,185)
7, 13  (165,52,173) (153,104,185)  (133,156,205) (105,208,233)
8, 15  (221,60,229) (209,120,241)    (0,0,0)     (161,240,289)
9, 17  (285,68,293) (273,136,305)  (253,204,325) (225,272,353)
10,19  (357,76,365) (345,152,377)  (325,228,397) (297,304,425)
...
Array continued:
Array of triples (a(n,m)=A208854,b(n,m)=A208855,c(n,m)):
......m|     5            6               7           8  ...
......v|    10           12              14          16  ...
n,  u
1,  1   (99,20,101)  (143,24,145)  (195,28,197)  (255,32,257)
2   3   (91,60,109)     (0,0,0)    (187,84,205)  (247,96,265)
3,  5    (0,0,0)     (119,120,169) (171,140,221) (231,160,281)
4,  7  (51,140,149)  (95,168,193)     (0,0,0)    (207,224,305)
5,  9  (19,180,181)     (0,0,0)    (115,252,277) (175,288,337)
6, 11  (21,220,221)  (23,264,265)  (75,308,317)  (135,352,377)
7, 13  (69,260,269)  (25,312,313)  (27,364,365)  (87,416,425)
8, 15     (0,0,0)       (0,0,0)    (29,420,421)  (31,480,481)
9, 17  (189,340,389) (145,408,433) (93,476,485)  (33,544,545)
10,19  (261,380,461) (217,456,505) (165,532,557) (105,608,617)
...
(0,0,0) indicates that no primitive Pythagorean triangle exists for these (n,m) values. The corresponding scaled triples would be (a,b,c) = g*(a/g,b/g,c/g), with g=gcd(u,v)^2 for such non-primitive triangles. E.g., c(n,m) = c(5,3) = 0, (u,v) = (9,6), g = 3^2, (45,108,117) = 3^2*(45/9,108/9,117/9) = 9*(5,12,13). The scaling factor for the primitive triangle (5,12,13), tabulated for c(n,m)=(2,1), is here 9.

I. Niven, H. S. Zuckerman and H.L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley & Sons, New York, 1991

Ron Knott, Pythagorean Triples and Online Calculators.
E. S. Rowland, Primitive Solutions to x^2 + y^2 = z^2
Michael Somos, Table of primitive Pythagorean triples and related parameters
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A020882, A002144, A208854, A208855.

T(n,m) = c(n-m+1,m), n >= m >= 1, with c(n,m) := (2*n-1)^2 + (2*m)^2, if gcd(2*n-1, 2*m) = 1 and 0 otherwise.

A282284 Least common multiple of 3n+1 and 3n-1.

Original entry on oeis.org

1, 4, 35, 40, 143, 112, 323, 220, 575, 364, 899, 544, 1295, 760, 1763, 1012, 2303, 1300, 2915, 1624, 3599, 1984, 4355, 2380, 5183, 2812, 6083, 3280, 7055, 3784, 8099, 4324, 9215, 4900, 10403, 5512, 11663, 6160, 12995, 6844, 14399, 7564, 15875, 8320, 17423

Offset: 0

Colin Barker, Feb 11 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000466, A136017, A141759, A282285, A282286.

Table[LCM@@{3n+1,3n-1},{n,0,50}] (* or *) LinearRecurrence[{0,3,0,-3,0,1},{1,4,35,40,143,112,323},60] (* Harvey P. Dale, Sep 05 2020 *)

PARI
```
vector(60, n, n--; lcm(3*n+1, 3*n-1))
```

Vec((1+4*x+32*x^2+28*x^3+41*x^4+4*x^5-2*x^6) / ((1-x)^3*(1+x)^3) + O(x^60))

a(n) = 9*n^2-1 for n>0 and even.
a(n) = (9*n^2-1)/2 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
G.f.: (1+4*x+32*x^2+28*x^3+41*x^4+4*x^5-2*x^6) / ((1-x)^3*(1+x)^3).

A282285 Least common multiple of 5n+1 and 5n-1.

Original entry on oeis.org

1, 12, 99, 112, 399, 312, 899, 612, 1599, 1012, 2499, 1512, 3599, 2112, 4899, 2812, 6399, 3612, 8099, 4512, 9999, 5512, 12099, 6612, 14399, 7812, 16899, 9112, 19599, 10512, 22499, 12012, 25599, 13612, 28899, 15312, 32399, 17112, 36099, 19012, 39999, 21012

Offset: 0

Colin Barker, Feb 11 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000466, A136017, A141759, A282284, A282286.

PARI
```
vector(60, n, n--; lcm(5*n+1, 5*n-1))
```

Vec((1+12*x+96*x^2+76*x^3+105*x^4+12*x^5-2*x^6) / ((1-x)^3*(1+x)^3) + O(x^60))

a(n) = 25*n^2-1 for n>0 and even.
a(n) = (25*n^2-1)/2 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
G.f.: (1+12*x+96*x^2+76*x^3+105*x^4+12*x^5-2*x^6) / ((1-x)^3*(1+x)^3).

A282286 Least common multiple of 7n+1 and 7n-1.

Original entry on oeis.org

1, 24, 195, 220, 783, 612, 1763, 1200, 3135, 1984, 4899, 2964, 7055, 4140, 9603, 5512, 12543, 7080, 15875, 8844, 19599, 10804, 23715, 12960, 28223, 15312, 33123, 17860, 38415, 20604, 44099, 23544, 50175, 26680, 56643, 30012, 63503, 33540, 70755, 37264, 78399

Offset: 0

Colin Barker, Feb 11 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000466, A136017, A141759, A282284, A282285.

PARI
```
vector(60, n, n--; lcm(7*n+1, 7*n-1))
```

Vec((1+24*x+192*x^2+148*x^3+201*x^4+24*x^5-2*x^6) / ((1-x)^3*(1+x)^3) + O(x^60))

a(n) = 49*n^2-1 for n>0 and even.
a(n) = (49*n^2-1)/2 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
G.f.: (1+24*x+192*x^2+148*x^3+201*x^4+24*x^5-2*x^6) / ((1-x)^3*(1+x)^3).

A128139 Triangle read by rows: matrix product A004736 * A128132.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 7, 5, 1, 6, 9, 10, 9, 6, 1, 7, 11, 13, 13, 11, 7, 1, 8, 13, 16, 17, 16, 13, 8, 1, 9, 15, 19, 21, 21, 19, 15, 9, 1, 10, 17, 22, 25, 26, 25, 22, 17, 10

Offset: 0

Gary W. Adamson, Feb 16 2007

A077028 with the final term in each row omitted.
Interchanging the factors in the matrix product leads to A128140 = A128132 * A004736.
From Gary W. Adamson, Jul 01 2012: (Start)
Alternatively, antidiagonals of an array A(n,k) of sequences with arithmetic progressions as follows:
  1, 2, 3,  4,  5,  6, ...
  1, 3, 5,  7,  9, 11, ...
  1, 4, 7, 10, 13, 16, ...
  1, 5, 9, 13, 17, 21, ...
  ... (End)
From Gary W. Adamson, Jul 02 2012: (Start)
A summation generalization for Sum_{k>=1} 1/(A(n,k)*A(n,k+1)) (formulas copied from A002378, A000466, A085001, A003185):
  1 = 1/(1)*(2) + 1/(2)*(3) + 1/(3)*(4)  + ...
  1 = 2/(1)*(3) + 2/(3)*(5) + 2/(5)*(7)  + ...
  1 = 3/(1)*(4) + 3/(4)*(7) + 3/(7)*(10) + ...
  1 = 4/(1)*(5) + 4/(5)*(9) + 4/(9)*(13) + ...
  ...
As a summation of terms equating to a definite integral:
  Integral_{0..1} dx/(1+x) = ... 1 - 1/2 + 1/3 - 1/4 + ... = log(2).
  Integral_{0..1} dx/(1+x^2) = 1 - 1/3 + 1/5 - 1/7  + ... = Pi/4 (see A157142)
  Integral_{0..1} dx/(1+x^3) = 1 - 1/4 + 1/7 - 1/10 + ... (see A016777)
  Integral_{0..1} dx/(1+x^4) = 1 - 1/5 + 1/9 - 1/13 + ... (see A016813). (End)

			First few rows of the triangle:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  5,  4;
  1,  5,  7,  7,  5;
  1,  6,  9, 10,  9,  6;
  1,  7, 11, 13, 13, 11,  7;
  1,  8, 13, 16, 17, 16, 13,  8;
  1,  9, 15, 19, 21, 21, 19, 15,  9;
  1, 10, 17, 22, 25, 26, 25, 22, 17, 10;
  ...

Cf. A004736, A128132, A128140, A004006 (row sums).

A004736 * A128132 as infinite lower triangular matrices.

T(n,k) = k*(1+n-k)+1 = 1 + A094053(n+1,1+n-k). - R. J. Mathar, Jul 09 2012

A157913 a(n) = 64*n^2 - 16.

Original entry on oeis.org

48, 240, 560, 1008, 1584, 2288, 3120, 4080, 5168, 6384, 7728, 9200, 10800, 12528, 14384, 16368, 18480, 20720, 23088, 25584, 28208, 30960, 33840, 36848, 39984, 43248, 46640, 50160, 53808, 57584, 61488, 65520, 69680, 73968, 78384, 82928, 87600, 92400, 97328, 102384

Offset: 1

Vincenzo Librandi, Mar 09 2009

The identity (8*n^2 - 1)^2 - (64*n^2 - 16)*n^2 = 1 can be written as A157914(n)^2 - a(n)*n^2 = 1. - Vincenzo Librandi, Feb 09 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000466, A157914.

I:=[48, 240, 560]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 09 2012

LinearRecurrence[{3, -3, 1}, {48, 240, 560}, 50] (* Vincenzo Librandi, Feb 09 2012 *)
64*Range[40]^2-16 (* Harvey P. Dale, Jul 27 2012 *)

for(n=1, 40, print1(64*n^2 - 16", ")); \\ Vincenzo Librandi, Feb 09 2012

From  Vincenzo Librandi, Feb 09 2012: (Start)
G.f.: -16*x*(3 + 6*x - x^2)/(x - 1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 07 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/32.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi-2)/64. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 16*(exp(x)*(4*x^2 + 4*x - 1) + 1).
a(n) = 16*A000466(n). (End)

A173039 Odd numerators of the fractions (1/4-1/n^2), n>= -2.

Original entry on oeis.org

-3, -1, -3, 5, 3, 21, 45, 15, 77, 117, 35, 165, 221, 63, 285, 357, 99, 437, 525, 143, 621, 725, 195, 837, 957, 255, 1085, 1221, 323, 1365, 1517, 399, 1677, 1845, 483, 2021, 2205, 575, 2397, 2597, 675

Offset: 1

Paul Curtz, Nov 21 2010

sign

Odd numbers in A061037, which is extended to negative n, and uses -1 to represent -1/0 at n=0. Trisections are apparently A003185, A000466 and A085027.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, -3, 0, 0, 1).

Conjecture: a(n)= +3*a(n-3) -3*a(n-6) +a(n-9). G.f.: ( 3+x+3*x^2-14*x^3-6*x^4-30*x^5-21*x^6-3*x^7-5*x^8 ) / ( (x-1)^3*(1+x+x^2)^3 ). - R. J. Mathar, Dec 02 2010

A173200 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 - 1 for a = 0,1,2, ... (solutions x are given by A053755).

Original entry on oeis.org

0, 11, 70, 225, 524, 1015, 1746, 2765, 4120, 5859, 8030, 10681, 13860, 17615, 21994, 27045, 32816, 39355, 46710, 54929, 64060, 74151, 85250, 97405, 110664, 125075, 140686, 157545, 175700, 195199, 216090, 238421, 262240, 287595, 314534, 343105

Offset: 1

Michel Lagneau, Feb 12 2010

For many values of k for the equation y^2 = x^3 + k, all the solutions are known. For example, we have solutions for k=-2: (x,y) = (3,-5) and (3,5). A complete resolution for all integers k is unknown. Theorem: Let k be < -1, free of square factors, with k == 2 or 3 (mod 4). Suppose that the number of classes h(Q(sqrt(k))) is not divisible by 3. Then the equation y^2 = x^3 + k admits integer solutions if and only if k = 1 - 3a^2 or 1 - 3a^2 where a is an integer. In this case, the solutions are x = a^2 - k, y = a(a^2 + 3k) or -a(a^2 + 3k) (the first reference gives the proof of this theorem). With k = -1 - 3a^2, we obtain the solutions x = 4a^2 + 1, y = a(8a^2 + 3) or -a(8a^2 + 3). For the case k = 1 - 3a^2, we obtain the solution x = 4a^2 - 1 given by the sequence A000466.

			With a=3, x =37 and y = 225, and then 225^2 = 37^2 - 28.

T. Apostol, Introduction to Analytic Number Theory, Springer, 1976.
D. Duverney, Théorie des nombres (2e edition), Dunod, 2007, p. 151.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
W. J. Ellison, F. Ellison, J. Pesek, C. E. Stall, and D. S. Stall, The diophantine equation y^2 + k = x^3, J. Number Theory, Vol. 4 (1972), pp. 107-117.
John J. O'Connor and Edmund F. Robertson, Louis Joel Mordell.
Helmut Richter, Solutions of Mordell's equation y^2 = x^3 + k. (solutions for 0
Eric Weisstein's World of Mathematics, Mordell Curve.
David J. Wright, Mordell's Equation.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000466.

I:=[0, 11, 70, 225]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 02 2012

for a from 0 to 150 do : z := evalf(a*(8*a^2 + 3)) : print (z) :od :

CoefficientList[Series[x*(11+26*x+11*x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,11,70,225},40] (* Harvey P. Dale, Dec 21 2016 *)

for n in range(1,20): print(8*n**3 - 24*n**2 + 27*n - 11, end=', ') # Stefano Spezia, Dec 05 2018

y = a*(8*a^2 + 3).
From  Colin Barker, Apr 26 2012: (Start)
a(n) = 8*n^3 - 24*n^2 + 27*n - 11.
G.f.: x^2*(11 + 26*x + 11*x^2)/(1 - x)^4. (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 02 2012
E.g.f.: 11 + exp(x)*(-11 + 11*x + 8*x^3). - Elmo R. Oliveira, Aug 15 2025

cp's OEIS Frontend

A173294 Values of 16*n^2+24*n+7, n>=0, each duplicated.

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A191567 Four interlaced 2nd order polynomials: a(4*k) = k*(1+2*k); a(1+2*k) = 2*(1+2*k)*(3+2*k); a(2+4*k) = 4*(1+k)*(1+2*k).

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A208853 Array of hypotenuses of primitive Pythagorean triangles when read by SW-NE diagonals.

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A282284 Least common multiple of 3*n+1 and 3*n-1.

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A282285 Least common multiple of 5*n+1 and 5*n-1.

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A282286 Least common multiple of 7*n+1 and 7*n-1.

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A128139 Triangle read by rows: matrix product A004736 * A128132.

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A157913 a(n) = 64*n^2 - 16.

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A173294 Values of 16n^2+24n+7, n>=0, each duplicated.

A191567 Four interlaced 2nd order polynomials: a(4k) = k(1+2k); a(1+2k) = 2(1+2k)(3+2k); a(2+4k) = 4(1+k)(1+2k).

A282284 Least common multiple of 3n+1 and 3n-1.

A282285 Least common multiple of 5n+1 and 5n-1.

A282286 Least common multiple of 7n+1 and 7n-1.