cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A164131 Numbers k such that k^2 == 2 (mod 31).

Original entry on oeis.org

8, 23, 39, 54, 70, 85, 101, 116, 132, 147, 163, 178, 194, 209, 225, 240, 256, 271, 287, 302, 318, 333, 349, 364, 380, 395, 411, 426, 442, 457, 473, 488, 504, 519, 535, 550, 566, 581, 597, 612, 628, 643, 659, 674, 690, 705, 721, 736, 752, 767, 783, 798, 814
Offset: 1

Views

Author

Vincenzo Librandi, Aug 11 2009

Keywords

Comments

Sequences of the type n^2 == 2 (mod m) are basically defined for each m of A057126. See A047341 (m=7), A113804 (m=14), A155449 (m=17), A155450 (m=23), A158803 (m=41) etc. - R. J. Mathar, Aug 26 2009

Examples

			At n= 4, a(4)=(31-1+186)/4=54. At n=5, a(5)=(31+1+248)/4=70.

Links

Crossrefs

Cf. A057126, A047341, A113804, A155449, A155450, A158803.

Programs

  • Mathematica
    Select[Range[850],Mod[#^2,31]==2&]  (* Harvey P. Dale, Feb 04 2011 *)
  • PARI
    isok(k) = Mod(k, 31)^2 == 2; \\ Michel Marcus, Nov 22 2022

Formula

a(n) = a(n-1)+a(n-2)-a(n-3).
a(n) = (31+(-1)^(n-1)+62(n-1))/4.
G.f.: x*(8+15*x+8*x^2)/((1+x)*(x-1)^2). - R. J. Mathar, Aug 26 2009
a(n) = 31*(n-1)-a(n-1) with n>1, a(1)=8. - Vincenzo Librandi, Nov 30 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(15*Pi/62)*Pi/31. - Amiram Eldar, Feb 28 2023

Extensions

Entries checked by R. J. Mathar, Aug 26 2009

A204769 a(n) = 151*(n-1) - a(n-1) with n>1, a(1)=46.

Original entry on oeis.org

46, 105, 197, 256, 348, 407, 499, 558, 650, 709, 801, 860, 952, 1011, 1103, 1162, 1254, 1313, 1405, 1464, 1556, 1615, 1707, 1766, 1858, 1917, 2009, 2068, 2160, 2219, 2311, 2370, 2462, 2521, 2613, 2672, 2764, 2823
Offset: 1

Views

Author

Vincenzo Librandi, Mar 08 2012

Keywords

Comments

Positive numbers k such that k^2 == 2 (mod 151), where the prime 151 == -1 (mod 8).
Equivalently, numbers k such that k == 46 or 105 (mod 151). - Bruno Berselli, Mar 08 2012

Links

Crossrefs

Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k-1: A047341, A155450, A164131, A164135, A167533, A167534, A177044, A177046, A204766.
Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k+1: A155449, A158803, A159007, A159008, A176010, A206525.

Programs

  • Magma
    [(-151-33*(-1)^n+302*n)/4: n in [1..60]];
  • Mathematica
    LinearRecurrence[{1,1,-1}, {46,105,197}, 40] (* or *) CoefficientList[Series[x*(46+59*x+46*x^2)/((1+x)*(x-1)^2),{x,0,33}],x] (* or *) a[1] = 46; a[n_] := a[n] = 151*(n-1) - a[n-1]; Table[a[n], {n, 1, 40}]
  • PARI
    a(n)=(-151-33*(-1)^n+302*n)/4 \\ Charles R Greathouse IV, Oct 16 2015

Formula

G.f.: x*(46+59*x+46*x^2)/((1+x)*(x-1)^2).
a(n) = (-151-33*(-1)^n+302*n)/4.
a(n) = a(n-1) +a(n-2) -a(n-3).
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(59*Pi/302)*Pi/151. - Amiram Eldar, Feb 28 2023

A206525 a(n) = 113*(n-1) - a(n-1) with n>1, a(1)=51.

Original entry on oeis.org

51, 62, 164, 175, 277, 288, 390, 401, 503, 514, 616, 627, 729, 740, 842, 853, 955, 966, 1068, 1079, 1181, 1192, 1294, 1305, 1407, 1418, 1520, 1531, 1633, 1644, 1746, 1757, 1859, 1870, 1972, 1983, 2085, 2096, 2198, 2209
Offset: 1

Views

Author

Vincenzo Librandi, Mar 09 2012

Keywords

Comments

Positive numbers k such that k^2 == 2 (mod 113), where the prime 113 == 1 (mod 8).
Equivalently, numbers k such that k == 51 or 62 (mod 113).

Links

Crossrefs

Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k+1: A155449, A158803, A159007, A159008, A176010.
Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k-1: A047341, A155450, A164131, A164135, A167533, A167534, A177044, A177046, A204769.

Programs

  • Magma
    [(-113-91*(-1)^n+226*n)/4: n in [1..60]];
  • Mathematica
    LinearRecurrence[{1, 1, -1}, {51, 62, 164}, 40] (* or *) CoefficientList[Series[x*(51+11*x+51*x^2)/((1+x)*(x-1)^2), {x, 0, 40}], x] (* or *) a[1] = 51; a[n_] := a[n] = 113*(n-1) - a[n-1]; Table[a[n], {n, 1, 40}]

Formula

a(n) = a(n-2) + 113.
G.f.: x*(51+11*x+51*x^2)/((1+x)*(x-1)^2).
a(n) = (-113-91*(-1)^n+226*n)/4.
a(n) = a(n-1)+a(n-2)-a(n-3).
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(11*Pi/226)*Pi/113. - Amiram Eldar, Feb 28 2023

A204766 a(n) = 167*(n-1)-a(n-1) with n>1, a(1)=13.

Original entry on oeis.org

13, 154, 180, 321, 347, 488, 514, 655, 681, 822, 848, 989, 1015, 1156, 1182, 1323, 1349, 1490, 1516, 1657, 1683, 1824, 1850, 1991, 2017, 2158, 2184, 2325, 2351, 2492, 2518, 2659, 2685, 2826, 2852, 2993, 3019, 3160
Offset: 1

Views

Author

Vincenzo Librandi, Mar 09 2012

Keywords

Comments

Positive numbers k such that k^2 == 2 (mod 167), where the prime 167 == -1 (mod 8).
Equivalently, numbers k such that k == 13 or 154 (mod 167).

Links

Crossrefs

Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k-1: A047341, A155450, A164131, A164135, A167533, A167534, A177044, A177046, A204769.
Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k+1: A155449, A158803, A159007, A159008, A176010, A206525, A206526.

Programs

  • Magma
    [(-167+115*(-1)^n+334*n)/4: n in [1..60]];
  • Mathematica
    CoefficientList[Series[x*(13+141*x+13*x^2)/((1+x)*(x-1)^2), {x, 0, 40}], x] (* or *) LinearRecurrence[{1, 1, -1}, {13, 154, 180}, 40]

Formula

G.f.: x*(13+141*x+13*x^2)/((1+x)*(x-1)^2).
a(n) = (-167+115*(-1)^n+334*n)/4.
a(n) = a(n-1)+a(n-2)-a(n-3).
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(13*Pi/167)*Pi/167. - Amiram Eldar, Feb 28 2023

A206526 a(n) = 137*(n-1) - a(n-1) with n>1, a(1)=31.

Original entry on oeis.org

31, 106, 168, 243, 305, 380, 442, 517, 579, 654, 716, 791, 853, 928, 990, 1065, 1127, 1202, 1264, 1339, 1401, 1476, 1538, 1613, 1675, 1750, 1812, 1887, 1949, 2024, 2086, 2161, 2223, 2298, 2360, 2435, 2497, 2572, 2634, 2709, 2771, 2846, 2908, 2983
Offset: 1

Views

Author

Vincenzo Librandi, Mar 09 2012

Keywords

Comments

Positive numbers k such that k^2 == 2 (mod 137), where the prime 137 == 1 (mod 8).
Equivalently, numbers k such that k == 31 or 106 (mod 137).
The subsequence of primes begins: 31, 853, 1613, 1949, 2161. - Jonathan Vos Post, Mar 09 2012

Links

Crossrefs

Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k+1: A155449, A158803, A159007, A159008, A176010, A206525.
Sequences of the type n^2 == 2 (mod p), where p is a prime of the form 8k-1: A047341, A155450, A164131, A164135, A167533, A167534, A177044, A177046, A204769.

Programs

  • Magma
    [(-137+13*(-1)^n+274*n)/4: n in [1..60]];
  • Magma
    [n: n in [1..3000] | n^2 mod 137 eq 2]; // Vincenzo Librandi, Mar 31 2016
  • Mathematica
    LinearRecurrence[{1, 1, -1}, {31, 106, 168}, 40] (* or *) CoefficientList[Series[x*(31+75*x+31*x^2)/((1+x)*(x-1)^2), {x, 0, 50}], x] (* or *) a[1] = 31; a[n_] := a[n] = 137*(n-1) - a[n-1]; Table[a[n], {n, 1, 40}]

Formula

a(n) = a(n-2) + 137.
G.f.: x*(31+75*x+31*x^2)/((1+x)*(x-1)^2).
a(n) = (-137+13*(-1)^n+274*n)/4.
a(n) = a(n-1)+a(n-2)-a(n-3).
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(31*Pi/137)*Pi/137. - Amiram Eldar, Feb 28 2023
Showing 1-5 of 5 results.

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