A047471 -id:A047471 - cp's OEIS frontend

A339774 a(n) is the least k such that 3^k == A047471(n) (mod 2^A047471(n)).

0, 1, 2, 39, 23988, 2685, 1079830, 3, 1798749736, 7936950713, 314244766442, 895397198495, 65283613526364, 203550894972341, 27025091041430142, 54487836217255419, 2756442714229679952, 34856858877609547377, 2262552012902592868562, 4616799241038411627031, 4, 116433218705414728492013

Offset: 1

J. M. Bergot and Robert Israel, Dec 16 2020

nonn

For n >= 3, 3^x == y (mod 2^n) has solutions x if and only if y is in A047471.

			a(4) = 39 because A047471(4) = 11 and 3^39 == 11 (mod 2^11).

Robert Israel, Table of n, a(n) for n = 1..831

Cf. A047471.

f:= proc(n) local k,v;
  v:= subs(msolve(3^k=n,2^n),k);
  subs(op(indets(v))=0,v)
end proc:
seq(seq(f(8*i+j),j=[1,3]),i=0..10);

a((3^k - (-1)^k)/4 + 1) = k.

A033200 Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.

Original entry on oeis.org

3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(-2)). - N. J. A. Sloane, Dec 25 2017
Fermat knew of the relationship between a prime being congruent to 1 or 3 mod 8 and its being the sum of a square and twice a square, and claimed to have a firm proof of this fact. These numbers are not primes in Z[sqrt(-2)], as they have x - y sqrt(-2) as a divisor. - Alonso del Arte, Dec 07 2012
Terms m in A047471 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012
This sequence gives the primes p which satisfy norm(rho(p)) = + 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For the minimal polynomial C see A187360. For p == 1 (mod 8) the norm is C(p, 0) (see a comment on 4*A005123) and for p == 3 (mod 8) the norm is -C(p, 0) (see a comment on A186297). For the primes with norm(rho(p)) = -1 see A003628. - Wolfdieter Lang, Oct 24 2013
If p is a member then it has a unique representation as x^2+2y^2 [Frei, Theorem 3]. - N. J. A. Sloane, May 30 2014
Primes that are the quarter perimeter of a Heronian triangle. Such primes are unique to the Heronian triangle (see Yiu link). - Frank M Jackson, Nov 30 2014

			Since 11 is prime and 11 == 3 (mod 8), 11 is in the sequence. (Also 11 = 3^2 + 2 * 1^2 = (3 + sqrt(-2))(3 - sqrt(-2)).)
Since 17 is prime and 17 == 1 (mod 8), 17 is in the sequence.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from T. D. Noe]
G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), p. 56.
Zak Seidov, Table of n, a(n), x and y for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Paul Yiu, CRUX, Problem 2331, Proposed by Paul Yiu
Paul Yiu and Jill S. Taylor, CRUX, Problem 2331, Solution pp 185-186
Index to sequences related to decomposition of primes in quadratic fields

Cf. A033203.

a033200 n = a033200_list !! (n-1)
a033200_list = filter ((== 1) . a010051) a047471_list
-- Reinhard Zumkeller, Dec 29 2012

[p: p in PrimesUpTo(600) | p mod 8 in [1, 3]]; // Vincenzo Librandi, Aug 04 2012

Rest[QuadPrimes2[1, 0, 2, 10000]] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,3},Mod[#,8]]&] (* Harvey P. Dale, Jun 09 2017 *)

is(n)=n%8<4 && n%2 && isprime(n) \\ Charles R Greathouse IV, Feb 09 2017

a(n) = A033203(n+1). - Zak Seidov, May 29 2014
A007519 UNION A007520. - R. J. Mathar, Jun 09 2020
L(-2, a(n)) = +1, n >= 1, with the Legendre symbol L. -Wolfdieter Lang, Jul 24 2024

A047404 Numbers that are congruent to {1, 2, 3, 6} mod 8.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 11, 14, 17, 18, 19, 22, 25, 26, 27, 30, 33, 34, 35, 38, 41, 42, 43, 46, 49, 50, 51, 54, 57, 58, 59, 62, 65, 66, 67, 70, 73, 74, 75, 78, 81, 82, 83, 86, 89, 90, 91, 94, 97, 98, 99, 102, 105, 106, 107, 110, 113, 114, 115, 118, 121, 122, 123

Offset: 1

N. J. A. Sloane

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

Cf. A016825, A047471, A056594.

[n : n in [0..150] | n mod 8 in [1, 2, 3, 6]]; // Wesley Ivan Hurt, May 30 2016

A047404:=n->(4*n-4+I^(1-n)-I^(1+n))/2: seq(A047404(n), n=1..100); # Wesley Ivan Hurt, May 30 2016

Table[(4n-4+I^(1-n)-I^(1+n))/2, {n, 80}] (* Wesley Ivan Hurt, May 30 2016 *)
LinearRecurrence[{2,-2,2,-1},{1,2,3,6},70] (* Harvey P. Dale, Sep 15 2024 *)

a(n)=(n-1)\4*8+[6,1,2,3][n%4+1] \\ Charles R Greathouse IV, Jun 11 2015

[lucas_number1(n,0,1)+2*n-2 for n in range(1,56)] # Zerinvary Lajos, Jul 06 2008

a(n) = A056594(n) + 2*n-2. - Zerinvary Lajos, Jul 06 2008
G.f.: x*(1+x)*(2*x^2-x+1)/((x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 30 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (4*n-4+i^(1-n)-i^(1+n))/2 where i = sqrt(-1).
a(2k) = A016825(k-1) k>0, a(2k-1) = A047471(k). (End)
E.g.f.: 2 + sin(x) + 2*(x - 1)*exp(x). - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/8 + log(2)/4. - Amiram Eldar, Dec 23 2021

A047476 Numbers that are congruent to {0, 1, 2, 3} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 24, 25, 26, 27, 32, 33, 34, 35, 40, 41, 42, 43, 48, 49, 50, 51, 56, 57, 58, 59, 64, 65, 66, 67, 72, 73, 74, 75, 80, 81, 82, 83, 88, 89, 90, 91, 96, 97, 98, 99, 104, 105, 106, 107, 112, 113, 114, 115, 120, 121, 122

Offset: 1

N. J. A. Sloane

Primes of this sequence are in A033203. All of these numbers satisfy the condition that n XOR 4 = n + 4. - Brad Clardy, Jul 24 2012

Numbers k such that floor(k/4) = 2*floor(k/8). - Bruno Berselli, Oct 05 2017

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

Cf. A033203 (primes), A047467, A047471.

a047476 n = a047476_list !! (n-1)
a047476_list = [n | n <- [1..], mod n 8 <= 3]
-- Reinhard Zumkeller, Dec 29 2012

I:=[0, 1, 2, 3, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 16 2012

A047476:=n->(-7-(-1)^n-(1-I)*(-I)^n-(1+I)*I^n+4*n)/2: seq(A047476(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,300], MemberQ[{0,1,2,3}, Mod[#,8]]&] (* Vincenzo Librandi, May 16 2012 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,3,8},100] (* G. C. Greubel, Jun 01 2016 *)

x='x+O('x^100); concat(0, Vec(x^2*(1+x+x^2+5*x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Dec 24 2015

a(n) = 8 * floor(n/4) + (n mod 4), with offset 0.. a(0)=0. - Gary Detlefs, Mar 09 2010
From Colin Barker, May 14 2012: (Start)
a(n) = (-7 - (-1)^n - (1-i)*(-i)^n - (1+i)*i^n + 4*n)/2, where i=sqrt(-1).
G.f.: x^2*(1 + x + x^2 + 5*x^3)/((1 - x)^2*(1 + x)*(1 + x^2)). (End)
a(n) = a(n-1) + a(n-4) - a(n-5). - Vincenzo Librandi, May 16 2012
a(2*k) = A047471, a(2*k-1) = A047467(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: 5 + sin(x) - cos(x) + (2*x - 3)*sinh(x) + 2*(x - 2)*cosh(x). - Ilya Gutkovskiy, Jun 01 2016
Sum_{n>=2} (-1)^n/a(n) = (2*sqrt(2)-1)*Pi/16 + 5*log(2)/8. - Amiram Eldar, Dec 19 2021

A153860 Triangle by columns: leftmost column = (1, 0, 1, -1, 1, -1, 1, ...); columns >1 = (1, 1, 0, 0, 0, ...).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, -1, 0, 1, 1, 1, 0, 0, 1, 1, -1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Jan 03 2009

As an infinite lower triangular matrix M; M * [1,2,3,...] = A063210: (1, 2, 6, 6, 10, 10, 14, 14, ...).
M * [1, 3, 5, 7, ...] = A047471, {1,3} mod 8.
Eigensequence of the triangle = A066983 starting (1, 1, 3, 3, 7, 9, 17, 25, ...).
Binomial transform of the triangle = A153861.
Row sums = A153284: (1, 1, 3, 1, 3, 1, 3, 1, ...).

			First few rows of the triangle:
   1;
   0, 1;
   1, 1, 1;
  -1, 0, 1, 1;
   1, 0, 0, 1, 1;
  -1, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 1, 1;
  -1, 0, 0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 0, 0, 1, 1;
  ...

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A153861 (binomial transform), A153284 (row sums), A063210, A047471, A066983.

a153860 n k = a153860_tabl !! (n-1) !! (k-1)
a153860_row n = a153860_tabl !! (n-1)
a153860_tabl = [1] : [0, 1] : iterate (\(x:xs) -> -x : 0 : xs) [1, 1, 1]
-- Reinhard Zumkeller, Dec 16 2013

Triangle by columns: leftmost column = (1, 0, 1, -1, 1, ...); columns > 1 = (1, 1, 0, 0, 0, ...).

A255840 a(n) = (4n^2 - 4n + 1 - (-1)^n)/2.

Original entry on oeis.org

0, 1, 4, 13, 24, 41, 60, 85, 112, 145, 180, 221, 264, 313, 364, 421, 480, 545, 612, 685, 760, 841, 924, 1013, 1104, 1201, 1300, 1405, 1512, 1625, 1740, 1861, 1984, 2113, 2244, 2381, 2520, 2665, 2812, 2965, 3120, 3281, 3444, 3613, 3784, 3961, 4140, 4325, 4512

Offset: 0

Wesley Ivan Hurt, Mar 07 2015

Take an n X n square grid and add unit squares along each side except for the corners --> do this repeatedly along each side with the same restriction until no squares can be added. a(n) is the total area of each figure. The perimeter, P, of each figure is given by P(n) = 4*A042963(n), n>0 (see example).

For n>0, partial sums of a(n) are in A056640.

			                                                                 _
                                                               _|_|_
                            _              _ _               _|_|_|_|_
                          _|_|_          _|_|_|_           _|_|_|_|_|_|_
              _ _       _|_|_|_|_      _|_|_|_|_|_       _|_|_|_|_|_|_|_|_
    _        |_|_|     |_|_|_|_|_|    |_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
   |_|       |_|_|       |_|_|_|      |_|_|_|_|_|_|       |_|_|_|_|_|_|_|
                           |_|          |_|_|_|_|           |_|_|_|_|_|
                                          |_|_|               |_|_|_|
                                                                |_|
   n=1        n=2          n=3             n=4                  n=5

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000290 (squares), A002620 (quarter-squares), A042963.

Cf. A056640, A085046, A255876.

[(4*n^2 - 4*n + 1 - (-1)^n)/2 : n in [0..100]];

A255840:=n->(4*n^2 - 4*n + 1 - (-1)^n)/2: seq(A255840(n), n=0..100);

CoefficientList[Series[x (1 + 2 x + 5 x^2)/((1 + x) (1 - x)^3), {x, 0, 50}], x]

vector(100,n,(4*(n-1)^2 - 4*(n-1) + 1 + (-1)^n)/2) \\ Derek Orr, Mar 09 2015

G.f.: x*(1+2*x+5*x^2)/((1+x)*(1-x)^3).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = A000290(n) + 4*A002620(n).
a(n) - a(n-1) = A047471(n). - Wesley Ivan Hurt, Apr 28 2017

A047454 Numbers that are congruent to {1, 2, 3, 4} mod 8.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 11, 12, 17, 18, 19, 20, 25, 26, 27, 28, 33, 34, 35, 36, 41, 42, 43, 44, 49, 50, 51, 52, 57, 58, 59, 60, 65, 66, 67, 68, 73, 74, 75, 76, 81, 82, 83, 84, 89, 90, 91, 92, 97, 98, 99, 100, 105, 106, 107, 108, 113, 114, 115, 116, 121, 122, 123

Offset: 1

N. J. A. Sloane

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

Cf. A047463, A047471.

I:=[1, 2, 3, 4, 9]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 15 2012

A047454:=n->(-5-I^(2*n)-(1-I)*(-I)^n-(1+I)*I^n+4*n)/2: seq(A047454(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,300], MemberQ[{1,2,3,4}, Mod[#,8]]&] (* Vincenzo Librandi, May 15 2012 *)

my(x='x+O('x^100)); Vec(x*(1+x+x^2+x^3+4*x^4)/((1-x)^2*(1+x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

From Colin Barker, May 14 2012: (Start)
a(n) = (-5-(-1)^n-(1-i)*(-i)^n-(1+i)*i^n+4*n)/2 where i=sqrt(-1).
G.f.: x*(1+x+x^2+x^3+4*x^4)/((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Vincenzo Librandi, May 15 2012
a(2k) = A047463(k), a(2k-1) = A047471(k). - Wesley Ivan Hurt, Jun 01 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 + 3*log(2)/8. - Amiram Eldar, Dec 23 2021

A047533 Numbers that are congruent to {1, 2, 3, 7} mod 8.

Original entry on oeis.org

1, 2, 3, 7, 9, 10, 11, 15, 17, 18, 19, 23, 25, 26, 27, 31, 33, 34, 35, 39, 41, 42, 43, 47, 49, 50, 51, 55, 57, 58, 59, 63, 65, 66, 67, 71, 73, 74, 75, 79, 81, 82, 83, 87, 89, 90, 91, 95, 97, 98, 99, 103, 105, 106, 107, 111, 113, 114, 115, 119, 121, 122, 123

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047471, A047524.

[n : n in [0..150] | n mod 8 in [1, 2, 3, 7]]; // Wesley Ivan Hurt, May 29 2016

A047533:=n->(8*n-7+I^(2*n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4: seq(A047533(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Table[(8n-7+I^(2n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 29 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 7, 9}, 50] (* G. C. Greubel, May 29 2016 *)

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+x+x^2+4*x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-7+i^(2*n)+(1+2*i)*i^(-n)+(1-2*i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047524(k), a(2k-1) = A047471(k). (End)
E.g.f.: (2 + 2*sin(x) + cos(x) + 4*(x - 1)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 29 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*sqrt(2)*Pi/16 + log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 23 2021

A047606 Numbers that are congruent to {1, 2, 3, 5} mod 8.

Original entry on oeis.org

1, 2, 3, 5, 9, 10, 11, 13, 17, 18, 19, 21, 25, 26, 27, 29, 33, 34, 35, 37, 41, 42, 43, 45, 49, 50, 51, 53, 57, 58, 59, 61, 65, 66, 67, 69, 73, 74, 75, 77, 81, 82, 83, 85, 89, 90, 91, 93, 97, 98, 99, 101, 105, 106, 107, 109, 113, 114, 115, 117, 121, 122, 123

Offset: 1

N. J. A. Sloane

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

Cf. A047471, A047617.

[n: n in [1..120] | n mod 8 in [1,2,3,5]];

A047606:=n->2*n-3+(3-I^(2*n))*(1-I^(n*(n+1)))/4: seq(A047606(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[120], MemberQ[{1, 2, 3, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 5, 9}, 60] (* Bruno Berselli, Jul 17 2012 *)

makelist(2*n-3+(3-(-1)^n)*(1-%i^(n*(n+1)))/4,n,1,60);

Vec((1+x+x^2+2*x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

From Bruno Berselli, Jul 17 2012: (Start)
G.f.: x*(1+x+x^2+2*x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).
a(n) = 2*n-3+(3-(-1)^n)*(1-i^(n*(n+1)))/4, where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A047617(k), a(2k-1) = A047471(k). (End)
E.g.f.: (6 + 2*sin(x) - cos(x) + 4*(x - 1)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, Jun 03 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-2)*Pi/16 + (2-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 23 2021

A047473 Numbers that are congruent to {2, 3} mod 8.

Original entry on oeis.org

2, 3, 10, 11, 18, 19, 26, 27, 34, 35, 42, 43, 50, 51, 58, 59, 66, 67, 74, 75, 82, 83, 90, 91, 98, 99, 106, 107, 114, 115, 122, 123, 130, 131, 138, 139, 146, 147, 154, 155, 162, 163, 170, 171, 178, 179, 186, 187, 194, 195, 202, 203, 210, 211, 218, 219, 226, 227, 234

Offset: 1

N. J. A. Sloane

Numbers k such that k and k+2 have the same digital binary sum. - Benoit Cloitre, Dec 01 2002

Also, numbers k such that k*(3*k + 1)/8 + 1/4 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

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

Union of A017089 and A017101.

Cf. A047467, A047468, A047470, A047471.

Flatten[# + {2,3} &/@ (8 Range[0, 30])] (* or *) LinearRecurrence[{1, 1, -1}, {2, 3, 10}, 60] (* Harvey P. Dale, Sep 28 2012 *)

a(n) = 8*n - a(n-1) - 11 for n>1, a(1)=2. - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 7/2 - 3*(-1)^n/2.
G.f.: x*(2 + x + 5*x^2)/((1 + x)*(1 - x)^2). (End)
a(1)=2, a(2)=3, a(3)=10; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Sep 28 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/16 + sqrt(2)*log(sqrt(2)+1)/8 - log(2)/8. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

cp's OEIS Frontend

A339774 a(n) is the least k such that 3^k == A047471(n) (mod 2^A047471(n)).

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A033200 Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.

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A047404 Numbers that are congruent to {1, 2, 3, 6} mod 8.

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A047476 Numbers that are congruent to {0, 1, 2, 3} mod 8.

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A153860 Triangle by columns: leftmost column = (1, 0, 1, -1, 1, -1, 1, ...); columns >1 = (1, 1, 0, 0, 0, ...).

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A255840 a(n) = (4*n^2 - 4*n + 1 - (-1)^n)/2.

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A047454 Numbers that are congruent to {1, 2, 3, 4} mod 8.

A255840 a(n) = (4n^2 - 4n + 1 - (-1)^n)/2.