A064680 -id:A064680 - cp's OEIS frontend

A302690 a(n) is the smallest integer m such that m*n is a sum of two squares but not one.

2, 1, 6, 2, 1, 3, 14, 1, 2, 1, 22, 6, 1, 7, 3, 2, 1, 1, 38, 1, 42, 11, 46, 3, 2, 1, 6, 14, 1, 3, 62, 1, 66, 1, 7, 2, 1, 19, 3, 1, 1, 21, 86, 22, 1, 23, 94, 6, 2, 1, 3, 1, 1, 3, 11, 7, 114, 1, 118, 3, 1, 31, 14, 2, 1, 33, 134, 1, 138, 7, 142, 1, 1, 1, 6, 38, 154, 3, 158

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2018

nonn

Previous name was: a(n) is the smallest integer m such that A002828(m*n) = 2.
All terms are squarefree.
Using the sum of two squares theorem it is easy to see that a(n) is either A363340(n) (if A363340(n)*n is not a square) or 2*A363340(n) (if A363340(n)*n is a square). - Peter Schorn, Jul 20 2023

Cf. A002828, A005117, A007913, A064680, A099304, A302694, A363340.

A302690 := proc(n)
    local k ;
    for k from 1 do
        if A002828(k*n) = 2 then
            return k;
        end if;
    end do:
end proc:
seq(A302690(n),n=1..100) ; # R. J. Mathar, Apr 16 2018

a363340(n) = my(r=1); foreach(mattranspose(factor(n)), f, if(f[1]%4==3&&f[2]%2==1, r*=f[1])); r;
a(n) = my(p=a363340(n)); if(issquare(p*n), 2*p, p); \\ Peter Schorn, Jul 20 2023

a(n^2) = 2.

Name corrected and more terms added by Michel Marcus, Apr 12 2018

Better name from Peter Schorn, Jul 20 2023

A320431 The number of tiles inside a regular n-gon created by lines that run from each of the vertices of the n edges orthogonal to these edges.

Original entry on oeis.org

1, 1, 31, 13, 71, 25, 127, 41, 199, 61, 287, 85, 391, 113, 511, 145, 647, 181, 799, 221, 967, 265, 1151, 313, 1351, 365, 1567, 421, 1799, 481, 2047, 545, 2311, 613, 2591, 685, 2887, 761, 3199, 841, 3527, 925, 3871, 1013, 4231, 1105, 4607, 1201, 4999, 1301, 5407, 1405, 5831, 1513, 6271, 1625, 6727, 1741

Offset: 3

R. J. Mathar, Jan 08 2019

Sequence proposed by Thomas Young: draw the regular n-gon and construct 2*n lines that run from both ends of the n edges perpendicular into the n-gon until they hit an opposite edge. (For n even the lines actually hit another vertex, so there are only n additional lines). a(n) is the number of non-overlapping tiles inside the n-gon with edges that are sections of the lines or n-gon edges.

R. J. Mathar, OEIS A320431
Thomas Young, R. J. Mathar, The surfer and the hut: a polygon dissection (2019)
Index to sequences on drawing diagonals in regular polygons
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A165217, A320422

a(2n) = 2*n^2+2*n+1 = A001844(n), n>1. a(2n+1) = 8*n^2-1 = A157914(n), n>1. - Thomas Young (tyoung(AT)district16.org), Nov 11 2017
G.f.: x^3 +x^4 -x^5*(31+13*x-22*x^2-14*x^3+7*x^4+5*x^5) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Jan 21 2019
a(n) = 1+n*A064680(n-2), n>=5. - R. J. Mathar, Jan 21 2019

A174029 a(n) = 3(3n+1)*(5 - (-1)^n)/4.

Original entry on oeis.org

3, 18, 21, 45, 39, 72, 57, 99, 75, 126, 93, 153, 111, 180, 129, 207, 147, 234, 165, 261, 183, 288, 201, 315, 219, 342, 237, 369, 255, 396, 273, 423, 291, 450, 309, 477, 327, 504, 345, 531, 363, 558, 381, 585, 399, 612, 417, 639, 435, 666, 453

Offset: 0

Paul Curtz, Mar 06 2010

All entries are multiples of 3.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

[3*(3*n+1)*(5-(-1)^n)/4: n in [0..50]]; // Vincenzo Librandi, Aug 05 2011

LinearRecurrence[{0,2,0,-1},{3,18,21,45},60] (* Harvey P. Dale, Mar 19 2015 *)

vector(50, n, n--; 3*(3*n+1)*(5-(-1)^n)/4) \\ G. C. Greubel, Nov 02 2018

a(2n) = 18*n + 3 = 3*A016921(n);
a(2n+1) = 27*n + 18 = A124388(n).
a(n) = A064680(3n) + A064680(3n+1) + A064680(3n+2).
G.f.: ( 3 + 18*x + 15*x^2 + 9*x^3 ) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Jul 02 2011
a(n) = 2*a(n-2) - a(n-4); a(0)=3, a(1)=18, a(2)=21, a(3)=45. - Harvey P. Dale, Mar 19 2015
E.g.f.: (3/4)*(5*(1+3*x)*exp(x) - (1-3*x)*exp(-x)). - G. C. Greubel, Nov 02 2018

A280579 Square array read by antidiagonals downwards giving the first differences A261327(n+p) - A261327(n), with p >= 0.

Original entry on oeis.org

0, 0, 4, 0, -3, 1, 0, 11, 8, 12, 0, -8, 3, 0, 4, 0, 24, 16, 27, 24, 28, 0, -19, 5, -3, 8, 5, 9, 0, 43, 24, 48, 40, 51, 48, 52, 0, -36, 7, -12, 12, 4, 15, 12, 16, 0, 68, 32, 75, 56, 80, 72, 83, 80, 84, 0, -59, 9, -27

Offset: 0

Paul Curtz, Jan 05 2017

Successive rows:
p
0:  0,   0,   0,   0,   0,   0,   0, ...
1:  4,  -3,  11,  -8,  24, -19,  43, ...
2:  1,   8,   3,  16,   5,  24,   7, ...
3: 12,   0,  27,  -3,  48, -12,  75, ...
4:  4,  24,   8,  40,  12,  56,  16, ...
5: 28,   5,  51,   4,  80,  -3, 115, ...
6:  9,  48,  15,  72,  21,  96,  27, ...
... .
Main diagonal: alternate 3*n^2, -3.
From p>0, the rows are multiples of 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, ... . Sequences appearing after division: shifted A144433 or A195161, A064680. For p=3, we have (n+2)^2, -n^2.
First column: alternate n^2, 4*(n^2 + n + 1). Its first differences (4, -3, 11, -8, 24, ...) is the sequence of the square array for p=1.
Third column: 0, 3, 8, 15, ... is A005563(n).
Fifth column: 5, 21, 45, 77, ... is a bisection of A061037(n).
Seventh column: 7, 16, 40, 55, 91, 112, ... is a subsequence of A061039(n).
Etc. From the Rydberg spectra of the hydrogen atom (mentioned in A261327).
Starting for instance from p=-3,at the main antidiagonal,yields:
-3:              -12,   0,  -27,  3, ... see p=3
-2:          -1,  -8,  -3,  -16, -5, ...     p=2
-1:     -4,   3, -11,   8,  -24, 19, ...     p=1.

Cf. A000290, A002061, A005563, A061037, A061039, A064680, A112087, A144433, A195161, A261327.

A281098 a(n) is the GCD of the sequence d(n) = A261327(k+n) - A261327(k) for all k.

Original entry on oeis.org

0, 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, 9, 1, 20, 3, 11, 1, 24, 1, 13, 3, 28, 1, 15, 1, 32, 3, 17, 1, 36, 1, 19, 3, 40, 1, 21, 1, 44, 3, 23, 1, 48, 1, 25, 3, 52, 1, 27, 1, 56, 3, 29, 1, 60, 1, 31, 3, 64, 1, 33, 1, 68, 3, 35, 1, 72, 1, 37, 3, 76, 1, 39, 1

Offset: 0

Paul Curtz, Jan 14 2017

Successive sequences:
   0,   0,  0,    0, ...    = 0 * ( )
   4,  -3,  11,  -8, ...    = 1 * ( )
   1,   8,   3,  16, ...    = 1 * ( )                 A195161
  12,   0,  27,  -3, ...    = 3 * (4, 0, 9, -1, ...)
   4,  24,   8,  40, ...    = 4 * (1, 6, 2, 10, ...)  A064680
5;   28,   5,  51,   4, ...    = 1 * ( )
   9,  48,  15,  72, ...    = 3 * (3, 16, 5, 24, ...) A195161
  52,  12,  83,  13, ...    = 1 * ( )
  16,  80,  24, 112, ...    = 8 * (2, 10, 3, 14, ...) A064080
  84   21, 123,  24, ...    = 3 * (28, 7, 41, 8, ...)
 25, 120,  35, 160, ...    = 5 * (5, 24, 7, 32, ...) A195161

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,1,0,2,0,1,0,-1,0,-1).

Cf. A064680, A144433 or A195161.

Cf. A000012, A005408, A008586, A010701, A109007 (bisection), A016825, A165988 (via A022998), A166138, A166304, A280579.

CoefficientList[Series[(-x (-1 - x - 4 x^2 - 5 x^3 - 3 x^4 - 6 x^5 + 3 x^6 - 5 x^7 + 4 x^8 - x^9 + x^10))/((x^2 - x + 1) (1 + x + x^2) (x - 1)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 79}], x] (* Michael De Vlieger, Feb 02 2017 *)

f(n) = numerator((4 + n^2)/4);
a(n) = gcd(vector(1000, k, f(k+n) - f(k))); \\ Michel Marcus, Jan 15 2017

A281098(n) = if(n%2, gcd((n\2)-1,3), n>>(bitand(n,2)/2)); \\ Antti Karttunen, Feb 15 2023

G.f.: -x*( -1 - x - 4*x^2 - 5*x^3 - 3*x^4 - 6*x^5 + 3*x^6 - 5*x^7 + 4*x^8 - x^9 + x^10 )/( (x^2 - x + 1)*(1 + x + x^2)*(x - 1)^2*(1 + x)^2*(1 + x^2)^2 ). - R. J. Mathar, Jan 31 2017
a(2*k)   = A022998(k).
a(2*k+1) = A109007(k-1).
a(3*k)   = interleave 3*k*(3 +(-1)^k)/2, 3.
a(3*k+1) = interleave 1, A166304(k).
a(3*k+2) = interleave A166138(k), 1.
a(4*k)   = 4*k.
a(4*k+1) = period 3: repeat [1, 1, 3].
a(4*k+2) = 1 + 2*k.
a(4*k+3) = period 3: repeat [3, 1, 1].
a(n+12) - a(n) = 6*A131743(n+3).
a(n) = (18*n + 40 - 16*cos(n*Pi/3) + 9*n*cos(n*Pi/2) + 32*cos(2*n*Pi/3) + (18*n - 40)*cos(n*Pi) + 3*n*cos(3*n*Pi/2) - 16*cos(5*n*Pi/3))/48. - Wesley Ivan Hurt, Oct 04 2018

Corrected and extended by Michel Marcus, Jan 15 2017

A174007 a(2n+1)=2. a(2n)= 1-n.

Original entry on oeis.org

2, 0, 2, -1, 2, -2, 2, -3, 2, -4, 2, -5, 2, -6, 2, -7, 2, -8, 2, -9, 2, -10, 2, -11, 2, -12, 2, -13, 2, -14, 2, -15, 2, -16, 2, -17, 2, -18, 2, -19, 2, -20, 2, -21, 2, -22, 2, -23, 2, -24, 2, -25, 2, -26, 2, -27, 2, -28, 2, -29, 2, -30, 2, -31, 2, -32, 2, -33, 2, -34, 2, -35, 2

Offset: 1

Paul Curtz, Mar 05 2010

A064680(n)+A022998(n-1) =c(n) = 2, 2, 10, 5, 18, 8, 26, 11,.. has differences c(2n)-c(2n-1) = -5*(n-1) = -A008587(n-1).

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

Cf. A147657.

a(n) = n/4+3/2-(-1)^n*(n/4+1/2).
a(n)= +2*a(n-2) -a(n-4). G.f.: -x*(-2+2*x^2+x^3) / ( (x-1)^2*(1+x)^2 ).
a(n+1)-a(n) = (-1)^n*A008619(n+1).
a(n) = A064680(n)-A022998(n-1).

cp's OEIS Frontend

A302690 a(n) is the smallest integer m such that m*n is a sum of two squares but not one.

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A320431 The number of tiles inside a regular n-gon created by lines that run from each of the vertices of the n edges orthogonal to these edges.

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A174029 a(n) = 3*(3*n+1)*(5 - (-1)^n)/4.

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A280579 Square array read by antidiagonals downwards giving the first differences A261327(n+p) - A261327(n), with p >= 0.

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A281098 a(n) is the GCD of the sequence d(n) = A261327(k+n) - A261327(k) for all k.

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Mathematica

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A174007 a(2n+1)=2. a(2n)= 1-n.

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Formula

A174029 a(n) = 3(3n+1)*(5 - (-1)^n)/4.