A181890 -id:A181890 - cp's OEIS frontend

A160050 Numerator of the Harary number for the star graph s_n.

0, 1, 5, 9, 7, 10, 27, 35, 22, 27, 65, 77, 45, 52, 119, 135, 76, 85, 189, 209, 115, 126, 275, 299, 162, 175, 377, 405, 217, 232, 495, 527, 280, 297, 629, 665, 351, 370, 779, 819, 430, 451, 945, 989, 517, 540, 1127, 1175, 612, 637, 1325, 1377, 715, 742, 1539

Offset: 1

Eric W. Weisstein, Apr 30 2009

			0, 1, 5/2, 9/2, 7, 10, 27/2, 35/2, 22, 27, ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Harary Index.
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A130658 (denominators), A033954 (quadrisection), A001107 (quadrisection), A181890 (quadrisection).

Cf. A000217, A060819, A064038.

m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3))); // G. C. Greubel, Sep 21 2018

f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 3] &, 58]
Rest[CoefficientList[Series[x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3), {x, 0, 50}], x]] (* G. C. Greubel, Sep 21 2018 *)

s=vector(40,n,1/4*(n+2)*(n-1)) /* fractions */
vector(#s,n,numerator(s[n])) /* this sequence */ \\ Joerg Arndt, Jan 04 2011

x='x+O('x^50); concat([0], Vec(x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3))) \\ G. C. Greubel, Sep 21 2018

Numerator of (1/4)*(n+2)*(n-1). - Joerg Arndt, Jan 04 2011
It appears that a(n + 1) = A060819(n-1) * A060819(n + 2). - Paul Curtz, Dec 23 2010 [Corrected by Joerg Arndt, Jan 04 2011]
G.f.: x^2*(-1-2*x-5*x^4+3*x^5-2*x^6+x^7) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Jan 04 2011
a(1+4*n) = (A000217(n+1)-1)/2, a(2+4*n) = (A000217(n+2)-1)/2, a(3+4*n) = A000217(n+3)-1, a(4+4*n) = A000217(n+4)-1. - Paul Curtz, Dec 23 2010.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). This is not the shortest recurrence. -Paul Curtz, Mar 27 2011
a(1+3*n) = numerator of 9*n*(n+1)/4 = 9*A064038(1+n). - Paul Curtz, Apr 06 2011
a(n) = (n-1)*(n+2)*(3-i^((n-2)*(n-1)))/8, where i=sqrt(-1).  - Bruno Berselli, Apr 07 2011, corrected by Vaclav Kotesovec, Aug 09 2022
Sum_{n>=2} 1/a(n) = 13/9 + Pi/6. - Amiram Eldar, Aug 09 2022

Edited by N. J. A. Sloane, Dec 23 2010

A187857 T(n,k)=Number of n-step one space for components leftwards or up, two space for components rightwards or down asymmetric quasi-queen's tours (antidiagonal moves become knight moves) on a kXk board summed over all starting positions.

Original entry on oeis.org

1, 4, 0, 9, 5, 0, 16, 27, 2, 0, 25, 65, 81, 0, 0, 36, 119, 254, 216, 0, 0, 49, 189, 578, 968, 486, 0, 0, 64, 275, 1030, 2754, 3320, 846, 0, 0, 81, 377, 1610, 5428, 11986, 9932, 1206, 0, 0, 100, 495, 2318, 9237, 26836, 47962, 26584, 1008, 0, 0, 121, 629, 3154, 14040

Offset: 1

R. H. Hardin Mar 14 2011

Table starts
.1.4....9.....16......25.......36........49.......64.......81.....100.....121
.0.5...27.....65.....119......189.......275......377......495.....629.....779
.0.2...81....254.....578.....1030......1610.....2318.....3154....4118....5210
.0.0..216....968....2754.....5428......9237....14040....19837...26628...34413
.0.0..486...3320...11986....26836.....50378....81124...120051..166504..220483
.0.0..846...9932...47962...126397....262409...452766...707541.1017934.1387600
.0.0.1206..26584..180750...568870...1314428..2456614..4062007.6094090
.0.0.1008..61668..636102..2432312...6343874.12918800.22675997
.0.0..414.124880.2090520..9934272..29607932.65963326
.0.0....0.219008.6387404.38766870.133665550

			Some n=4 solutions for 4X4
..0..0..1..0....4..0..0..0....0..0..0..0....0..3..0..4....0..0..0..0
..0..3..0..0....0..3..0..0....0..2..1..0....0..0..2..1....3..2..0..0
..0..0..2..0....0..0..2..0....0..4..0..0....0..0..0..0....0..1..0..0
..0..0..0..4....0..0..0..1....0..3..0..0....0..0..0..0....0..0..4..0

R. H. Hardin, Table of n, a(n) for n = 1..130

Row 2 is A181890(n-2)

Empirical: T(1,k) = k^2
Empirical: T(2,k) = 8*k^2 - 18*k + 9 for k>1
Empirical: T(3,k) = 64*k^2 - 252*k + 238 for k>3
Empirical: T(4,k) = 497*k^2 - 2652*k + 3448 for k>5
Empirical: T(5,k) = 3763*k^2 - 25044*k + 40644 for k>7
Empirical: T(6,k) = 28294*k^2 - 224508*k + 433614 for k>9
Empirical: T(7,k) = 211612*k^2 - 1941340*k + 4328678 for k>11
Empirical: T(8,k) = 1575830*k^2 - 16367550*k + 41250447 for k>13
Empirical: T(9,k) = 11710007*k^2 - 135575032*k + 380311550 for k>15
Empirical: T(10,k) = 86897560*k^2 - 1108193530*k + 3420011978 for k>17

A159231 Primes p such that 8p^2-2p-1 divides Fibonacci(p).

Original entry on oeis.org

37, 97, 577, 727, 1297, 3037, 3067, 4447, 4567, 5557, 7507, 7867, 8647, 9067, 9157, 12967, 17257, 20107, 20407, 21787, 22147, 23677, 25447, 27817, 28687, 29347, 30187, 32587, 33487, 35617, 38377, 42157, 42667, 42967, 43207, 45697, 46447, 47497, 49477

Offset: 1

Arkadiusz Wesolowski, Apr 06 2009

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Chris Caldwell, The Prime Glossary, Fibonacci number
C. K. Caldwell, "Top Twenty" page, Fibonacci cofactor

Subsequence of A159259. Supersequence of A215158.

Cf. A000045, A023172, A159234, A181890.

[p : p in PrimesUpTo(49477) | IsZero(Fibonacci(p) mod (8*p^2-2*p-1))]; // Arkadiusz Wesolowski, Nov 09 2013

Select[Prime@Range[5084], Mod[Fibonacci[#], 8*#^2 - 2*# - 1] == 0 &] (* Arkadiusz Wesolowski, Dec 12 2011 *)

forprime(p=2, 49477, if(Mod(fibonacci(p), 8*p^2-2*p-1)==0, print1(p, ", "))); \\ Arkadiusz Wesolowski, Nov 09 2013

A159259 Positive numbers n such that 8n^2-2n-1 divides Fibonacci(n).

Original entry on oeis.org

27, 37, 97, 577, 687, 727, 777, 807, 1297, 1707, 1917, 2067, 2487, 2787, 2977, 3027, 3037, 3067, 3277, 3367, 3417, 3507, 3837, 4047, 4257, 4377, 4447, 4567, 4717, 5137, 5557, 5637, 5677, 5917, 5967, 6057, 6187, 6327, 7077, 7087, 7357, 7407, 7507, 7597

Offset: 1

Arkadiusz Wesolowski, Apr 07 2009

nonn

The prime numbers of this sequence are in A159231.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Chris Caldwell, The Prime Glossary, Fibonacci number

Cf. A000045, A023172, A159234, A159231, A181890.

[n : n in [1..7597] | IsZero(Fibonacci(n) mod (8*n^2-2*n-1))] // Arkadiusz Wesolowski, Nov 09 2013

Select[Range[7597], Mod[Fibonacci[#], 8*#^2 - 2*# - 1] == 0 &] (* Arkadiusz Wesolowski, Dec 12 2011 *)

for(n=1, 7597, if(Mod(fibonacci(n), 8*n^2-2*n-1)==0, print1(n, ", "))); \\ Arkadiusz Wesolowski, Nov 09 2013

A159234 Composite numbers n such that 8n^2-2n-1 divides the primitive part U(n) of Fibonacci(n).

Original entry on oeis.org

27, 807, 1707, 2977, 3027, 3277, 4717, 5137, 5677, 5917, 5967, 6187, 7087, 7357, 7597, 7707, 8217, 9117, 9297, 9387, 9667, 9877, 9927, 9997, 10387, 11097, 11647, 11797, 12727, 13407, 13867, 15757, 15987, 16327, 16507, 16857, 17347, 17767, 18237, 18817, 18997

Offset: 1

Arkadiusz Wesolowski, Apr 06 2009

nonn

Chris Caldwell, The Prime Glossary, Fibonacci number

Subsequence of A159259.

Cf. A000045, A023172, A061446, A159231, A181890.

lst = {1}; Do[f = Fibonacci[a]; Do[f = f/GCD[f, lst[[d]]], {d, Most[Divisors[a]]}]; AppendTo[lst, f], {a, 2, 19000}]; Flatten[Table[If[! PrimeQ[n] && Mod[lst[[n]], 8*n^2 - 2*n - 1] == 0, n, {}], {n, 19000}]] (* Arkadiusz Wesolowski, Dec 12 2011 *)

A220021 Number of cyclotomic cosets of 11 mod 10^n.

Original entry on oeis.org

10, 27, 65, 119, 189, 275, 377, 495, 629, 779, 945, 1127, 1325, 1539, 1769, 2015, 2277, 2555, 2849, 3159, 3485, 3827, 4185, 4559, 4949, 5355, 5777, 6215, 6669, 7139, 7625, 8127, 8645, 9179, 9729, 10295, 10877, 11475, 12089, 12719, 13365, 14027, 14705, 15399, 16109, 16835, 17577, 18335, 19109, 19899

Offset: 1

V. Raman, Jan 27 2013

How is this related to A181890? - R. J. Mathar, Apr 11 2013

			a(2) = 27 because there are 27 cyclotomic cosets of 11 mod 100:
{1, 11, 21, 31, 41, 51, 61, 71, 81, 91}
{3, 33, 63, 93, 23, 53, 83, 13, 43, 73}
{7, 77, 47, 17, 87, 57, 27, 97, 67, 37}
{9, 99, 89, 79, 69, 59, 49, 39, 29, 19}
{2, 22, 42, 62, 82}
{12, 32, 52, 72, 92}
{4, 44, 84, 24, 64}
{14, 54, 94, 34, 74}
{6, 66, 26, 86, 46}
{16, 76, 36, 96, 56}
{8, 88, 68, 48, 28}
{18, 98, 78, 58, 38}
{5, 55}
{15, 65}
{25, 75}
{35, 85}
{45, 95}
{0}
{10}
{20}
{30}
{40}
{50}
{60}
{70}
{80}
{90}

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

Cf. A006694, A220468.

a[n_] := DivisorSum[10^n, EulerPhi[#] / MultiplicativeOrder[11, #] &]; Array[a, 50] (* Jean-François Alcover, Dec 18 2015 *)

for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(11, d)))", "))

Conjecture: a(n) = 8*n^2-2*n-1 for n>1. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>4. G.f.: x*(5*x^3-14*x^2+3*x-10) / (x-1)^3. - Colin Barker, Apr 13 2013

cp's OEIS Frontend

A160050 Numerator of the Harary number for the star graph s_n.

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A187857 T(n,k)=Number of n-step one space for components leftwards or up, two space for components rightwards or down asymmetric quasi-queen's tours (antidiagonal moves become knight moves) on a kXk board summed over all starting positions.

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Formula

A159231 Primes p such that 8*p^2-2*p-1 divides Fibonacci(p).

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Magma

Mathematica

PARI

A159259 Positive numbers n such that 8*n^2-2*n-1 divides Fibonacci(n).

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Programs

Magma

Mathematica

PARI

A159234 Composite numbers n such that 8*n^2-2*n-1 divides the primitive part U(n) of Fibonacci(n).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

A220021 Number of cyclotomic cosets of 11 mod 10^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A159231 Primes p such that 8p^2-2p-1 divides Fibonacci(p).

A159259 Positive numbers n such that 8n^2-2n-1 divides Fibonacci(n).

A159234 Composite numbers n such that 8n^2-2n-1 divides the primitive part U(n) of Fibonacci(n).