A296359 -id:A296359 - cp's OEIS frontend

A296360 Number of monohedral disk tilings of type C^t_{2n+1,3}.

116, 6402, 446930, 34121322, 2741227176, 227759341712, 19382568941318, 1679333068357460, 147541888215426742, 13107891004266127974, 1175188298096727647322, 106164291028322202227232, 9652457243380891557169712, 882443342536355491502025678

Offset: 1

N. J. A. Sloane, Dec 15 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..100
Joel Anthony Haddley, Stephen Worsley, Infinite families of monohedral disk tilings, arXiv:1512.03794v2 [math.MG], 2015-2016.

Cf. A241926, A296359, A296361, A296362.

U[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[(n + k)/#, n/#]/(n + k) &];
a[n_] := 2*Sum[U[i, 3*(4*n + 2 - i)], {i, 0, 4*n + 2}];
Array[a, 16] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

\\ here U is A241926
U(n,k)={sumdiv(gcd(n,k), d, eulerphi(d)*binomial((n+k)/d, n/d)/(n+k))}
a(n)={2*sum(i=0, 4*n+2, U(i,3*(4*n+2-i)))} \\ Andrew Howroyd, Jan 09 2018

a(n) = 2*Sum_{i=0..4*n+2} A241926(i, 3*(4*n+2-i)). - Andrew Howroyd, Jan 09 2018

Terms a(6) and beyond from Lars Blomberg, Jan 09 2018

A296361 Number of monohedral disk tilings of type C^t_{3,n}.

Original entry on oeis.org

2, 62, 116, 200, 318, 476, 682, 946, 1272, 1674, 2152, 2724, 3394, 4176, 5078, 6110, 7284, 8614, 10108, 11784, 13646, 15716, 18002, 20522, 23288, 26314, 29616, 33212, 37114, 41344, 45910, 50838, 56140, 61838, 67948, 74488, 81478, 88940, 96890, 105354, 114344

Offset: 1

N. J. A. Sloane, Dec 15 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1000
Joel Anthony Haddley, Stephen Worsley, Infinite families of monohedral disk tilings, arXiv:1512.03794v2 [math.MG], 2015-2016.

Cf. A241926, A296359, A296360, A296362.

U[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[(n + k)/#, n/#]/(n + k) &];
a[1] = 2; a[n_] := 2*Sum[U[i, n*(6 - i)], {i, 0, 6}];
Array[a, 50] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

\\ here U is A241926
U(n,k)={sumdiv(gcd(n,k), d, eulerphi(d)*binomial((n+k)/d, n/d)/(n+k))}
a(n)={2*if(n<2, n==1, sum(i=0, 6, U(i,n*(6-i))))} \\ Andrew Howroyd, Jan 09 2018

Conjectures from Colin Barker, Jan 09 2018: (Start)
G.f.: 2*x*(1 + 28*x - 33*x^2 - 10*x^3 + 34*x^4 - 16*x^5 - 26*x^6 + 35*x^7 + 8*x^8 - 32*x^9 + 13*x^10) / ((1 - x)^5*(1 + x)*(1 + x + x^2 + x^3 + x^4)).
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - 3*a(n-6) + 2*a(n-7) + 2*a(n-8) - 3*a(n-9) + a(n-10) for n>11.
(End)
a(n) = 2*Sum_{i=0..6} A241926(i, n*(6-i)) for n > 1. - Andrew Howroyd, Jan 09 2018

Terms a(6) and beyond from Lars Blomberg, Jan 09 2018

A296362 Number of monohedral disk tilings of type C^t_{5,n}.

Original entry on oeis.org

2, 1532, 6402, 19884, 51128, 115188, 235180, 445096, 792822, 1343814, 2185396, 3431466, 5227806, 7758398, 11251894, 15989150, 22311572, 30630012, 41434474, 55305224, 72924016, 95086728, 122717140, 156881256, 198802766, 249880274, 311704608, 386077910

Offset: 1

N. J. A. Sloane, Dec 15 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1000
Joel Anthony Haddley, Stephen Worsley, Infinite families of monohedral disk tilings, arXiv:1512.03794v2 [math.MG], 2015-2016.

Cf. A241926, A296359, A296360, A296361.

U[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[(n+k)/#, n/#]/(n+k) &];
a[1] = 2; a[n_] := 2*Sum[ U[i, n*(10 - i)], {i, 0, 10}];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

\\ here U is A241926
U(n,k)={sumdiv(gcd(n,k), d, eulerphi(d)*binomial((n+k)/d, n/d)/(n+k))}
a(n)={2*if(n<2, n==1, sum(i=0, 10, U(i,n*(10-i))))} \\ Andrew Howroyd, Jan 09 2018

a(n) = 2*Sum_{i=0..10} A241926(i, n*(10-i)) for n > 1. - Andrew Howroyd, Jan 09 2018

G.f.: 2*x*(1 + 763*x + 905*x^2 + 1871*x^3 + 2142*x^4 + 2318*x^5 + 2333*x^6 + 1022*x^7 + 602*x^8 - 348*x^9 - 1422*x^10 - 1599*x^11 - 2949*x^12 - 3041*x^13 - 2413*x^14 - 2329*x^15 - 316*x^16 - 538*x^17 + 175*x^18 + 703*x^19 + 562*x^20 + 1446*x^21 + 852*x^22 + 147*x^23 + 48*x^24 - 646*x^25 - 6*x^26 + 224*x^27 + 16*x^28 + 184*x^29 - 310*x^30 + 107*x^31) / ((1 - x)^9*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^3*(1 + x^3 + x^6)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) (conjectured). - Colin Barker, Jan 09 2018

Terms a(6) and beyond from Lars Blomberg, Jan 09 2018

cp's OEIS Frontend

A296360 Number of monohedral disk tilings of type C^t_{2n+1,3}.

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A296361 Number of monohedral disk tilings of type C^t_{3,n}.

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A296362 Number of monohedral disk tilings of type C^t_{5,n}.

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