A289722
Irregular triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the n-Apollonian network.
Original entry on oeis.org
6, 15, 6, 42, 72, 6, 123, 522, 258, 366, 2970, 3894, 396, 1095, 14838, 37332, 13680, 216, 3282, 68736, 278490, 224928, 24624, 9843, 303918, 1779678, 2517228, 754704, 22032, 29526, 1303938, 10269150, 22233096, 13114656, 1489104, 7776
Offset: 1
Triangle begins:
6;
15, 6;
42, 72, 6;
123, 522, 258;
366, 2970, 3894, 396;
1095, 14838, 37332, 13680, 216;
3282, 68736, 278490, 224928, 24624;
9843, 303918, 1779678, 2517228, 754704, 22032;
29526, 1303938, 10269150, 22233096, 13114656, 1489104, 7776;
...
-
R[dp_, peq_, p1_, p2_] := {3*(dp - x + peq^2 + (2 + 7*x)*p1^2 + (7 + 2*x)*p2^2 + (4 + 2*x)*peq*p1 + 6*peq*p2 + 2*(4 + 5*x)*p1*p2 + x*(peq + 3*p1 + 3*p2)), x*(1 + 3*p1), 2*(p1 + p2), peq + p2};
A[n_] := (v = {6*x, x, 0, 0}; For[i = 2, i <= n, i++, v = R @@ v]; v[[1]]);
Table[CoefficientList[A[n], x] // Rest, {n, 1, 10}] // Flatten (* Jean-François Alcover, Dec 28 2017, after Andrew Howroyd *)
-
R(dp, peq,p1,p2,x) = {[3*(dp - x + peq^2 + (2+7*x)*p1^2 + (7+2*x)*p2^2 + (4+2*x)*peq*p1 + 6*peq*p2 + 2*(4+5*x)*p1*p2 + x*(peq+3*p1+3*p2)), x*(1+3*p1), 2*(p1+p2), peq+p2]}
A(n,x) = {my(v=[6*x,x,0,0,x]); for(i=2, n, v=R(v[1],v[2],v[3],v[4],x)); v[1]}
for (n=1,10,print(Vec(polrecip(A(n,x))),";" ))
A192792
Molecular topological indices of the Apollonian graphs.
Original entry on oeis.org
72, 360, 2556, 22572, 219636, 2204244, 22197420, 222257988, 2207645892, 21754722852, 212845625820, 2069408197476, 20010127994676, 192565336573476, 1845376043710284, 17619057807964452, 167667905660138532, 1590879916369856484, 15054743317985652924
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..200
- Eric Weisstein's World of Mathematics, Apollonian Network
- Eric Weisstein's World of Mathematics, Molecular Topological Index
- Index entries for linear recurrences with constant coefficients, signature (29, -320, 1676, -4109, 2537, 9182, -26346, 43920, -49896, 23328).
-
Table[(3 (3025 + 605 2^(2 + n) + 3 (-1)^n 2^(4 + n) + 605 2^(3 + 2 n) + 5 9^n (63 + 44 n) + 4 3^n (-277 + 55 n) + 35 2^(2 + n/2) Cos[n Pi/2] - 15 2^((7 + n)/2) Sin[n Pi/2]))/1210, {n, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{29, -320, 1676, -4109, 2537, 9182, -26346, 43920, -49896, 23328}, {72, 360, 2556, 22572, 219636, 2204244, 22197420, 222257988, 2207645892, 21754722852}, 20] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[36 (2 - 48 x + 421 x^2 - 1584 x^3 + 2096 x^4 + 1960 x^5 - 9573 x^6 + 17670 x^7 - 25056 x^8 + 15552 x^9)/((1 - x) (1 - 2 x) (1 - 3 x)^2 (1 - 4 x) (1 - 9 x)^2 (1 + 2 x) (1 + 2 x^2)), {x, 0, 50}], x] (* Eric W. Weisstein, Sep 08 2017 *)
-
Rec(mti, peq, p1, p2, weq, w1, w2, t, x) = {[3*(mti + 2*weq*peq + 2*(2+7*x)*w1*p1 + 2*(7+2*x)*w2*p2 + (4+2*x)*(weq*p1+peq*w1) + 6*(weq*p2+peq*w2) + 2*(4+5*x)*(w1*p2+p1*w2) + x*(weq+3*w1+3*w2) + 3*t*(peq+p1+2*p2) + 3*t*x*(t+1+2*p1+p2)), x*(1+3*p1), 2*(p1+p2), peq+p2, x*(3*t+3*w1), 2*(w1+w2), weq+w2, 2*t]}
Fin(peq, p1, p2, t, x) = {(t+1)*(peq + p1 + 2*p2 + x*(t + 3 + 2*p1 + p2))}
a(n) = { my(v=[18*x,x,0,0,3*x,0,0,2,x]);
for(i=2, n, v=Rec(v[1],v[2],v[3],v[4],v[5],v[6],v[7],v[8],x));
subst(deriv(v[1] + 3*Fin(v[2],v[3],v[4],v[8],x)), x, 1);
} \\ Andrew Howroyd, Sep 03 2017
Showing 1-2 of 2 results.
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