A262990 G.f. A(x) satisfies: a([n/r^2]) = [x^n] A(x)^2/x and a([n/r^3]) = [x^n] A(x)^3/x^2, for n>=1, where r^2 + r^3 = 1.

1, 1, 2, 3, 5, 9, 10, 20, 22, 40, 51, 67, 114, 126, 203, 230, 354, 468, 571, 885, 908, 1486, 1674, 2250, 3045, 3586, 5418, 5322, 8186, 9560, 12234, 16341, 17976, 27912, 26970, 38435, 46383, 57024, 76794, 80805, 125205, 116376, 165914, 201580, 232352, 322980, 332388, 508710, 456154, 645661, 800495, 886018, 1241190, 1226382, 1916682, 1693454, 2290704, 2935492

Offset

1,3

Comments

The integer floor values, [n/r^2] and [n/r^3] where r^2 + r^3 = 1, form Beatty sequences and thus together contain all the positive integers without repetition.

Here r = 6 / ( (108 + 12*sqrt(69))^(1/3) + (108 - 12*sqrt(69))^(1/3) ) = 0.75487766624669276.... satisfies r^2 + r^3 = 1.

Not equal to A090845.

What is the rate of growth of this sequence?

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 10*x^7 + 20*x^8 + 22*x^9 + 40*x^10 + 51*x^11 + 67*x^12 + 114*x^13 + 126*x^14 + 203*x^15 +...
where the terms are formed from the union of coefficients in A(x)^2 and A(x)^3.
The coefficients of A(x)^2 begin:
A^2 = [1, 2, 5, 10, 20, 40, 67, 126, 203, 354, 571, 908, 1486, 2250, 3586, 5322, 8186, 12234, 17976, 26970, 38435, 57024, 80805, 116376, 165914, 232352,...]
and form the terms of this sequence at positions [n/r^2] for n>=1:
{[n/r^2]} = [1, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 21, 22, 24, 26, 28, 29, 31, 33, 35, 36, 38, 40, 42, 43, 45, 47, 49, 50, 52, 54, 56, 57, 59, ...].
The coefficients of A(x)^3 begin:
A^3 = [1, 3, 9, 22, 51, 114, 230, 468, 885, 1674, 3045, 5418, 9560, 16341, 27912, 46383, 76794, 125205, 201580, 322980, 508710, 800495, 1241190, ...]
and form the terms of this sequence at positions [n/r^3] for n>=1:
{[n/r^3]} = [2, 4, 6, 9, 11, 13, 16, 18, 20, 23, 25, 27, 30, 32, 34, 37, 39, 41, 44, 46, 48, 51, 53, 55, 58, 60, 62, 65, 67, 69, 72, ...].

Prog

(PARI) {a(n) = local(A=vector(n+1), B=A, C=A, r=6/((108+12*sqrt(69))^(1/3)+(108-12*sqrt(69))^(1/3))); A[1]=1; A[2]=1;
for(i=1, ceil(log(#A)/log(1/r)),
B=vector(floor(#A/r^2)); for(n=1, #A, m=floor(n/r^2); if(m<=#B, B[m]=Vec(Ser(A)^2)[n]));
C=vector(floor(#A/r^3)); for(n=1, #A, m=floor(n/r^3); if(m<=#C, C[m]=Vec(Ser(A)^3)[n]));
A=vector(#A, n, if(C[n]==0, B[n], C[n])); ); A[n]}
for(n=1, 80, print1(a(n), ", "))

Crossrefs

Cf. A075778.

Sequence in context: A273139 A303435 A090845 * A058108 A174512 A056144

Adjacent sequences: A262987 A262988 A262989 * A262991 A262992 A262993

Keyword

nonn

Author

Paul D. Hanna, Oct 06 2015

Status

approved