A180711 a(n) = [x^n] A(x) = floor(sqrt( b(n) )) where b(n) = [x^n] A(x)^2.

1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 15, 17, 19, 22, 25, 28, 32, 35, 39, 44, 48, 53, 58, 64, 70, 76, 83, 90, 97, 105, 113, 122, 131, 140, 150, 161, 172, 183, 195, 208, 221, 235, 249, 264, 280, 296, 313, 331, 349, 368, 387, 408, 429, 451, 474, 497, 522, 547, 573, 600

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n), n = 0..500.

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 +...
The square of the g.f. A(x) begins:
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 + 42*x^7 + 61*x^8 + 88*x^9 + 124*x^10 + 168*x^11 + 227*x^12 + 302*x^13 + 394*x^14 +...
The integer part of the square-root of the coefficients of the square of the g.f. A(x) equals the respective coefficients of A(x), as illustrated by:
[sqrt(1)]=1; [sqrt(2)]=1; [sqrt(3)]=1; [sqrt(6)]=2; [sqrt(11)]=3;
[sqrt(18)]=4; [sqrt(28)]=5; [sqrt(42)]=6; [sqrt(61)]=7; [sqrt(88)]=9;
[sqrt(124)]=11; [sqrt(168)]=12; [sqrt(227)]=15; [sqrt(302)]=17; ...

Prog

(PARI) {a(n)=local(A=[1, 1, 1, t], T); for(i=1, n-2, for(N=A[ #A-1], 2*A[ #A-1], T=subst(Vec(Ser(A)^2)[ #A], t, N); if(sqrtint(T)==N, A[ #A]=N; A=concat(A, t); break))); A[n+1]}

Crossrefs

Cf. A180712.

Sequence in context: A337133 A062490 A211543 * A363938 A032957 A362132

Adjacent sequences: A180708 A180709 A180710 * A180712 A180713 A180714

Keyword

nonn

Author

Paul D. Hanna, Sep 22 2010

Status

approved