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n appears 3n times.
6

%I #32 Oct 05 2024 10:57:19

%S 1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,

%T 5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,

%U 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8

%N n appears 3n times.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%H Kevin Ryde, <a href="/A111651/b111651.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (q/(1-q))psi(q^3) in powers of q where psi() is a Ramanujan theta function. - _Michael Somos_, Aug 31 2006

%F G.f.: x/(1-x)*Product_{k>0} (1-x^(3k))^((-1)^k).

%F a(n) = round(sqrt((2/3)*n)) = A002024(ceiling(n/3)). - _Kevin Ryde_, Aug 31 2024

%t Table[PadRight[{},3n,n],{n,10}]//Flatten (* _Harvey P. Dale_, Sep 15 2021 *)

%o (PARI) {a(n)=if(n<1, 0, polcoeff( x/(1-x)*prod(k=1, n\3, (1-x^(3*k))^(-1)^k, 1+O(x^n)), n))} /* _Michael Somos_, Aug 31 2006 */

%o (PARI) a(n) = sqrtint(24*n) \/ 6; \\ _Kevin Ryde_, Aug 31 2024

%o (Python)

%o from math import isqrt

%o def A111651(n): return isqrt((n<<3)//3)+1>>1 # _Chai Wah Wu_, Oct 05 2024

%Y Cf. A000194, A002024, A008585, A111650, A111652.

%K easy,nonn

%O 1,4

%A _Jonathan Vos Post_, Aug 12 2005