A199015 G.f.: 1/(1-x) * Product_{n>=1} (1 - x^(2*n))^2/(1 - x^(2*n-1))^2.

1, 3, 4, 6, 8, 8, 11, 13, 13, 15, 17, 19, 20, 22, 22, 24, 28, 28, 30, 30, 31, 35, 37, 37, 39, 41, 41, 43, 45, 47, 48, 52, 52, 52, 54, 54, 58, 60, 62, 64, 64, 64, 67, 69, 69, 71, 75, 75, 77, 79, 79, 83, 83, 83, 83, 87, 90, 92, 94, 94, 96, 98, 98, 98, 100, 102, 106, 108, 108, 110, 112, 112, 115, 117, 117, 117, 121, 121, 123

Offset

0,2

Comments

Equals the partial sums of A008441, where A008441(n) is the number of ways of writing n as the sum of 2 triangular numbers.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

Lim_{n->infinity} a(n)/n = Pi/2.

a(n) = Sum_{k=0..n} Sum_{d|4*k+1} (-1)^floor(d/2). - Michael Somos [see A008441]

G.f.: 1/(1-x) * Sum_{n>=0} x^n/(1 - x^(4*n + 1)). - Michael Somos [see A008441]

G.f.: theta_2(sqrt(x))^2/(4*x^(1/4)*(1 - x)), where theta_2() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

Example

G.f.: A(x) = 1 + 3*x + 4*x^2 + 6*x^3 + 8*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + ...
where the g.f. equals the product:
A(x) = 1/(1-x) * (1-x^2)^2/(1-x)^2 * (1-x^4)^2/(1-x^3)^2 * (1-x^6)^2/(1-x^5)^2 * ...
Illustrate the limit a(n)/n = Pi/2:
a(10)/10 = 1.7, a(10^2)/10^2 = 1.58, a(10^3)/10^3 = 1.574, a(10^4)/10^4 = 1.5704, a(10^5)/10^5 = 1.57086, a(10^6)/10^6 = 1.570784, a(10^7)/10^7 = 1.5707972, ...

Mathematica

CoefficientList[Series[EllipticTheta[2, 0, Sqrt[x]]^2/(4*x^(1/4)*(1 - x)), {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2018 *)
a = Compile[{{n, _Integer}}, Block[{c = 0}, Do[ c ++, {i, 0, (1 + Sqrt[1 + 8 n])/4}, {j, 0, (1 + Sqrt[1 + 8(n - i (i + 1))])/2}]; c]]; Array[a, 79, 0] (* Robert G. Wilson v, Aug 28 2025 *)

Prog

(PARI) {a(n)=sum(k=0, n, sumdiv(4*k+1, d, (-1)^(d\2)))}
(PARI) {a(n)=polcoeff(1/(1-x)*prod(m=1, n\2+1, (1-x^(2*m))/(1-x^(2*m-1)+x*O(x^n)))^2, n)}

Crossrefs

Cf. A008441, A008443.

Sequence in context: A295996 A240675 A072152 * A196098 A099356 A377431

Adjacent sequences: A199012 A199013 A199014 * A199016 A199017 A199018

Keyword

nonn

Author

Paul D. Hanna, Nov 02 2011

Status

approved