A369677 a(n) = Product_{k=0..n} (2^k + 5^(n-k)).

2, 18, 910, 275562, 509528318, 5782203860202, 403066704971309470, 172986911139059942455818, 457494980583771669025834718462, 7445459859979605380607238308201858858, 746155118699551878624986638597659812003763550, 461066589238234272286243169377378506495126815749310922

Offset

0,1

Links

Table of n, a(n) for n=0..11.

Formula

a(n) = Product_{k=0..n} (2^k + 5^(n-k)).

a(n) = 10^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/5^(n-k)).

a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/10^k).

a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/10^k).

a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 10^k).

a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 10^k).

a(n) = 2^(n*(n+1)/2)*QPochhammer(-5^n, 1/10, n + 1). - Stefano Spezia, Feb 07 2024

Limit_{n->oo} a(n)^(1/n^2) = 2^(1/(2*(1 + log(5)/log(2)))) * sqrt(5) = 5^(1/(2*(1 + log(2)/log(5)))) * sqrt(2) = 2.481958590195459039209137154563963236753327... - Vaclav Kotesovec, Feb 07 2024

Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(2)^2 + log(2)*log(5) + log(5)^2) / log(10)). - Paul D. Hanna, Feb 08 2024

Example

a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(2 + 1) = 18;
a(2) = (1 + 5^2)*(2 + 5)*(2^2 + 1) = 910;
a(3) = (1 + 5^3)*(2 + 5^2)*(2^2 + 5)*(2^3 + 1) = 275562;
a(4) = (1 + 5^4)*(2 + 5^3)*(2^2 + 5^2)*(2^3 + 5)*(2^4 + 1) = 509528318;
a(5) = (1 + 5^5)*(2 + 5^4)*(2^2 + 5^3)*(2^3 + 5^2)*(2^4 + 5)*(2^5 + 1) = 5782203860202;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/5^(n-k)) = 2 + 18/10 + 910/10^3 + 275562/10^6 + 509528318/10^10 + 5782203860202/10^15 + ... + a(n)/10^(n*(n+1)/2) + ... = 5.0427178660718059961260933841217518099...

Prog

(PARI) {a(n) = prod(k=0, n, 2^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369678, A369679, A369680.

Sequence in context: A015190 A180606 A334553 * A013040 A268051 A268074

Adjacent sequences: A369674 A369675 A369676 * A369678 A369679 A369680

Keyword

nonn

Author

Paul D. Hanna, Feb 07 2024

Status

approved