A369679 -id:A369679 - cp's OEIS frontend

A369680 a(n) = Product_{k=0..n} (2^k + 3^(n-k)).

2, 12, 250, 19404, 5780918, 6691848108, 30261978906250, 535757771934053916, 37171553237849766044342, 10113067879819381109893992732, 10789224041146220828897229003906250, 45150513047221188662211059385153001179564, 741117672560101894851755994230829254062662140918

Offset: 0

Paul D. Hanna, Feb 06 2024

nonn

For p > 1, q > 1, limit_{n->oo} (Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = q^(1/2)*p^(1/(2*(1 + log(q)/log(p)))) = p^(1/2)*q^(1/(2*(1 + log(p)/log(q)))). - Vaclav Kotesovec, Feb 07 2024

Equivalently, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)). - Paul D. Hanna, Feb 08 2024

			a(0) = (1 + 1) = 2;
a(1) = (1 + 3)*(2 + 1) = 12;
a(2) = (1 + 3^2)*(2 + 3)*(2^2 + 1) = 250;
a(3) = (1 + 3^3)*(2 + 3^2)*(2^2 + 3)*(2^3 + 1) = 19404;
a(4) = (1 + 3^4)*(2 + 3^3)*(2^2 + 3^2)*(2^3 + 3)*(2^4 + 1) = 5780918;
a(5) = (1 + 3^5)*(2 + 3^4)*(2^2 + 3^3)*(2^3 + 3^2)*(2^4 + 3)*(2^5 + 1) = 6691848108;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/3^(n-k)) = 2 + 12/6 + 250/6^3 + 19404/6^6 + 5780918/6^10 + 6691848108/6^15 + ... + a(n)/6^(n*(n+1)/2) + ... = 5.6846111010137973166832330595516662115250385271...

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679.

Cf. A323715.

Table[Product[2^k+3^(n-k),{k,0,n}],{n,0,12}] (* James C. McMahon, Feb 07 2024 *)

{a(n) = prod(k=0, n, 2^k + 3^(n-k))}
for(n=0, 15, print1(a(n), ", "))

a(n) = Product_{k=0..n} (2^k + 3^(n-k)).
a(n) = 6^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/3^(n-k)).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/6^k).
a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/6^k).
a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 6^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 6^k).
a(n) = 2^(n*(n+1)/2)*QPochhammer(-3^n, 1/6, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 2^(1/(2*(1 + log(3)/log(2)))) * sqrt(3) = 3^(1/(2*(1 + log(2)/log(3)))) * sqrt(2) = 1.9805589654474717155611061670180902111915926... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(2)^2 + log(2)*log(3) + log(3)^2) / log(6)). - Paul D. Hanna, Feb 08 2024

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

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Feb 07 2024

nonn

			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...

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

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

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

A369678 a(n) = Product_{k=0..n} (3^k + 5^(n-k)).

Original entry on oeis.org

2, 24, 2080, 1382976, 7148699648, 287041728769536, 90391546425391144960, 221202979125273147766738944, 4237647337376998325597017538035712, 633933934421036224259931934460116571357184, 738292285249623417870561091674252758908330993254400

Offset: 0

Paul D. Hanna, Feb 07 2024

nonn

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(3 + 1) = 24;
a(2) = (1 + 5^2)*(3 + 5)*(3^2 + 1) = 2080;
a(3) = (1 + 5^3)*(3 + 5^2)*(3^2 + 5)*(3^3 + 1) = 1382976;
a(4) = (1 + 5^4)*(3 + 5^3)*(3^2 + 5^2)*(3^3 + 5)*(3^4 + 1) = 7148699648;
a(5) = (1 + 5^5)*(3 + 5^4)*(3^2 + 5^3)*(3^3 + 5^2)*(3^4 + 5)*(3^5 + 1) = 287041728769536;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/3^k + 1/5^(n-k)) = 2 + 24/15 + 2080/15^3 + 1382976/15^6 + 7148699648/15^10 + 287041728769536/15^15 + ... + a(n)/15^(n*(n+1)/2) + ... = 4.3507806549816093424129450104392682482776...

Cf. A369673, A369674, A369675, A369676, A369677, A369679, A369680.

{a(n) = prod(k=0, n, 3^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

a(n) = Product_{k=0..n} (3^k + 5^(n-k)).
a(n) = 15^(n*(n+1)/2) * Product_{k=0..n} (1/3^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/15^k).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/15^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 15^k).
a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 15^k).
a(n) = 3^(n*(n+1)/2)*QPochhammer(-5^n, 1/15, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 3^(1/(2*(1 + log(5)/log(3)))) * sqrt(5) = 5^(1/(2*(1 + log(3)/log(5)))) * sqrt(3) = 2.794249622709633938040980858655052416325961... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(3)^2 + log(3)*log(5) + log(5)^2) / log(15)). - Paul D. Hanna, Feb 08 2024

A369681 a(n) = Product_{k=0..n} (4^k + 5^(n-k)).

Original entry on oeis.org

2, 30, 3978, 4987710, 58712437962, 6601051349841150, 7017151861981535193738, 70966047508527496843460412990, 6820716704126571481897874317127918922, 6205644698427009393117687864650447521113942270, 53916867047490616763228279441645027173409633988839675658

Offset: 0

Paul D. Hanna, Feb 07 2024

nonn

For p > 1, q > 1, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)); formula due to Vaclav Kotesovec (cf. A369680).

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(4 + 1) = 30;
a(2) = (1 + 5^2)*(4 + 5)*(4^2 + 1) = 3978;
a(3) = (1 + 5^3)*(4 + 5^2)*(4^2 + 5)*(4^3 + 1) = 4987710;
a(4) = (1 + 5^4)*(4 + 5^3)*(4^2 + 5^2)*(4^3 + 5)*(4^4 + 1) = 58712437962;
a(5) = (1 + 5^5)*(4 + 5^4)*(4^2 + 5^3)*(4^3 + 5^2)*(4^4 + 5)*(4^5 + 1) = 6601051349841150;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/4^k + 1/5^(n-k)) = 2 + 30/20 + 3978/20^3 + 4987710/20^6 + 58712437962/20^10 + 6601051349841150/20^15 + ... + a(n)/20^(n*(n+1)/2) + ... = 4.0811214259450988699292249336017494522520...

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679, A369680.

{a(n) = prod(k=0, n, 4^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

a(n) = Product_{k=0..n} (4^k + 5^(n-k)).
a(n) = 20^(n*(n+1)/2) * Product_{k=0..n} (1/4^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/20^k).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/20^k).
a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 20^k).
a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 20^k).
Limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(4)^2 + log(4)*log(5) + log(5)^2) / log(20)) = 3.0816872899745614612763875038173884057052077... [from a formula by Vaclav Kotesovec].

A370014 a(n) = Product_{k=0..n} (2^k + 4^(n-k)).

Original entry on oeis.org

2, 15, 510, 84240, 69204960, 284844384000, 5892302096179200, 613826012249992396800, 322003239202740297793536000, 850857971372280730568060043264000, 11334246342025651164429104024534384640000, 760681528794595483313206024106936185273712640000

Offset: 0

Paul D. Hanna, Feb 08 2024

nonn

For p > 1, q > 1, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)); formula due to Vaclav Kotesovec (cf. A369680). For this sequence, p = 2 and q = 4.

			a(0) = (1 + 1) = 2;
a(1) = (1 + 4)*(2 + 1) = 15;
a(2) = (1 + 4^2)*(2 + 4)*(2^2 + 1) = 510;
a(3) = (1 + 4^3)*(2 + 4^2)*(2^2 + 4)*(2^3 + 1) = 84240;
a(4) = (1 + 4^4)*(2 + 4^3)*(2^2 + 4^2)*(2^3 + 4)*(2^4 + 1) = 69204960;
a(5) = (1 + 4^5)*(2 + 4^4)*(2^2 + 4^3)*(2^3 + 4^2)*(2^4 + 4)*(2^5 + 1) = 284844384000;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/4^(n-k)) = 2 + 15/8 + 510/8^3 + 84240/8^6 + 69204960/8^10 + 284844384000/8^15 + 5892302096179200/8^21 + ... + a(n)/8^(n*(n+1)/2) + ... = 5.2656633442570372661094196585300212123165...

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679, A369680, A369681.

{a(n) = prod(k=0, n, 2^k + 4^(n-k))}
for(n=0, 15, print1(a(n), ", "))

a(n) = Product_{k=0..n} (2^k + 4^(n-k)).
a(n) = 8^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/4^(n-k)).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/8^k).
a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/8^k).
a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 8^k).
a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 8^k).
Limit_{n->oo} a(n)^(1/n^2) = 2^(7/6) = 2.244924096618745962867... [using the formula by Vaclav Kotesovec given in the comments section].

cp's OEIS Frontend

A369680 a(n) = Product_{k=0..n} (2^k + 3^(n-k)).

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

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A369678 a(n) = Product_{k=0..n} (3^k + 5^(n-k)).

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A369681 a(n) = Product_{k=0..n} (4^k + 5^(n-k)).

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A370014 a(n) = Product_{k=0..n} (2^k + 4^(n-k)).

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