A301804 Sequence satisfies: 1 = Sum_{n>=1} 2^(n*(n-1)) / a(n)^n, with a(1) = 2, by a greedy algorithm.

2, 3, 11, 28, 67, 181, 540, 1605, 4500, 12770, 37773, 127030, 444950, 1379830, 4396237, 13632772, 45274296, 158468043, 530856473, 1800446217, 6279909810, 24743271334, 85850127322, 290413108775, 977634838108, 3306283825369, 11185652749615, 38838048510711, 133953360029207, 459989927920851, 1634605692726786, 5918370893528999, 20465922530700426, 71980481052561822, 265331221445369542

Offset

1,1

Comments

It appears that the limit a(n+1)/a(n) exists and is near 4.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Example

1 = 1/2 + 2^2/3^2 + 2^6/11^3 + 2^12/28^4 + 2^20/67^5 + 2^30/181^6 + 2^42/540^7 + 2^56/1605^8 + 2^72/4500^9 + 2^90/12770^10 + 2^110/37773^11 + 2^132/127030^12 + 2^156/444950^13 + 2^182/1379830^14 + 2^210/4396237^15 + 2^240/13632772^16 + 2^272/45274296^17 + 2^306/158468043^18 + 2^342/530856473^19 + 2^380/1800446217^20 + 2^420/6279909810^21 + ... + ( 2^(n-1)/a(n) )^n + ...
Incidentally,
Sum_{n>=1} 2^(n-1)/a(n) = 2.594806011516631787617662898514062588686879234...
Sum_{n>=1} 2^(n^2)/a(n)^n = 3.295922872490926100815120594347157182861242917...
Sum_{n>=1} 2^(n*(n+1))/a(n)^n = 14.82031378016256272989741456078817533736...
Sum_{n>=1} 1/a(n) = 0.983220030675069959469784597542593767565029822764...

Prog

(PARI) /* Must have appropriate precision for N terms: */ N = 100;
{A=[2]; for(i=1, N, A=concat(A, 1 + floor((1/(1 - sum(n=1, #A, (2^n)^(n-1)/A[n]^n *1.))*2^((#A)*(#A+1)) )^(1/(#A+1))) ) ; print1(#A, ", ")); A}

Crossrefs

Sequence in context: A232212 A232219 A265095 * A335856 A335816 A330979

Adjacent sequences: A301801 A301802 A301803 * A301805 A301806 A301807

Keyword

nonn

Author

Paul D. Hanna, Mar 27 2018

Status

approved