A326804 a(n) = floor( Sum_{k>=0} n^(k*Phi) / Gamma(k*Phi + 1) ), where Gamma(x) is Euler's gamma function and Phi = (sqrt(5) + 1)/2 is the golden ratio.

1, 1, 4, 12, 33, 91, 249, 677, 1842, 5007, 13613, 37004, 100587, 273426, 743250, 2020363, 5491918, 14928581, 40580092, 110308128, 299848580, 815072948, 2215597985, 6022619743, 16371177808, 44501475147, 120967551231, 328823896346, 893836022201, 2429698216773, 6604604511293, 17953176427208

Offset

0,3

Comments

a(n) ~ exp(n) * (sqrt(5) - 1)/2.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Example

a(n) = floor(1 + n^Phi/Gamma(Phi+1) + n^(2*Phi)/Gamma(2*Phi+1) + n^(3*Phi)/Gamma(3*Phi+1) + n^(4*Phi)/Gamma(4*Phi+1) + n^(5*Phi)/Gamma(5*Phi+1) +  n^(6*Phi)/Gamma(6*Phi+1) + ...) where Phi = (sqrt(5) + 1)/2.
Sample of actual values:
n  | Sum_{k>=0} n^(k*Phi) / gamma(k*Phi + 1)
---+-------------------------------------------------------
0  | 1
1  | 1.8242186970142293482811907601740004481582060664172...
2  | 4.6411499876354997135766745161861025619606931664053...
3  | 12.458033110486227334005490900044868760156182746758...
4  | 33.772827636718664725840622695536754388698539389035...
5  | 91.745113003487592378804753255758770103015001327903...
6  | 249.34817457235336415492091487792502753925733493304...
7  | 677.76858274342862965002639720303758277745743924364...
8  | 1842.3429933125542741837067795825174364552910048629...
9  | 5007.9892086067535847763792612960555750439505128201...
10 | 13613.111168260512427104600576371603333751797005903...
11 | 37004.260308189872733759090203849389739567140765181...
12 | 100587.99784632291116677566445122445081000355383781...
13 | 273426.51764650200721445954967770694747115532054823...
14 | 743250.32644008297802053398664331007984602544468769...
15 | 2020363.8494013344669144695353969448664316126612264...
16 | 5491918.3325183103355711358264332049304062770315708...
17 | 14928581.801123374012617260715780658136872007285332...
18 | 40580092.629657118039112467300633253591763899054431...
19 | 110308128.38784153558518623157638223371637437763422...
20 | 299848580.92385254560063174684553555104528664675905...
...
A related sequence of reals is illustrated as follows.
n  | b(n) = Sum_{k>=0} n^(k*Phi^2) / gamma(k*Phi^2 + 1)
---+-------------------------------------------------------
0  | 1,
1  | 1.2691417325369672809494103877123683653869189945372...
2  | 2.8349264477075750702951828589498469958147669043997...
3  | 7.6215048740220802425661954678079405560302002328126...
4  | 20.810397450404851456972409967004647596399203114777...
5  | 56.665103160045222668730139461429760451637612915609...
6  | 154.08709140369276431167898906830429367193349362321...
7  | 418.87429543418361482002934947840910901846510740386...
8  | 1138.6242482530971673557246344204771570040748636998...
9  | 3095.1023393187185229274783722049122276431310815904...
10 | 8413.3607170679453255258268268600975366662603763685...
11 | 22869.886402120801865609787207907788563054674234515...
12 | 62166.797837233242465719906369407713455052154704129...
13 | 168986.87811732291342764025923884927468323316136175...
14 | 459353.96108792478359267146723300337256336487359646...
15 | 1248653.5261082684248139179322937590691116457584312...
16 | 3394192.1907460189802851077375600340756265270991557...
17 | 9226370.9549636770014282607395061821517481507477216...
18 | 25079876.509972016087541582886154522979484703033513...
19 | 68174172.577458556101197864531288440510284242910732...
20 | 185316614.48788799735040675456090504454285328687386...
...
where b(n) ~ exp(n)/Phi^2 so that a(n) + b(n) ~ exp(n).

Prog

(PARI) /* Requires adequate precision */
{a(n) = my(Phi=(sqrt(5) + 1)/2); if(n==0, 1, floor( suminf(k=0, n^(k*Phi) / gamma(k*Phi + 1)  ) ) )}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A293064 A219092 A135254 * A000754 A317974 A119683

Adjacent sequences: A326801 A326802 A326803 * A326805 A326806 A326807

Keyword

nonn

Author

Paul D. Hanna, Sep 16 2019

Status

approved