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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, 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 11 13:02 EDT 2020. Contains 336428 sequences. (Running on oeis4.)