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%I #11 Sep 21 2019 14:36:15
%S 1,1,4,12,33,91,249,677,1842,5007,13613,37004,100587,273426,743250,
%T 2020363,5491918,14928581,40580092,110308128,299848580,815072948,
%U 2215597985,6022619743,16371177808,44501475147,120967551231,328823896346,893836022201,2429698216773,6604604511293,17953176427208
%N 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.
%C a(n) ~ exp(n) * (sqrt(5) - 1)/2.
%H Paul D. Hanna, <a href="/A326804/b326804.txt">Table of n, a(n) for n = 0..300</a>
%e 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.
%e Sample of actual values:
%e n | Sum_{k>=0} n^(k*Phi) / gamma(k*Phi + 1)
%e ---+-------------------------------------------------------
%e 0 | 1
%e 1 | 1.8242186970142293482811907601740004481582060664172...
%e 2 | 4.6411499876354997135766745161861025619606931664053...
%e 3 | 12.458033110486227334005490900044868760156182746758...
%e 4 | 33.772827636718664725840622695536754388698539389035...
%e 5 | 91.745113003487592378804753255758770103015001327903...
%e 6 | 249.34817457235336415492091487792502753925733493304...
%e 7 | 677.76858274342862965002639720303758277745743924364...
%e 8 | 1842.3429933125542741837067795825174364552910048629...
%e 9 | 5007.9892086067535847763792612960555750439505128201...
%e 10 | 13613.111168260512427104600576371603333751797005903...
%e 11 | 37004.260308189872733759090203849389739567140765181...
%e 12 | 100587.99784632291116677566445122445081000355383781...
%e 13 | 273426.51764650200721445954967770694747115532054823...
%e 14 | 743250.32644008297802053398664331007984602544468769...
%e 15 | 2020363.8494013344669144695353969448664316126612264...
%e 16 | 5491918.3325183103355711358264332049304062770315708...
%e 17 | 14928581.801123374012617260715780658136872007285332...
%e 18 | 40580092.629657118039112467300633253591763899054431...
%e 19 | 110308128.38784153558518623157638223371637437763422...
%e 20 | 299848580.92385254560063174684553555104528664675905...
%e ...
%e A related sequence of reals is illustrated as follows.
%e n | b(n) = Sum_{k>=0} n^(k*Phi^2) / gamma(k*Phi^2 + 1)
%e ---+-------------------------------------------------------
%e 0 | 1,
%e 1 | 1.2691417325369672809494103877123683653869189945372...
%e 2 | 2.8349264477075750702951828589498469958147669043997...
%e 3 | 7.6215048740220802425661954678079405560302002328126...
%e 4 | 20.810397450404851456972409967004647596399203114777...
%e 5 | 56.665103160045222668730139461429760451637612915609...
%e 6 | 154.08709140369276431167898906830429367193349362321...
%e 7 | 418.87429543418361482002934947840910901846510740386...
%e 8 | 1138.6242482530971673557246344204771570040748636998...
%e 9 | 3095.1023393187185229274783722049122276431310815904...
%e 10 | 8413.3607170679453255258268268600975366662603763685...
%e 11 | 22869.886402120801865609787207907788563054674234515...
%e 12 | 62166.797837233242465719906369407713455052154704129...
%e 13 | 168986.87811732291342764025923884927468323316136175...
%e 14 | 459353.96108792478359267146723300337256336487359646...
%e 15 | 1248653.5261082684248139179322937590691116457584312...
%e 16 | 3394192.1907460189802851077375600340756265270991557...
%e 17 | 9226370.9549636770014282607395061821517481507477216...
%e 18 | 25079876.509972016087541582886154522979484703033513...
%e 19 | 68174172.577458556101197864531288440510284242910732...
%e 20 | 185316614.48788799735040675456090504454285328687386...
%e ...
%e where b(n) ~ exp(n)/Phi^2 so that a(n) + b(n) ~ exp(n).
%o (PARI) /* Requires adequate precision */
%o {a(n) = my(Phi=(sqrt(5) + 1)/2); if(n==0, 1, floor( suminf(k=0, n^(k*Phi) / gamma(k*Phi + 1) ) ) )}
%o for(n=0, 40, print1(a(n), ", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 16 2019
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