A115523 - OEIS

%I #23 Apr 06 2024 15:09:29

%S 1,2,5,12,33,60,111,176,287,440,637,864,1237,1652,2147,2752,3555,4428,

%T 5517,6700,8177,9878,11785,13824,16441,19214,22265,25676,29685,33900,

%U 38715,43776,49595,55964,62821,69984,78445,87248,96647,106800,118167,129948,142905,156332

%N Number of ordered quadruples (i,j,k,l) in range [0..n] satisfying i == j (mod 2), j == k (mod 3) and k == l (mod 4).

%C Quasipolynomial of order 12. - _Charles R Greathouse IV_, Dec 03 2014

%H <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,-1,0,-1,1,0,0,0,2,-2,0,-2,0,2,0,2,-2,0,0,0,-1,1,0,1,0,-1,0,-1,1).

%F a(n) = binomial(n+1,4) - presumably quadratic (PORC) correction term which depends on n mod 24.

%F From _Charles R Greathouse IV_, Dec 03 2014: (Start)

%F n ==  0 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 24*n + 24)/24

%F n ==  1 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 20*n + 11)/24

%F n ==  2 (mod 12): a(n) = (n^4 + 4*n^3 + 10*n^2 + 12*n +  8)/24

%F n ==  3 (mod 12): a(n) = (n^4 + 4*n^3 +  8*n^2 +  8*n +  3)/24

%F n ==  4 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 20*n +  8)/24

%F n ==  5 (mod 12): a(n) = (n^4 + 4*n^3 + 10*n^2 + 12*n +  5)/24

%F n ==  6 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 12*n     )/24

%F n ==  7 (mod 12): a(n) = (n^4 + 4*n^3 +  8*n^2 +  8*n +  3)/24

%F n ==  8 (mod 12): a(n) = (n^4 + 4*n^3 + 10*n^2 + 12*n +  8)/24

%F n ==  9 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 12*n +  3)/24

%F n == 10 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 +  8*n +  8)/24

%F n == 11 (mod 12): a(n) = (n^4 + 4*n^3 +  6*n^2 +  4*n +  1)/24

%F (End)

%F a(n) = (19958400*(n^4+4*n^3+12*n^2+24*n+24) - (1235*n^2+2*1127*n+215)*m^11 +(74987*n^2+2*69047*n+13541)*m^10 -(1983300*n^2+2*1844700*n+377520)*m^9 +(29983800*n^2+2*28201800*n+6115890)*m^8 - (285731655*n^2+2*272034411*n+63415275)*m^7 +(1784142591*n^2+2*1720539051*n+436295013)*m^6 -(7344548530*n^2+2*7175131810*n+1995595030)*m^5 +(19515989350*n^2+2*19301456350*n+5911801060)*m^4 -(31672473360*n^2+2*31658103312*n+10685562360)*m^3 +(27907182072*n^2+2*28127231352*n+10490664096)*m^2 -(9932634720*n^2+2*10110299040*n+4359398400)*m)/479001600 where m=n-12*floor(n/12). - _Luce ETIENNE_, Sep 27 2017

%o (PARI) a(n)=my(s);for(i=0,n,forstep(j=i%2,n,2,forstep(k=j%3,n,3,s+=(n-(k%4))\4+1)));s \\ naive; _Charles R Greathouse IV_, Dec 03 2014

%Y Cf. A115520, A010881.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_ and _Vinay Vaishampayan_, Mar 09 2006