A115523 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).
1, 2, 5, 12, 33, 60, 111, 176, 287, 440, 637, 864, 1237, 1652, 2147, 2752, 3555, 4428, 5517, 6700, 8177, 9878, 11785, 13824, 16441, 19214, 22265, 25676, 29685, 33900, 38715, 43776, 49595, 55964, 62821, 69984, 78445, 87248, 96647, 106800, 118167, 129948, 142905, 156332
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, 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).
Programs
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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
Formula
a(n) = binomial(n+1,4) - presumably quadratic (PORC) correction term which depends on n mod 24.
From Charles R Greathouse IV, Dec 03 2014: (Start)
n == 0 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 24*n + 24)/24
n == 1 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 20*n + 11)/24
n == 2 (mod 12): a(n) = (n^4 + 4*n^3 + 10*n^2 + 12*n + 8)/24
n == 3 (mod 12): a(n) = (n^4 + 4*n^3 + 8*n^2 + 8*n + 3)/24
n == 4 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 20*n + 8)/24
n == 5 (mod 12): a(n) = (n^4 + 4*n^3 + 10*n^2 + 12*n + 5)/24
n == 6 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 12*n )/24
n == 7 (mod 12): a(n) = (n^4 + 4*n^3 + 8*n^2 + 8*n + 3)/24
n == 8 (mod 12): a(n) = (n^4 + 4*n^3 + 10*n^2 + 12*n + 8)/24
n == 9 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 12*n + 3)/24
n == 10 (mod 12): a(n) = (n^4 + 4*n^3 + 12*n^2 + 8*n + 8)/24
n == 11 (mod 12): a(n) = (n^4 + 4*n^3 + 6*n^2 + 4*n + 1)/24
(End)
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
Comments