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a(n) = Product_{i=0..6} floor((n+i)/7).
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%I #39 Jan 10 2023 01:49:32

%S 0,0,0,0,0,0,0,1,2,4,8,16,32,64,128,192,288,432,648,972,1458,2187,

%T 2916,3888,5184,6912,9216,12288,16384,20480,25600,32000,40000,50000,

%U 62500,78125,93750,112500,135000,162000,194400,233280,279936,326592,381024

%N a(n) = Product_{i=0..6} floor((n+i)/7).

%C For n >= 7, a(n) is the maximal product of seven positive integers with sum n. - _Wesley Ivan Hurt_, Jun 29 2022

%H M. El-Mikkawy and T. Sogabe, <a href="https://doi.org/10.1016/j.amc.2009.12.069">A new family of k-Fibonacci numbers</a>, Appl. Math. Comput. 215 (2010) 4456-4461, Table 1 k=7.

%H <a href="/index/Rec#order_44">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 0, 0, 0, 6, -12, 6, 0, 0, 0, 0, -15, 30, -15, 0, 0, 0, 0, 20, -40, 20, 0, 0, 0, 0, -15, 30, -15, 0, 0, 0, 0, 6, -12, 6, 0, 0, 0, 0, -1, 2, -1).

%F a(n) = 2*a(n-1) - a(n-2) + 6*a(n-7) - 12*a(n-8) + 6*a(n-9) - 15*a(n-14) + 30*a(n-15) - 15*a(n-16) + 20*a(n-21) - 40*a(n-22) + 20*a(n-23) - 15*a(n-28) + 30*a(n-29) - 15*a(n-30) + 6*a(n-35) - 12*a(n-36) + 6*a(n-37) - a(n-42) + 2*a(n-43) - a(n-44). - _Wesley Ivan Hurt_, Jun 29 2022

%F a(7*n) = n^7 (A001015). - _Bernard Schott_, Nov 04 2022

%F Sum_{n>=7} 1/a(n) = 1 + zeta(7). - _Amiram Eldar_, Jan 10 2023

%o (PARI) a(n) = prod(k=0, 6, (n+k)\7); \\ _Georg Fischer_, Nov 07 2019

%Y Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), this sequence (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

%Y Cf. A001015, A013665.

%K nonn,easy

%O 0,9

%A _N. J. A. Sloane_

%E a(40)-a(44) from _Georg Fischer_, Nov 07 2019