A354600 a(n) = Product_{k=0..9} floor((n+k)/10).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1536, 2304, 3456, 5184, 7776, 11664, 17496, 26244, 39366, 59049, 78732, 104976, 139968, 186624, 248832, 331776, 442368, 589824, 786432, 1048576, 1310720, 1638400, 2048000, 2560000, 3200000, 4000000

Offset

0,12

Comments

For n >= 10, a(n) is the maximal product of ten positive integers with sum n.

Links

Table of n, a(n) for n=0..46.

Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 9, -18, 9, 0, 0, 0, 0, 0, 0, 0, -36, 72, -36, 0, 0, 0, 0, 0, 0, 0, 84, -168, 84, 0, 0, 0, 0, 0, 0, 0, -126, 252, -126, 0, 0, 0, 0, 0, 0, 0, 126, -252, 126, 0, 0, 0, 0, 0, 0, 0, -84, 168, -84, 0, 0, 0, 0, 0, 0, 0, 36, -72, 36, 0, 0, 0, 0, 0, 0, 0, -9, 18, -9, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

Formula

a(n) = 2*a(n-1) - a(n-2) + 9*a(n-10) - 18*a(n-11) + 9*a(n-12) - 36*a(n-20) + 72*a(n-21) - 36*a(n-22) + 84*a(n-30) - 168*a(n-31) + 84*a(n-32) - 126*a(n-40) + 252*a(n-41) - 126*a(n-42) + 126*a(n-50) - 252*a(n-51) + 126*a(n-52) - 84*a(n-60) + 168*a(n-61) - 84*a(n-62) + 36*a(n-70) - 72*a(n-71) + 36*a(n-72) - 9*a(n-80) + 18*a(n-81) - 9*a(n-82) + a(n-90) - 2*a(n-91) + a(n-92).

Sum_{n>=10} 1/a(n) = 1 + zeta(10). - Amiram Eldar, Jan 10 2023

a(10*n) = n^10 (A008454). - Bernard Schott, Feb 02 2023

Mathematica

Table[Product[Floor[(n + k)/10], {k, 0, 9}], {n, 0, 50}]

Prog

(PARI) a(n) = prod(k=0, 9, (n+k)\10); \\ Michel Marcus, Jul 09 2022

Crossrefs

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), A009641 (k=7), A009694 (k=8), A009714 (k=9), this sequence (k=10).

Cf. A008454 (subsequence), A013668.

Sequence in context: A366855 A330127 A292568 * A056767 A008863 A145117

Adjacent sequences: A354597 A354598 A354599 * A354601 A354602 A354603

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 08 2022

Status

approved