Philip Newton - cp's OEIS frontend

A090205 a(n) = n^n * (n+1)^(n+1).

1, 4, 108, 6912, 800000, 145800000, 38423222208, 13816758796288, 6499837226778624, 3874204890000000000, 2853116706110000000000, 2543867514227405321404416, 2700464873652770753916960768, 3365550251744726983606907650048, 4865879898782263021500000000000000

Offset: 0

Philip Newton, Jan 22 2004

I was made aware of this sequence by M. Frazer.

These are the denominators of the rapidly converging approximation to e given by (n+2)^(n+2)/(n+1)^(n+1) - (n+1)^(n+1)/n^n (see A111130). - Harlan J. Brothers, Aug 25 2023

			a(4) = 4^4 * 5^5 = 256 * 3125 = 800000.

H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, Math. Intelligencer, 20 (1998), 25-29.

Cf. A000312, A111130.

Table[n^n (n+1)^(n+1),{n,20}] (* Harvey P. Dale, May 29 2016 *)
Denominator[Table[(n + 2)^(n + 2)/(n + 1)^(n + 1) - (n + 1)^(n + 1)/n^n, {n, 20}]] (* Harlan J. Brothers, Aug 25 2023 *)
Join[{1},Times@@@Partition[Table[n^n,{n,20}],2,1]] (* Harvey P. Dale, Feb 04 2024 *)

A090205[n]:=n^n*(n+1)^(n+1)$ makelist(A090205[n],n,1,30); /* Martin Ettl, Oct 29 2012 */

a(n) = A000312(n) * A000312(n+1).

a(n) ~ e^(1 + 1/(2*n)) * n^(2*n+1). - Harlan J. Brothers, Aug 25 2023

More terms from Harlan J. Brothers, Aug 25 2023

A060852 Sum of number of pips on dice in shape of Platonic solids.

Original entry on oeis.org

10, 21, 36, 78, 210

Offset: 1

Philip Newton, Aug 09 2001

This is obviously a finite sequence, since there are only five Platonic solids.

			The third number in the sequence is 36 because the third Platonic solid has eight faces and the pips on an eight-faced die add up to 36 [by sum_1^n(n) = 1/2 * n * (n+1), so 1/2 * 8 * 9 = 36 for n = 8].

Wikipedia, Dice.

Cf. A053016.

# (# + 1)/2 & /@ {4, 6, 8, 12, 20} (* Robert G. Wilson v, Apr 18 2008 *)

for $n (4, 6, 8, 12, 20) { print $n*($n+1)/2 }

A060512 Number of units of time making up the next larger unit: seconds, minutes, hours, days, weeks, years, centuries, millennia.

Original entry on oeis.org

60, 60, 24, 7, 52, 100, 10

Offset: 1

Philip Newton, Mar 23 2001

This assumes 52 weeks in a year, which is not exact. (All other conversions to next higher unit are exact.)

			The third term is 24 because there are 24 hours in one day.

Index entries for sequences related to calendars

Cf. A060513.

A060513 Number of units of time making up the next larger unit: seconds, minutes, hours, days, years, centuries, millennia.

Original entry on oeis.org

60, 60, 24, 365, 100, 10

Offset: 1

Philip Newton, Mar 23 2001

This assumes 365 days in a year, which is not exact. (All other conversions to next higher unit are exact.)

			The third term is 24 because there are 24 hours in one day.

Index entries for sequences related to calendars

Cf. A060512.

A058525 Odd numbers z by which not all integers y, 0 <= y < 2^64, can be divided using "high multiplication" followed by a right shift.

Original entry on oeis.org

7, 21, 23, 25, 29, 31, 39, 47, 49, 53, 55, 61, 63, 71, 81, 89, 91, 93, 95, 97, 99, 101, 103, 107, 111, 115, 119, 121, 123, 125, 127, 137, 147, 161, 169, 181, 183, 199, 201, 207, 213, 223, 225, 233, 235, 237, 239, 243, 251, 253, 259, 273, 281, 285, 313, 315, 323

Offset: 0

Philip Newton, Dec 22 2000

nonn

For many odd numbers z, it is possible to compute the integer division of y / z for 0 <= y < 2^64 (that is, floor(y/z)) by multiplying by a suitable constant a and shifting right: floor((a*y)/(2^(64+e))). a is computed as a = ceiling((2^(64+e))/z), where e is such that 2^e < z < 2^(e+1).
Knuth showed that the formula floor((a*y)/(2^(64+e))) = floor(y/z) holds for all y, 0 <= y < 2^64, if and only if it holds for the single value y = 2^64 - 1 - (2^64 mod z).
There are 189 odd divisors z less than 1000 for which this method cannot be used to find the division result for all y, 0 <= y < 2^64.

			For the first term in the sequence, 7, floor(ay/(2^(64+e))) = 2635249153387078802 for y = 2^64 - 1 - (2^64 mod z) = 18446744073709551613, while floor(y/z) = 2635249153387078801.
Example for a term not in the sequence: for 9, both floor(ay/(2^(64+e))) and floor(y/z) are 2049638230412172400 for y = 2^64 - 1 - (2^64 mod z) = 18446744073709551608.

Donald Ervin Knuth, The Art of Computer Programming, fascicle 1, MMIX. Addison Wesley Longman, 1999. Zeroth printing (revision 8), 24 December 1999. Exercise 19 in section 1.3.1', page 25 and the answer on page 95.

Donald E. Knuth, The fascicle as a compressed PostScript file

Name corrected by Boon Suan Ho, Feb 27 2023

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A090205 a(n) = n^n * (n+1)^(n+1).

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A060852 Sum of number of pips on dice in shape of Platonic solids.

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A060512 Number of units of time making up the next larger unit: seconds, minutes, hours, days, weeks, years, centuries, millennia.

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A060513 Number of units of time making up the next larger unit: seconds, minutes, hours, days, years, centuries, millennia.

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A058525 Odd numbers z by which not all integers y, 0 <= y < 2^64, can be divided using "high multiplication" followed by a right shift.

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