A130877 -id:A130877 - cp's OEIS frontend

A263941 Minimal most likely sum for a roll of n 8-sided dice.

1, 9, 13, 18, 22, 27, 31, 36, 40, 45, 49, 54, 58, 63, 67, 72, 76, 81, 85, 90, 94, 99, 103, 108, 112, 117, 121, 126, 130, 135, 139, 144, 148, 153, 157, 162, 166, 171, 175, 180, 184, 189, 193, 198, 202, 207, 211, 216, 220, 225

Offset: 1

Gianmarco Giordano, Oct 30 2015

			For n=1, there are eight equally likely outcomes, 1,2,3,4,5,6,7,8 and the smallest of these is 1, so a(1)=1.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A030123, A130877, A256680.

Join[{1}, Table[(18 n + (-1)^n - 1)/4, {n, 2, 50}]]

PARI
```
a(n)=if(n<2,1,9*n\2);
vector(50,n,a(n))
```

G.f.: x*(1 + 8*x + 3*x^2 - 3*x^3)/((1 - x)^2*(1 + x)).
a(n) = floor(9*n/2) = (18*n + (-1)^n - 1)/4 with n>1, a(1)=1.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
a(n) = -A130877(-n+1) for n>1.

Edited by Bruno Berselli, Oct 30 2015

A382531 Number of n-digit base-10 numbers whose digit sum is equal to ceiling(9*n/2).

Original entry on oeis.org

1, 9, 70, 615, 5520, 50412, 468448, 4379055, 41395240, 392406145, 3748943890, 35866068766, 345143007910, 3323483518810, 32150758083580, 311088525668335, 3021445494584902, 29344719005694973, 285904843977651598, 2785022004925340460, 27203012941819689340

Offset: 1

Miquel Cerda, Mar 30 2025

Digit sum ceiling(9*n/2) = A130877(n+1) has highest frequency among all n-digit base-10 numbers.

The count excludes numbers with leading zeros.

			a(2) = 9, the 2-digit numbers with digit sum 9 are: 18, 27, 36, 45, 54, 63, 72, 81, 90.

Alois P. Heinz, Table of n, a(n) for n = 1..1002

Cf. A210736 (analogous for base-2 digits).
Cf. A025015 (maximal coefficient of (1+...+x^9)^n).
Cf. A007953, A071817, A071976, A130877.

b:= proc(n, s, t) option remember; `if`(9*n b(n, ceil(9*n/2), 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Apr 12 2025

a(n) = [x^ceiling(9*n/2)] (f^n - f^(n-1)) with f = (x^10-1)/(x-1). - Alois P. Heinz, Apr 12 2025

A120740 Numbers n such that n = Sum_digits[k*abs(n-k)] for some k>=0.

Original entry on oeis.org

0, 4, 5, 9, 14, 18, 23, 27, 32, 36, 41, 45, 50, 54, 59, 63, 68, 72, 77, 81, 86, 90, 95, 99, 104, 108, 113, 117, 122, 126, 131, 135, 140, 144, 149, 153, 158, 162, 167, 171, 176, 180, 185, 189, 194, 198, 203, 207, 212, 216, 221, 225, 230, 234, 239, 243, 248, 252

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 26 2007

The first difference is eventually 2-periodic: 4, 1, 4, 5, 4, 5, 4, etc. The minimum numbers k associated to the first elements of the sequence are (n,k): (0,0), (4,2), (5,7), (9,3), (14,19), (18,33), (23,67), (27,69), etc.

			n = 36 -> k = 279 -> 279*abs(36-279)=279*243=67797 -> 6+7+7+9+7 = 36

Cf. A130877.

P:=proc(n) local i, j, k, w; for i from 0 by 1 to n do for j from 0 by 1 to 100*n do w:=0; k:=j*abs(i-j); while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; if w=i then print(i); break; fi; od; od; end: P(100000);

Conjecture: a(n) = (18*n-(-1)^n-35)/4 for n>2. a(n) = a(n-1)+a(n-2)-a(n-3) for n>5. G.f.: x^2*(4+x+4*x^3)/((1-x)^2*(1+x)). [Colin Barker, Apr 10 2012]

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A263941 Minimal most likely sum for a roll of n 8-sided dice.

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A382531 Number of n-digit base-10 numbers whose digit sum is equal to ceiling(9*n/2).

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A120740 Numbers n such that n = Sum_digits[k*abs(n-k)] for some k>=0.

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