OFFSET
0,2
COMMENTS
a(n) is the sum of the smallest n+3 multiples of n. For example, a(4) = 112 since 112 = 4 + 8 + 12 + 16 + 20 + 24 + 28.
a(n) = row sum of the n+4 row in the triangular view of A001477 (excluding the last 4 entries) - T(n-1), where T(n) is the n-th triangular number. For example, a(2) = 15 + 16 - T(1) = 31 - 1 = 30; a(4) = 28 + 29 + 30 + 31 - T(3) = 118 - 6 = 112.
a(n)*binomial(n+2,2) is the number of strings of length n+4 defined on {0,1,2,3} that contain exactly two 0's, exactly two 1's, exactly one 2 and any number of 3's (see Examples).
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = n*A000217(n+3).
a(n) - a(n-1) = A140229(n+2), n >= 1.
G.f.: x*(10 - 10*x + 3*x^2)/(1-x)^4.
E.g.f.: exp(x)*x*(20 + 10*x + x^2)/2. - Stefano Spezia, Feb 12 2026
From Amiram Eldar, Mar 01 2026: (Start)
Sum_{n>=1} 1/a(n) = 13/72.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/3 - 61/72. (End)
EXAMPLE
For n=1, a(1)*binomial(3,2) = 10*3 is the number of strings of length 5 described in the comment since there are 30 permutations of the string 00112.
For n=4, a(4)*binomial(6,2) = 112*15 is the number of strings of length 8 since there are 1680 permutations of the string 00112333.
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {0, 10, 30, 63}, 50] (* Paolo Xausa, Feb 23 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Feb 11 2026
STATUS
approved