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A267370 Partial sums of A140091. 6
0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
After 0, this sequence is the third column of the array in A185874.
Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.
LINKS
FORMULA
O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)
EXAMPLE
The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
. 0;
. 1, 5;
. 4, 7, 10;
. 9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.
MATHEMATICA
Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4, -6, 4, -1}, {0, 6, 21, 48}, 50] (* Harvey P. Dale, Jul 18 2019 *)
PROG
(PARI) vector(50, n, n--; n*(n+1)*(n+5)/2)
(Sage) [n*(n+1)*(n+5)/2 for n in (0..50)]
(Magma) [n*(n+1)*(n+5)/2: n in [0..50]];
CROSSREFS
Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).
Sequence in context: A212656 A051941 A212707 * A213388 A372226 A163715
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Jan 13 2016
STATUS
approved

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Last modified July 14 05:06 EDT 2024. Contains 374291 sequences. (Running on oeis4.)