The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A059270 a(n) is both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers. 24
0, 3, 15, 42, 90, 165, 273, 420, 612, 855, 1155, 1518, 1950, 2457, 3045, 3720, 4488, 5355, 6327, 7410, 8610, 9933, 11385, 12972, 14700, 16575, 18603, 20790, 23142, 25665, 28365, 31248, 34320, 37587, 41055, 44730, 48618, 52725, 57057, 61620 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Group the non-multiples of n as follows, e.g., for n = 4: (1,2,3), (5,6,7), (9,10,11), (13,14,15), ... Then a(n) is the sum of the members of the n-th group. Or, the sum of (n-1)successive numbers preceding n^2. - Amarnath Murthy, Jan 19 2004
Convolution of odds (A005408) and multiples of three (A008585). G.f. is the product of the g.f. of A005408 by the g.f. of A008585. - Graeme McRae, Jun 06 2006
Sums of rows of the triangle in A126890. - Reinhard Zumkeller, Dec 30 2006
Corresponds to the Wiener indices of C_{2n+1} i.e., the cycle on 2n+1 vertices (n > 0). - K.V.Iyer, Mar 16 2009
Also the product of the three numbers from A005843(n) up to A163300(n), divided by 8. - Juri-Stepan Gerasimov, Jul 26 2009
Partial sums of A033428. - Charlie Marion, Dec 08 2013
For n > 0, sum of multiples of n and (n+1) from 1 to n*(n+1). - Zak Seidov, Aug 07 2016
A generalization of Ianakiev's formula, a(n) = A005408(n)*A000217(n), follows. A005408(n+k)*A000217(n) is the sum of n+1 consecutive integers and, after skipping k integers, the sum of the n immediately higher consecutive integers. For example, for n = 3 and k = 2, 9*6 = 54 = 12+13+14+15 = 17+18+19. - Charlie Marion, Jan 25 2022
LINKS
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Roger B. Nelsen, Proof Without Words: Consecutive Sums of Consecutive Integers, Math. Mag., 63 (1990), 25.
FORMULA
a(n) = n*(n+1)*(2*n+1)/2.
a(n) = A000330(n)*3 = A006331(n)*3/2 = A055112(n)/2 = A000217(A002378(n)) - A000217(A005563(n-1)) = A000217(A005563(n)) - A000217(A002378(n)).
a(n) = A110449(n+1, n-1) for n > 1.
a(n) = Sum_{k=A000290(n) .. A002378(n)} k = Sum_{k=n^2..n^2+n} k.
a(n) = Sum_{k=n^2+n+1 .. n^2+2*n} k = Sum_{k=A002061(n+1) .. A005563(n)} k.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6 = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Ant King, Jan 03 2011
G.f.: 3*x*(1+x)/(1-x)^4. - Ant King, Jan 03 2011
a(n) = A000578(n+1) - A000326(n+1). - Ivan N. Ianakiev, Nov 29 2012
a(n) = A005408(n)*A000217(n) = a(n-1) + 3*A000290(n). -Ivan N. Ianakiev, Mar 08 2013
a(n) = n^3 + n^2 + A000217(n). - Charlie Marion, Dec 04 2013
From Ilya Gutkovskiy, Aug 08 2016: (Start)
E.g.f.: x*(6 + 9*x + 2*x^2)*exp(x)/2.
Sum_{n>=1} 1/a(n) = 2*(3 - 4*log(2)) = 0.4548225555204375246621... (End)
a(n) = Sum_{k=0..2*n} A001318(k). - Jacob Szlachetka, Dec 20 2021
a(n) = Sum_{k=0..n} A000326(k) + A005449(k). - Jacob Szlachetka, Dec 21 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(Pi-3). - Amiram Eldar, Sep 17 2022
EXAMPLE
a(5) = 25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35 = 165.
MAPLE
A059270 := proc(n) n*(n+1)*(2*n+1)/2 ; end proc: # R. J. Mathar, Jul 10 2011
MATHEMATICA
# (#+1)(2#+1)/2 &/@ Range[0, 39] (* Ant King, Jan 03 2011 *)
CoefficientList[Series[3 x (1 + x)/(x - 1)^4, {x, 0, 39}], x]
LinearRecurrence[{4, -6, 4, -1}, {0, 3, 15, 42}, 50] (* Vincenzo Librandi, Jun 23 2012 *)
PROG
(Sage) [bernoulli_polynomial(n+1, 3) for n in range(0, 41)] # Zerinvary Lajos, May 17 2009
(Magma) I:=[0, 3, 15, 42]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 23 2012
(PARI) a(n) = n*(n+1)*(2*n+1)/2 \\ Charles R Greathouse IV, Mar 08 2013
CROSSREFS
Cf. A059255 for analog for sum of squares.
Cf. A222716 for the analogous sum of triangular numbers.
Cf. A234319 for nonexistence of analogs for sums of n-th powers, n > 2. - Jonathan Sondow, Apr 23 2014
Cf. A098737 (first subdiagonal).
Bisection of A109900.
Sequence in context: A012222 A069267 A348411 * A219085 A346142 A366576
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jan 24 2001
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 20 05:07 EDT 2024. Contains 372703 sequences. (Running on oeis4.)