A059270 a(n) is both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers.

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

Offset: 0

Author

Henry Bottomley, Jan 24 2001

Keywords

• nonn
• easy

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

Examples

			a(5) = 25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35 = 165.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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.

Programs

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

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 *)

PARI

a(n) = n*(n+1)*(2*n+1)/2 \\ Charles R Greathouse IV, Mar 08 2013

Sage

[bernoulli_polynomial(n+1,3) for n in range(0, 41)] # Zerinvary Lajos, May 17 2009

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