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A059270 Numbers which are both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers. 14
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) = 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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013

R. 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).

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)

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 xrange(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

Sequence in context: A012256 A012222 A069267 * A219085 A093627 A192060

Adjacent sequences:  A059267 A059268 A059269 * A059271 A059272 A059273

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Jan 24 2001

STATUS

approved

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Last modified March 28 21:25 EDT 2017. Contains 284246 sequences.