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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. 16

%I

%S 0,3,15,42,90,165,273,420,612,855,1155,1518,1950,2457,3045,3720,4488,

%T 5355,6327,7410,8610,9933,11385,12972,14700,16575,18603,20790,23142,

%U 25665,28365,31248,34320,37587,41055,44730,48618,52725,57057,61620

%N Numbers which are both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers.

%C 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

%C 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

%C Sums of rows of the triangle in A126890. - _Reinhard Zumkeller_, Dec 30 2006

%C 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

%C Also the product of the three numbers from A005843(n) up to A163300(n), divided by 8. - _Juri-Stepan Gerasimov_, Jul 26 2009

%C Partial sums of A033428. - _Charlie Marion_, Dec 08 2013

%C For n > 0, sum of multiples of n and (n+1) from 1 to n*(n+1). - _Zak Seidov_, Aug 07 2016

%H Vincenzo Librandi, <a href="/A059270/b059270.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013

%H R. Nelsen, <a href="http://www.jstor.org/stable/2691505">Proof Without Words: Consecutive Sums of Consecutive Integers</a>, Math. Mag., 63 (1990), 25.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = n*(n+1)*(2*n+1)/2.

%F 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)).

%F a(n) = A110449(n+1, n-1) for n > 1.

%F a(n) = Sum_{k=A000290(n) .. A002378(n)} k = Sum_{k=n^2.. n^2+n} k.

%F a(n) = Sum_{k=n^2+n+1 .. n^2+2*n} k = Sum_{k=A002061(n+1) .. A005563(n)} k.

%F 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

%F G.f.: 3*x(1+x)/(1-x)^4. - _Ant King_, Jan 03 2011

%F a(n) = A000578(n+1) - A000326(n+1). - _Ivan N. Ianakiev_, Nov 29 2012

%F a(n) = A005408(n)*A000217(n) = a(n-1) + 3*A000290(n). -_Ivan N. Ianakiev_, Mar 08 2013

%F a(n) = n^3 + n^2 + A000217(n). - _Charlie Marion_, Dec 04 2013

%F From _Ilya Gutkovskiy_, Aug 08 2016: (Start)

%F E.g.f.: x*(6 + 9*x + 2*x^2)*exp(x)/2.

%F Sum_{n>=1} 1/a(n) = 2*(3 - 4*log(2)) = 0.4548225555204375246621... (End)

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

%p A059270 := proc(n) n*(n+1)*(2*n+1)/2 ; end proc: # _R. J. Mathar_, Jul 10 2011

%t # (#+1)(2#+1)/2 &/@ Range[0,39] (* _Ant King_, Jan 03 2011 *)

%t CoefficientList[Series[3 x (1 + x)/(x - 1)^4, {x, 0, 39}], x]

%t LinearRecurrence[{4,-6,4,-1},{0,3,15,42},50] (* _Vincenzo Librandi_, Jun 23 2012 *)

%o (Sage) [bernoulli_polynomial(n+1,3) for n in xrange(0, 41)] # _Zerinvary Lajos_, May 17 2009

%o (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

%o (PARI) a(n) = n*(n+1)*(2*n+1)/2 \\ _Charles R Greathouse IV_, Mar 08 2013

%Y Cf. A059255 for analog for sum of squares.

%Y Cf. A222716 for the analogous sum of triangular numbers.

%Y Cf. A234319 for nonexistence of analogs for sums of n-th powers, n > 2. - _Jonathan Sondow_, Apr 23 2014

%K nonn,easy

%O 0,2

%A _Henry Bottomley_, Jan 24 2001

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Last modified March 25 16:26 EDT 2019. Contains 321470 sequences. (Running on oeis4.)