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 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 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). 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 Adjacent sequences: A059267 A059268 A059269 * A059271 A059272 A059273 KEYWORD nonn,easy AUTHOR Henry Bottomley, Jan 24 2001 STATUS approved

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