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A059270
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Numbers which are both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers.
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7
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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; internal format)
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OFFSET
| 0,2
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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 (amarnath_murthy(AT)yahoo.com), Jan 19 2004
Convolution of odds (A005408) and multiples of three (A008585) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 06 2006
Sums of rows of the triangle in A126890. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006
Corresponds to the Wiener indices of C_{2n+1} i.e., the cycle on 2n+1 vertices (n > 0). [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Mar 16 2009]
Also the product of the three numbers from A005843(n) up to A163300(n), divided by 8. [Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jul 26 2009]
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
| a(n) = n(n+1)(2n+1)/2 = 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) = 3 a (n - 1) - 3 a (n - 2) + a (n - 3) + 6 = 4a(n-1) - 6a(n-2) + 4a(n-3) - a(n-4) [Added by Ant King, Jan 03 2011]
G.F.: 3x (1+x) / (x-1)^4 [Added by Ant King, Jan03 2011]
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EXAMPLE
| a(5)=25+26+27+28+29+30=31+32+33+34+35=165
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MAPLE
| with(finance):seq(add(futurevalue(3, k, 2), k=0..n), n=-1..38); # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 15 2008
A059270 := proc(n) n*(n+1)*(2*n+1)/2 ; end proc: # R. J. Mathar, Jul 10 2011
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MATHEMATICA
| f[n_]:=6*n+3; s1=s2=0; lst={}; Do[a=f[n]; s1+=a; s2+=s1; AppendTo[lst, s2], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 25 2009]
# (#+1)(2#+1)/2 &/@ Range[0, 39] (*from Ant King, Jan 03 2011*)
a [n_] := a[n] = 3 a[n - 1] - 3 a[n - 2] + a [n - 3] + 6; a[0] = 0; a[1] = 3; a[2] = 15; Array[a, 40, 0]
a [n_] := a[n] = 4 a[n - 1] - 6 a[n - 2] + 4 a[n - 3] - a[n - 4]; a[0] = 0; a[1] = 3; a[2] = 15; a[3] = 42; Array[a, 40, 0]
CoefficientList[Series[3 x (1 + x)/(x - 1)^4, {x, 0, 39}], x]
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PROG
| (Other) sage: [bernoulli_polynomial(n, 3) for n in xrange(1, 41)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
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CROSSREFS
| Sum of i from i=n^2 (A000290) through to n^2+n (A002378) and from i=n^2+n+1 (A002061 offset) through to n^2+2n (A005563). Cf. A059255 for analogue for sum of squares.
a(n) = A110449(n+1, n-1) for n>1.
Sequence in context: A012256 A012222 A069267 * A093627 A192060 A146805
Adjacent sequences: A059267 A059268 A059269 * A059271 A059272 A059273
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KEYWORD
| easy,nonn
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jan 24 2001
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