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A189835 Number of representations of n as a*b + b*c + c*d + d*e where a, b, d, e>0, c>=0 are integers. 3
0, 1, 4, 9, 16, 26, 36, 53, 64, 90, 100, 138, 144, 194, 200, 261, 256, 347, 324, 426, 416, 522, 484, 658, 576, 746, 712, 882, 784, 1060, 900, 1173, 1088, 1314, 1160, 1587, 1296, 1658, 1544, 1890, 1600, 2164, 1764, 2298, 2096, 2466, 2116, 2930, 2304, 2955, 2696 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Related to "Liouville's Last Theorem".

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 273.

George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_5(n).

E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380.

FORMULA

G.f.: Sum_{k>0} (x^k + x^(2*k)) / (1 - x^k)^3 - k * x^k / (1 - x^k)^2.

a(n) = A001157(n) - A038040(n) = sigma( n, 2) - n * sigma( n, 0) where sigma( n, k) is the sum of the k-th powers of the divisors of n.

EXAMPLE

G.f. = x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 26*x^6 + 36*x^7 + 53*x^8 + 64*x^9 + 90*x^10 + ...

a(3) = 4 since 3 = 1*1 + 1*0 + 0*1 + 1*2 = 1*1 + 1*0 + 0*2 + 2*1 = 1*2 + 2*0 + 0*1 + 1*1 = 2*1 + 1*0 + 0*1 + 1*1 are all 4 representations of 3.

MAPLE

with(numtheory); f:=n->sigma[2](n)-n*sigma[0](n);

MATHEMATICA

a[n_] := DivisorSigma[2, n] - n*DivisorSigma[0, n]; Table[a[n], {n, 51}] (* Jean-Fran├žois Alcover, Aug 31 2011 *)

PROG

(PARI) {a(n) = if( n<1, 0, sigma( n, 2) - n * sigma( n, 0))}

(Haskell)

a189835 n = a001157 n - a038040 n  -- Reinhard Zumkeller, Jan 21 2014

CROSSREFS

Cf. A001157, A038040, A191822.

Sequence in context: A230365 A274963 A050461 * A183764 A022334 A073171

Adjacent sequences:  A189832 A189833 A189834 * A189836 A189837 A189838

KEYWORD

nonn

AUTHOR

Michael Somos, Apr 28 2011

EXTENSIONS

Added references, comment, Maple program, cross-reference to A191822. - N. J. A. Sloane, Jun 17 2011

STATUS

approved

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Last modified November 26 07:49 EST 2020. Contains 338632 sequences. (Running on oeis4.)