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A078703
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Number of ways of subtracting twice a triangular number from a perfect square to obtain the integer n.
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2
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1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 1, 1, 4, 1, 2, 3, 1, 2, 2, 1, 1, 3, 3, 1, 2, 2, 1, 4, 1, 2, 3, 1, 2, 2, 1, 1, 4, 2, 1, 3, 2, 1, 4, 2, 1, 2, 1, 3, 3, 1, 2, 2, 2, 2, 2, 1, 1, 6, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 1, 3, 1, 2, 4, 1, 1, 3, 2, 2, 4, 2, 2, 2, 2, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Also number of symmetric unimodal consecutive integer sequences that sum to the integer n (e.g. 4+5+6+5+4=24=n). Also number of double trapezoidal arrangements of n objects; i.e. the number of ways to arrange n objects into symmetrically-placed, congruent isosceles trapezoids adjoined at overlapping largest bases.
Also number of divisors of 4*n-1 of form 4*k+1 (or 4*k+3). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 05 2004
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LINKS
| T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.
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FORMULA
| SDT(n)=((r1+1)*(r2+1)*...*(rk+1))/2, where ((p1^r1)*(p2^r2)*...*(pk^rk)) is the factorization of 4n-1 into (odd) primes.
G.f.: Sum_{n>0} x^n/(1-x^(4*n-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 05 2004
a(n) = A034178(4*n - 1). - Michael Somos May 11 2011
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EXAMPLE
| SDT(34) = 4 since we have 34 or 11+12+11 or 6+7+8+7+6 or 2+3+4+5+6+5+4+3+2 Also 4*34 - 1 = 135 = (3^3)*(5^1) so that r1=3 and r2=1 (p1=3 and p2=5), resulting in SDT(34) = (3+1)*(1+1)/2 = 4.
a(4) = 2 since 4 = 2^2 - 2*0 = 4^2 - 2*6. Also A034178(4*4 - 1) = 2 since 15 = 4^2 - 1^2 = 8^2 - 7^2. - Michael Somos May 11 2011x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 2*x^10 + x^11 + ...
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MATHEMATICA
| This defines SDT(n): SDT[n_] := Length[Cases[Range[1, n], j_ /; Cases[Range[1, j], k_ /; Plus @@ Join[Range[k, j], Range[j - 1, k, -1]] == n] != {}]] The restricted factorization technique for obtaining SDT(n) is encoded as follows: SDT[n_] := (Times @@ Cases[FactorInteger[4 n - 1], {p_, r_} -> r + 1])/2
Rest[ CoefficientList[ Series[ Sum[x^k/(1 - x^(4k - 1)), {k, 111}], {x, 0, 110}], x]] - Robert G. Wilson v (rgwv(at)rgwv.com), Sep 20 2005
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CROSSREFS
| Cf. A001227, A034178.
Sequence in context: A091591 A109374 A079706 * A090629 A086412 A192006
Adjacent sequences: A078700 A078701 A078702 * A078704 A078705 A078706
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KEYWORD
| nonn
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AUTHOR
| R. L. Coffman, K. W. McLaughlin and R. J. Dawson (robert.l.coffman(AT)uwrf.edu), Dec 19 2002
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