A007773 For any circular arrangement of 0..n-1, let S = sum of squares of every sum of two contiguous numbers; then a(n) = # of distinct values of S.

1, 1, 1, 3, 8, 21, 43, 69, 102, 145, 197, 261, 336, 425, 527, 645, 778, 929, 1097, 1285, 1492, 1721, 1971, 2245, 2542, 2865, 3213, 3589, 3992, 4425, 4887, 5381, 5906, 6465, 7057, 7685, 8348, 9049, 9787, 10565, 11382, 12241, 13141, 14085, 15072, 16105

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

K. S. Brown, The Dartboard Sequence

H. L. Montgomery, Kevin Brown's enumeration problem, Manuscript, 4 Oct. 1994. (Annotated scanned copy)

Formula

For n >= 7, a(n) = (n^3-16*n+27)/6 (n odd); (n^3-16*n+30)/6 (n even).

G.f.: x*(1-2*x+4*x^3+3*x^5-10*x^7+2*x^8+8*x^9-4*x^10)/((1-x)^3*(1-x^2)). - Michael Somos, May 03, 2002

a(2n) = A322594(n-4), n>=4. - R. J. Mathar, Mar 18 2019

Mathematica

Drop[CoefficientList[Series[x*(1-2*x+4*x^3+3*x^5-10*x^7+2*x^8+8*x^9 -4*x^10)/((1-x)^3*(1-x^2)), {x, 0, 60}], x], 1] (* G. C. Greubel, Mar 15 2019 *)

Prog

(PARI) a(n)=polcoeff(x*(1-2*x+4*x^3+3*x^5-10*x^7+2*x^8+8*x^9-4*x^10+O(x^n))/(1-x)^3/(1-x^2), n)
(PARI) A007773(n)=if(n>5, (n^3-max(16*n, 116)+31)\6, n>3, 5*n-17, 1) \\ M. F. Hasler, Mar 15 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( x*(1-2*x +4*x^3+3*x^5-10*x^7+2*x^8+8*x^9-4*x^10)/((1-x)^3*(1-x^2)) )); // G. C. Greubel, Mar 15 2019
(Sage) a=(x*(1-2*x+4*x^3+3*x^5-10*x^7+2*x^8+8*x^9-4*x^10)/((1-x)^3*(1-x^2))).series(x, 60).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Mar 15 2019

Crossrefs

Sequence in context: A103736 A172243 A101332 * A071078 A179903 A363601

Adjacent sequences: A007770 A007771 A007772 * A007774 A007775 A007776

Keyword

nonn,easy

Author

K. S. Brown (kevin2003(AT)delphi.com), Hugh L. Montgomery

Extensions

More terms from David W. Wilson, Oct 27 2000

Status

approved