A322594 -id:A322594 - cp's OEIS frontend

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

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

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, Oct 04 1994. (Annotated scanned copy)

R:=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

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 *)

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)

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

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

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

More terms from David W. Wilson, Oct 27 2000

A322595 a(n) = (n^3 + 9n + 14n + 9)/3.

Original entry on oeis.org

3, 11, 21, 35, 55, 83, 121, 171, 235, 315, 413, 531, 671, 835, 1025, 1243, 1491, 1771, 2085, 2435, 2823, 3251, 3721, 4235, 4795, 5403, 6061, 6771, 7535, 8355, 9233, 10171, 11171, 12235, 13365, 14563, 15831, 17171, 18585, 20075, 21643, 23291, 25021, 26835

Offset: 0

Franck Maminirina Ramaharo, Dec 19 2018

For n >= 6, a(n) is the number of evaluating points on the hypersphere in R^n in Stoyanovas's degree 7 cubature rule.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.
Srebra B. Stoyanova, Cubature of the seventh degree of accuracy for the hypersphere, Journal of Computational and Applied Mathematics Vol. 84 (1997), 15-21.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences: A027693.

Cf. A000292, A161680, A174794, A321124, A322594.

[(n^3 + 9*n + 14*n + 9)/3: n in [0..45]]; // Vincenzo Librandi, Jun 05 2019

Table[(n^3 + 9*n + 14*n + 9)/3, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{3,11,21,35},50] (* Harvey P. Dale, Aug 19 2020 *)

makelist((n^3 + 9*n + 14*n + 9)/3, n, 0, 50);

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
a(n) = 2*binomial(n + 1, 3) + 6*binomial(n + 1, 2) + 2*binomial(n + 1, 1) + 1.
G.f.: (3 - x - 5*x^2 + 5*x^3)/(1 - x)^4. [Corrected by Georg Fischer, May 23 2019]
E.g.f.: (1/3)*(9 + 24*x + 12*x^2 + x^3)*exp(x).

A321124 a(n) = (4n^3 - 6n^2 + 14*n + 3)/3.

Original entry on oeis.org

1, 5, 13, 33, 73, 141, 245, 393, 593, 853, 1181, 1585, 2073, 2653, 3333, 4121, 5025, 6053, 7213, 8513, 9961, 11565, 13333, 15273, 17393, 19701, 22205, 24913, 27833, 30973, 34341, 37945, 41793, 45893, 50253, 54881, 59785, 64973, 70453, 76233, 82321, 88725

Offset: 0

Franck Maminirina Ramaharo, Dec 17 2018

For n >= 5, a(n) is the number of evaluation points on the n-dimensional cube in Phillips-Dobrodeev's degree 7 cubature rule.

Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.
Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
L. N. Dobrodeev, Cubature formulas of the seventh order of accuracy for a hypersphere and a hypercube, USSR Computational Mathematics and Mathematical Physics Vol. 10 (1970), 252-253.
G. M. Phillips, Numerical integration over an N-dimensional rectangular region, The Computer Journal Vol. 10 (1967), 297-299.
Rudolf M. Schürer, High-Dimensional Numerical Integration on Parallel Computers, Phd Dissertation, 2001. p. 80.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A128445, A161680, A174794, A322594, A322595.

Table[(4*n^3 - 6*n^2 + 14*n + 3)/3, {n, 0, 50}]

makelist((4*n^3 - 6*n^2 + 14*n + 3)/3, n, 0, 50);

a(n) = 8*binomial(n, 3) + 4*binomial(n, 2) + 4*binomial(n, 1) + 1.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
a(n) = a(n-1) + A128445(n+1), n >= 1.
E.g.f.: (1/3)*(3 + 12*x + 6*x^2 + 4*x^3)*exp(x).
G.f.: (1 + x - x^2 + 7*x^3)/(1 - x)^4.

A322597 a(n) = (4n^3 - 6n^2 + 20*n + 3)/3.

Original entry on oeis.org

1, 7, 17, 39, 81, 151, 257, 407, 609, 871, 1201, 1607, 2097, 2679, 3361, 4151, 5057, 6087, 7249, 8551, 10001, 11607, 13377, 15319, 17441, 19751, 22257, 24967, 27889, 31031, 34401, 38007, 41857, 45959, 50321, 54951, 59857, 65047, 70529, 76311, 82401, 88807

Offset: 0

Franck Maminirina Ramaharo, Jan 23 2019

For n >= 2, a(n) gives the number of function evaluations for Dooren and Ridder's degree 5 and 7 cubature rule over an n-dimensional cube, with the exception of a(3) = 45 and a(4) = 97.

Ronald Cools, Encyclopaedia of Cubature Formulas
Paul van Dooren and Luc de Ridder, An adaptive algorithm for numerical integration over an n-dimensional cube, Journal of Computational and Applied Mathematics, Vol. 2 (1976), 207-217.
Alan C. Genz and Awais A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, Journal of Computational and Applied Mathematics, Vol. 6 (1980), 295-302.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences: 2*A093328.

Cf. A174794, A321124, A322594, A322595.

[(4*n^3-6*n^2+20*n+3)/3$n=0..50]; # Muniru A Asiru, Jan 23 2019

Table[(4*n^3 - 6*n^2 + 20*n + 3)/3, {n, 0, 50}]

makelist((4*n^3 - 6*n^2 + 20*n + 3)/3, n, 0, 50);

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
G.f.: (1 + 3*x - 5*x^2 + 9*x^3)/((1 - x)^4).
E.g.f.: (1/3)*(3 + 18*x + 6*x^2 + 4*x^3)*exp(x).

cp's OEIS Frontend

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.

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A322595 a(n) = (n^3 + 9*n + 14*n + 9)/3.

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A321124 a(n) = (4*n^3 - 6*n^2 + 14*n + 3)/3.

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A322597 a(n) = (4*n^3 - 6*n^2 + 20*n + 3)/3.

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A322595 a(n) = (n^3 + 9n + 14n + 9)/3.

A321124 a(n) = (4n^3 - 6n^2 + 14*n + 3)/3.

A322597 a(n) = (4n^3 - 6n^2 + 20*n + 3)/3.