A321118 T(n,k) = A321119(n) - (-1)^k*A321119(n-2*k)/2 for 0 < k < n, with T(0,0) = 0 and T(n,0) = T(n,n) = A002530(n+1) for n > 0, triangle read by rows; unreduced numerator of the weights of Holladay-Sard's quadrature formula.

0, 1, 1, 3, 10, 3, 4, 11, 11, 4, 11, 32, 26, 32, 11, 15, 43, 37, 37, 43, 15, 41, 118, 100, 106, 100, 118, 41, 56, 161, 137, 143, 143, 137, 161, 56, 153, 440, 374, 392, 386, 392, 374, 440, 153, 209, 601, 511, 535, 529, 529, 535, 511, 601, 209

Offset

0,4

Comments

The n-th row common denominator is factorized out and is given by A321119(n).

Given a continuous function f over the interval [0,n], the best quadrature formula in the sense of Holladay-Sard is given by Integral_{x=0..n} f(x) dx = Sum_{k=0..n} T(n,k)*f(k)/A321119(n). The formula is exact if f belongs to the class of natural cubic splines.

References

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.2.

Links

Franck Maminirina Ramaharo, Rows n = 0..150 of triangle, flattened

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, Chapter II The Cubic Spline, Mathematics in Science and Engineering Volume 38 (1967), p. 9-74.

John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.

Peter Köhler, On the weights of Sard's quadrature formulas, CALCOLO Vol. 25 (1988), 169-186.

Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.

Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.

Isaac J. Schoenberg, Spline interpolation and best quadrature formulae, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.

Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.

Formula

T(n,k)/A321119(n) = (alpha^(n + 1) - (-alpha)^(-(n + 1)))/(2*sqrt(6)*(alpha^n + (-alpha)^(-n))) if k = 0 or k = n, and 1 - (-1)^k*(alpha^(n - 2*k) + (-alpha)^(2*k - n))/(2*(alpha^n + (-alpha)^(-n))) if 0 < k < n, where alpha = (sqrt(2) + sqrt(6))/2.

T(n,k) = T(n,n-k).

T(n,k) = 4*T(n-2,k) - T(n-4,k), n >= k + 4.

T(2*n+2,k)*A001834(n+1) = A001834(n)*T(2*n,k) + 2*A003500(n)*T(2*n+1,k) for k < 2*n.

T(2*n+3,k)*A003500(n+1) = A003500(n)*T(2*n+1,k) + 2*A001834(n+1)*T(2*n+2,k) for k < 2*n + 1.

Sum_{k=0..n} T(n,k)/A321119(n) = n.

Example

Triangle begins (denominator is factored out):
    0;                                                 1/4
    1,   1;                                            1/2
    3,  10,   3;                                       1/8
    4,  11,  11,   4;                                  1/10
   11,  32,  26,  32,  11;                             1/28
   15,  43,  37,  37,  43,  15;                        1/38
   41, 118, 100, 106, 100, 118,  41;                   1/104
   56, 161, 137, 143, 143, 137, 161,  56;              1/142
  153, 440, 374, 392, 386, 392, 374, 440, 153;         1/388
  209, 601, 511, 535, 529, 529, 535, 511, 601, 209;    1/530
  ...
If f is a continuous function over the interval [0,3], then the quadrature formula yields Integral_{x=0..3} f(x) d(x) = (1/10)*(4*f(0) + 11*f(1) + 11*f(2) + 4*f(3)).

Mathematica

alpha = (Sqrt[2] + Sqrt[6])/2; T[0, 0] = 0;
T[n_, k_] := If[n > 0 && k == 0 || k == n, (alpha^(n + 1) - (-alpha)^(-(n + 1)))/(2*Sqrt[6]*(alpha^n + (-alpha)^(-n))), 1 - (-1)^k*(alpha^(n - 2*k) + (-alpha)^(2*k - n))/(2*(alpha^n + (-alpha)^(-n)))];
a321119[n_] := 2^(-Floor[(n - 1)/2])*((1 - Sqrt[3])^n + (1 + Sqrt[3])^n);
Table[FullSimplify[a321119[n]*T[n, k]], {n, 0, 10}, {k, 0, n}] // Flatten

Prog

(Maxima)
(b[0] : 0, b[1] : 1, b[2] : 1, b[3] : 3, b[n] := 4*b[n-2] - b[n-4])$ /* A002530 */
d(n) := 2^(-floor((n - 1)/2))*((1 - sqrt(3))^n + (1 + sqrt(3))^n) $ /* A321119 */
T(n, k) := if n = 0 and k = 0 then 0 else if n > 0 and k = 0 or k = n then b[n + 1] else d(n) - (-1)^k*d(n - 2*k)/2$
create_list(ratsimp(T(n, k)), n, 0, 10, k, 0, n);

Crossrefs

Cf. A093735, A093736, A100641, A002176, A100640, A321119, A321120, A321121, A321122.

Sequence in context: A111272 A124692 A091043 * A167790 A010708 A072988

Adjacent sequences: A321115 A321116 A321117 * A321119 A321120 A321121

Keyword

nonn,easy,tabl,frac

Author

Franck Maminirina Ramaharo, Nov 01 2018

Status

approved