cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A093049 n-1 minus exponent of 2 in n, a(0) = 0.

Original entry on oeis.org

0, 0, 0, 2, 1, 4, 4, 6, 4, 8, 8, 10, 9, 12, 12, 14, 11, 16, 16, 18, 17, 20, 20, 22, 20, 24, 24, 26, 25, 28, 28, 30, 26, 32, 32, 34, 33, 36, 36, 38, 36, 40, 40, 42, 41, 44, 44, 46, 43, 48, 48, 50, 49, 52, 52, 54, 52, 56, 56, 58, 57, 60, 60, 62, 57, 64, 64, 66, 65, 68
Offset: 0

Views

Author

Ralf Stephan, Mar 16 2004

Keywords

Examples

			G.f. = 2*x^3 + x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 4*x^8 + 8*x^9 + 8*x^10 + ... - _Michael Somos_, Jan 25 2020

Crossrefs

a(n) = n - A007814(n) - 1 = A093048(n) - 1, n>0.
a(n) is the exponent of 2 in A001761(n+1), A002105(n), A002682(n-1), A006963(n), A036770(n-1), A059837(n), A084623(n), |A003707(n)|, |A011859(n)|.

Programs

  • Mathematica
    a[ n_] := If[ n == 0, 0, n - 1 - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)
  • PARI
    a(n)=if(n<1,0,if(n%2==0,a(n/2)+n/2-1,n-1))
  • PARI
    {a(n) = if( n, n - 1 - valuation(n, 2))}; /* Michael Somos, Jan 25 2020 */
  • Python
    def A093049(n): return n-1-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Formula

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n.
G.f.: sum(k>=0, t^3(t+2)/(1-t^2)^2, t=x^2^k).

A002681 Numerators of coefficients for repeated integration.

Original entry on oeis.org

1, -1, 1, -23, 263, -133787, 157009, -16215071, 2689453969, -26893118531, 5600751928169, -3340626516019229, 885646796787371, -859202038021848149, 2766671664340938282413, -319473088311274492668499, 436677987276721765221113, -191960665849028069896950959123
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002195, A002196, A002682.

Programs

  • Maple
    M:=n->(2/(2*n+1)!)*int(t*product(t^2-k^2,k=1..n),t=0..1): A:=n->((n+1)/2)*M(n)+(2*n+2)*M(n+1): seq(numer(A(n)),n=0..18); # Emeric Deutsch, Jan 25 2005
  • Mathematica
    M[n_] := (2/(2n+1)!) Integrate[t Product[t^2-k^2, {k, 1, n}], {t, 0, 1}];
A[n_] := ((n+1)/2) M[n] + (2n+2) M[n+1];
Table[Numerator[A[n]], {n, 0, 18}] (* Jean-François Alcover, Oct 04 2021, after Maple code *)

Formula

a(n) is the numerator of ((n+1)/2)M(n) + (2n+2)M(n+1), where M(n) = (2/(2n+1)!)*Integral_{t=0..1} (t*Product_{k=1..n} (t^2 - k^2)). - Emeric Deutsch, Jan 25 2005

Extensions

More terms from Emeric Deutsch, Jan 25 2005

A348220 Numerators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).

Original entry on oeis.org

2, 2, 0, 2, 2, 1, 2, 4, 1, -1, 2, 6, 7, 0, 29, 2, 8, 19, 1, -1, -14, 2, 10, 37, 8, -1, 1, 1139, 2, 12, 61, 9, 29, 0, -37, -41, 2, 14, 91, 64, 269, -1, 1, 8, 32377, 2, 16, 127, 125, 1079, 14, 1, -1, -119, -3956, 2, 18, 169, 72, 2999, 33, -37, 0, 127, 9, 2046263
Offset: 0

Views

Author

Jean-François Alcover and Paul Curtz, Oct 08 2021

Keywords

Comments

It can be noticed that the sequence A002681/A002682 shows as these 4 subsequences: A(i, 2i+2), -A(i, 2i+3), A(i+1, 2i+2) and A(i+2, 2i+3), for i >= 0.
Columns: A007395, A005843, A003215 (numerators).

Examples

			Array begins:
2, 0,    1/3,  -1/3,    29/90, -14/45,  1139/3780,   -41/140, ...
2, 2,    1/3,     0,    -1/90,   1/90,   -37/3780,     8/945, ...
2, 4,    7/3,   1/3,    -1/90,      0,      1/756,    -1/756, ...
2, 6,   19/3,   8/3,    29/90,  -1/90,      1/756,         0, ...
2, 8,   37/3,     9,   269/90,  14/45,   -37/3780,     1/756, ...
2, 10,  61/3,  64/3,  1079/90,  33/10,  1139/3780,    -8/945, ...
2, 12,  91/3, 125/3,  2999/90, 688/45, 13613/3780,    41/140, ...
2, 14, 127/3,    72,  6749/90, 875/18,  14281/756,   736/189, ...
2, 16, 169/3, 343/3, 13229/90,  618/5,  51031/756, 17225/756, ...
...

References

  • Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, page 127, 1969, Arcueil. Later CELAR. Now DGA Maitrise de l'Information 35170 Bruz.

Crossrefs

Cf. A002681, A002682, A348221 (denominators).
Cf. A003215, A005843, A007395.

Programs

  • Mathematica
    A[i_ /; i >= 0, k_ /; k >= 0] := A[i, k] = If[i == 0, (1/k!) Integrate[ Product[u+j, {j, -k+1, 0}], {u, -1, 1}], A[i-1, k-1] + A[i-1, k]];
A[, ] = 0;
Table[A[i-k, k] // Numerator, {i, 0, 10}, {k, 0, i}] // Flatten
  • PARI
    array(nn) = {my(m = matrix(nn, nn)); for (k=0, nn-1, m[1, k+1] = bestappr(intnum(x=-1, 1, prod(j=1-k, 0, x+j)))/k!; ); for (j=1, nn-1, for (k=0, nn-1, m[j+1, k+1] = if (k>0, m[j,k], 0) + m[j, k+1];);); apply(numerator, m);} \\ Michel Marcus, Oct 08 2021

Formula

Numerators of A(i,k) where:
A(i,k) = (1/k!)*Integral_(-1,1) Product(u+j, (j, -k+1 .. 0)) du for i=0.
A(i,k) = A(i-1, k-1) + A(i-1, k) for i>0.

A348221 Denominators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 90, 1, 1, 3, 3, 90, 45, 1, 1, 3, 3, 90, 90, 3780, 1, 1, 3, 1, 90, 1, 3780, 140, 1, 1, 3, 3, 90, 90, 756, 945, 113400, 1, 1, 3, 3, 90, 45, 756, 756, 16200, 14175, 1, 1, 3, 1, 90, 10, 3780, 1, 113400, 1400, 7484400
Offset: 0

Views

Author

Jean-François Alcover and Paul Curtz, Oct 08 2021

Keywords

Comments

See A348220.

Examples

			Array begins:
2, 0,    1/3,  -1/3,    29/90, -14/45,  1139/3780,   -41/140, ...
2, 2,    1/3,     0,    -1/90,   1/90,   -37/3780,     8/945, ...
2, 4,    7/3,   1/3,    -1/90,      0,      1/756,    -1/756, ...
2, 6,   19/3,   8/3,    29/90,  -1/90,      1/756,         0, ...
2, 8,   37/3,     9,   269/90,  14/45,   -37/3780,     1/756, ...
2, 10,  61/3,  64/3,  1079/90,  33/10,  1139/3780,    -8/945, ...
2, 12,  91/3, 125/3,  2999/90, 688/45, 13613/3780,    41/140, ...
2, 14, 127/3,    72,  6749/90, 875/18,  14281/756,   736/189, ...
2, 16, 169/3, 343/3, 13229/90,  618/5,  51031/756, 17225/756, ...
...

References

  • Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, page 127, 1969.

Crossrefs

Cf. A002681, A002682, A348220 (numerators).

Programs

  • Mathematica
    A[i_ /; i >= 0, k_ /; k >= 0] := A[i, k] = If[i == 0, (1/k!)  Integrate[ Product[u + j, {j, -k + 1, 0}], {u, -1, 1}], A[i - 1, k - 1] + A[i - 1, k]]; A[, ] = 0;
Table[A[i - k, k] // Denominator, {i, 0, 10}, {k, 0, i}] // Flatten

Formula

Denominators of A(i,k) where:
A(i,k) = (1/k!)*Integral_(-1,1) Product(u+j, (j, -k+1 .. 0)) du for i=0.
A(i,k) = A(i-1, k-1) + A(i-1, k) for i>0.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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