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.

A196514 Partial sums of A100381.

Original entry on oeis.org

0, 4, 28, 124, 444, 1404, 4092, 11260, 29692, 75772, 188412, 458748, 1097724, 2588668, 6029308, 13893628, 31719420, 71827452, 161480700, 360710140, 801112060, 1769996284, 3892314108, 8522825724, 18589155324, 40399536124, 87509958652
Offset: 0

Views

Author

R. J. Mathar, Oct 03 2011

Keywords

Comments

Like any sequence with a linear recurrence, this has a Pisano period length modulo any k >= 1. The period lengths for this sequence are (modulo k >= 1) 1, 1, 6, 1, 20, 6, 21, 1, 18, 20, 110, 6, 156, 21, 60, 1, 136, 18, 342, 20, ....

References

  • Jolley, Summation of Series, Dover (1961), eq (53) page 10.

Links

Programs

Formula

G.f.: 4*x / ( (x-1)*(2*x-1)^3 ).
a(n) = (n^2 - n + 2)*2^(n+1) - 4 = 4*A055580(n-1).
a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4); a(0)=0, a(1)=4, a(2)=28, a(3)=124. - Harvey P. Dale, Jan 12 2016

A244009 Decimal expansion of 1 - log(2).

Original entry on oeis.org

3, 0, 6, 8, 5, 2, 8, 1, 9, 4, 4, 0, 0, 5, 4, 6, 9, 0, 5, 8, 2, 7, 6, 7, 8, 7, 8, 5, 4, 1, 8, 2, 3, 4, 3, 1, 9, 2, 4, 4, 9, 9, 8, 6, 5, 6, 3, 9, 7, 4, 4, 7, 4, 5, 8, 7, 9, 3, 1, 9, 9, 9, 0, 5, 0, 6, 6, 0, 6, 3, 7, 8, 0, 3, 0, 3, 0, 5, 2, 8, 4, 3, 9, 4, 1, 3, 6, 6, 7, 3, 0, 0, 3, 5, 8, 1, 3, 1, 2, 4, 5, 7, 9, 9, 8, 5
Offset: 0

Views

Author

Franklin T. Adams-Watters, Jun 17 2014

Keywords

Comments

Fraction of numbers which are sqrt-smooth, see A048098 and A063539. - Charles R Greathouse IV, Jul 14 2014
Asymptotic survival probability in the 100 prisoners problem. - Alois P. Heinz, Jul 08 2022

Examples

			0.30685281944005469058276787854...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Links

Crossrefs

Cf. A002162, A019739, A024168, A100381, A135002, A188859.
Essentially the same digits as A239354.

Programs

  • Maple
    f:= sum(1/(2*k*(2*k+1)), k=1..infinity):
s:= convert(evalf(f, 140), string):
seq(parse(s[i+1]), i=1..106);  # Alois P. Heinz, Jun 17 2014
  • Mathematica
    RealDigits[1-Log[2],10,120][[1]] (* Harvey P. Dale, Sep 23 2016 *)
  • PARI
    1-log(2) \\ Charles R Greathouse IV, Jul 14 2014

Formula

Equals Sum_{k>=0} 1/(2*k*(2*k+1)) = A239354 + 1/4 = A188859/2.
From Amiram Eldar, Aug 07 2020: (Start)
Equals Sum_{k>=1} 1/(k*(k+1)*2^k) = Sum_{k>=2} 1/A100381(k).
Equals Sum_{k>=2} (-1)^k * zeta(k)/2^k.
Equals Integral_{x=1..oo} 1/(x^2 + x^3) dx. (End)
Equals log(e/2) = log(A019739) = -log(2/e) = -log(A135002). - Wolfdieter Lang, Mar 04 2022
Equals lim_{n->oo} A024168(n)/n!. - Alois P. Heinz, Jul 08 2022
Equals 1/(4 - 4/(7 - 12/(10 - ... - 2*n*(n-1)/((3*n+1) - ...)))) (an equivalent continued fraction for 1 - log(2) was conjectured by the Ramanujan machine). - Peter Bala, Mar 04 2024
Equals Sum_{k>=1} zeta(2*k)/((2*k + 1)*2^(2*k-1)) (see Finch). - Stefano Spezia, Nov 02 2024

A162007 Third left hand column of the EG1 triangle A162005.

Original entry on oeis.org

1, 270, 36096, 4766048, 704357760, 120536980224, 24060789342208, 5590122715250688, 1503080384197754880, 464520829174515630080, 163839204411117787938816, 65500849343294249018327040
Offset: 3

Views

Author

Johannes W. Meijer, Jun 27 2009

Keywords

Crossrefs

Third left hand column of the EG1 triangle A162005.
Other left hand columns are A000182 and A162006.
Related to A094665, A083061 and A156919.
A000079, A036289 and A100381 appear in the a(n, 3) formula.
A001789, A003472, A054849, A002409, A054851, A140325 and A140354 (scaled by 2^(m-1)) appear one by one in the a(n, m) formulas for m= 4 and higher .

Programs

  • Maple
    nmax := 14; mmax := nmax: imax := nmax: T1(0, x) := 1: T1(0, x+1) := 1: for i from 1 to imax do T1(i, x) := expand((2*x+1)*(x+1)*T1(i-1, x+1) - 2*x^2*T1(i-1, x)): dx := degree(T1(i, x)): for k from 0 to dx do c(k) := coeff(T1(i, x), x, k) od: T1(i, x+1) := sum(c(j1)*(x+1)^(j1), j1 = 0..dx): od: for i from 0 to imax do for j from 0 to i do A083061(i, j) := coeff(T1(i, x), x, j) od: od: for n from 0 to nmax do for k from 0 to n do A094665(n+1, k+1) := A083061(n, k) od: od: A094665(0, 0) := 1: for n from 1 to nmax do A094665(n, 0) := 0 od: for m from 1 to mmax do A156919(0, m) := 0 end do: for n from 0 to nmax do A156919(n, 0) := 2^n end do: for n from 1 to nmax do for m from 1 to mmax do A156919(n, m) := (2*m+2)*A156919(n-1, m) + (2*n-2*m+1) * A156919(n-1, m-1) end do end do: m:=3; for n from m to nmax do a(n, m) := sum((-1)^(m-p1-1)*sum(2^(n-q-1)*binomial(n-q-1, m-p1-1) * A094665(n-1, q) * A156919(q, p1), q=1..n-m+p1), p1=0..m-1) od: seq(a(n, m), n = m..nmax);
# Maple program edited by Johannes W. Meijer, Sep 25 2012

Formula

a(n) = sum((-1)^(m-p-1)*sum(2^(n-q-1)*binomial(n-q-1,m-p-1)*A094665(n-1,q)* A156919(q,p),q=1..n-m+p), p=0..m-1) with m = 3.

A237765 Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 3, 9, 9, 3, 6, 24, 36, 24, 6, 10, 50, 100, 100, 50, 10, 15, 90, 225, 300, 225, 90, 15, 21, 147, 441, 735, 735, 441, 147, 21, 28, 224, 784, 1568, 1960, 1568, 784, 224, 28, 36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36
Offset: 0

Views

Author

Geoffrey Critzer, Feb 12 2014

Keywords

Comments

T(n,k) is the number of ways to underline exactly two elements of {1,2,...,n} and then circle exactly k elements. (The k elements that are circled are not necessarily different from the two underlined elements).
T(n,0) = T(n,n) = binomial(n,2) = A000217(n-1).
Row sums = 2^n*binomial(n,2) = A100381(n).

Examples

			0;
0,  0;
1,  2,   1;
3,  9,   9,    3;
6,  24,  36,   24,   6;
10, 50,  100,  100,  50,   10;
15, 90,  225,  300,  225,  90,   15;
21, 147, 441,  735,  735,  441,  147,  21;
28, 224, 784,  1568, 1960, 1568, 784,  224,  28;
36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36;

References

  • J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 14, problem #2.

Crossrefs

Cf. A134400.

Programs

  • Mathematica
    Table[Table[Binomial[n,2](Binomial[n-2,r]+2Binomial[n-2,r-1]+Binomial[n-2,r-2]),{r,0,n}],{n,0,9}]//Grid

Formula

E.g.f.: (x^2/2! + 2*y*x^2/2! + y^2*x^2/2!)*exp(y*x)*exp(x).
E.g.f. for column k: x^2/2!*exp(x)*(x^k/k! + 2*x^(k-1)/(k-1)! + x^(k-2)/(k-2)!).
T(n,k) = C(n,2)*( C(n-2,k) + 2*C(n-2,k-1) + C(n-2,k-2) ).
Showing 1-4 of 4 results.

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