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 1-5 of 5 results.

A241765 a(n) = n*(n + 1)*(n + 2)*(3*n + 17)/24.

Original entry on oeis.org

0, 5, 23, 65, 145, 280, 490, 798, 1230, 1815, 2585, 3575, 4823, 6370, 8260, 10540, 13260, 16473, 20235, 24605, 29645, 35420, 41998, 49450, 57850, 67275, 77805, 89523, 102515, 116870, 132680, 150040, 169048, 189805, 212415, 236985, 263625, 292448
Offset: 0

Views

Author

Bruno Berselli, Apr 28 2014

Keywords

Comments

Equivalently, Sum_{i=0..n} (i+4)*A000217(i).
Sequences of the type Sum_{i=0..n} (i+k)*A000217(i):
k = 0, A001296: 0, 1, 7, 25, 65, 140, 266, 462, ...
k = 1, A000914: 0, 2, 11, 35, 85, 175, 322, 546, ...
k = 2, A050534: 0, 3, 15, 45, 105, 210, 378, 630, ... (deleting two 0)
k = 3, A215862: 0, 4, 19, 55, 125, 245, 434, 714, ...
k = 4, a(n): 0, 5, 23, 65, 145, 280, 490, 798, ...
k = 5, A239568: 0, 6, 27, 75, 165, 315, 546, 882, ...
Antidiagonal sums (without 0) give A034263: 1, 9, 39, 119, 294, ...
Diagonal: 1, 11, 45, 125, 280, 546, ... is A051740.
Also: k = -1 gives A050534 deleting a 0; k = -2 gives 0 followed by A059302.
After 0, partial sums of A212343 and third column of A118788.
This sequence is even related to A005286 by a(n) = n*A005286(n) - Sum_{i=0..n-1} A005286(i).

Examples

			a(7) = 4*0 + 5*1 + 6*3 + 7*6 + 8*10 + 9*15 + 10*21 + 11*28 = 798.

Links

Crossrefs

Cf. A000217, A005286, A118788, A212343, A227342.
Cf. similar sequences A000914, A001296, A050534, A059302, A215862, A239568 (see table in Comments lines).

Programs

  • Magma
    /* By first comment: */ k:=4; A000217:=func; [&+[(i+k)*A000217(i): i in [0..n]]: n in [0..40]];
  • Maple
    A241765:=n->n*(n + 1)*(n + 2)*(3*n + 17)/24; seq(A241765(n), n=0..40); # Wesley Ivan Hurt, May 09 2014
  • Mathematica
    Table[n (n + 1) (n + 2) (3 n + 17)/24, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 23, 65, 145}, 40]
CoefficientList[Series[x (5 - 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
  • Maxima
    makelist(coeff(taylor(x*(5-2*x)/(1-x)^5, x, 0, n), x, n), n, 0, 40);
  • PARI
    a(n)=n*(n+1)*(n+2)*(3*n+17)/24 \\ Charles R Greathouse IV, Oct 07 2015
  • PARI
    x='x+O('x^99); concat(0, Vec(x*(5-2*x)/(1-x)^5)) \\ Altug Alkan, Apr 10 2016
  • Sage
    [n*(n+1)*(n+2)*(3*n+17)/24 for n in (0..40)]

Formula

G.f.: x*(5 - 2*x)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A227342(A055998(n+1)).
a(n) = Sum_{j=0..n+2} (-1)^(n-j)*binomial(-j,-n-2)*S1(j,n), S1 Stirling cycle numbers A132393. - Peter Luschny, Apr 10 2016

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

2, 9, 5, 24, 18, 8, 50, 42, 27, 11, 90, 80, 60, 36, 14, 147, 135, 110, 78, 45, 17, 224, 210, 180, 140, 96, 54, 20, 324, 308, 273, 225, 170, 114, 63, 23, 450, 432, 392, 336, 270, 200, 132, 72, 26, 605, 585, 540, 476, 399, 315
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2012

Keywords

Comments

Principal diagonal: A213820.
Antidiagonal sums: A153978.
Row 1, (1,2,3,4,...)**(2,5,8,11,...): A006002.
Row 2, (1,2,3,4,...)**(5,8,11,14,...): is it the sequence A212343?.
Row 3, (1,2,3,4,...)**(8,11,14,17,...): (k^3 + 8*k^2 + 7*k)/2.
For a guide to related arrays, see A212500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
2....9....24....50....90....147
5....18...42....80....135...210
8....27...60....110...180...273
11...36...78....140...225...336
14...45...96....170...270...399
17...54...114...200...315...462

Links

Crossrefs

Cf. A212500

Programs

  • Mathematica
    b[n_]:=n;c[n_]:=3n-1;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213819 *)
Table[t[n,n],{n,1,40}] (* A213820 *)
d/2 (* A002414 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A153978 *)
s1/2 (* A001296 *)

Formula

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x(3*n-1 - (3*n-4)*x) and g(x) = (1-x)^4.

A267370 Partial sums of A140091.

Original entry on oeis.org

0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510
Offset: 0

Views

Author

Bruno Berselli, Jan 13 2016

Keywords

Comments

After 0, this sequence is the third column of the array in A185874.
Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.

Examples

			The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
.  0;
.  1,  5;
.  4,  7, 10;
.  9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.

Links

Crossrefs

Cf. A005581, A051744, A105163, A140091, A185874.
Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).

Programs

  • Magma
    [n*(n+1)*(n+5)/2: n in [0..50]];
  • Mathematica
    Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,6,21,48},50] (* Harvey P. Dale, Jul 18 2019 *)
  • PARI
    vector(50, n, n--; n*(n+1)*(n+5)/2)
  • Sage
    [n*(n+1)*(n+5)/2 for n in (0..50)]

Formula

O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)

A123168 Continued fraction for c = sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1).

Original entry on oeis.org

0, 1, 6, 5, 14, 9, 22, 13, 30, 17, 38, 21, 46, 25, 54, 29, 62, 33, 70, 37, 78, 41, 86, 45, 94, 49, 102, 53, 110, 57, 118, 61, 126, 65, 134, 69, 142, 73, 150, 77, 158, 81, 166, 85, 174, 89, 182, 93, 190, 97, 198, 101, 206, 105, 214, 109, 222, 113
Offset: 1

Views

Author

Benoit Cloitre, Oct 02 2006

Keywords

Comments

This continued fraction shows exp(sqrt(2)) is irrational (see A274540).

References

  • J. Borwein and D. Bailey, Mathematics by experiment, plausible reasoning in the 21st Century, A. K. Peters, p. 77.

Links

Crossrefs

Cf. A000217, A000384, A212343.
Odd bisection of A062828 with 0 prepended.
Cf. A274540, A123251.

Programs

  • Mathematica
    $MinPrecision = 5 $MachinePrecision; ContinuedFraction[Sqrt[2]* (Exp[Sqrt[2]] - 1)/(Exp[Sqrt[2]] + 1), 100]  (* G. C. Greubel, Aug 17 2018 *) (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 6, 5, 14}, 100] (* Georg Fischer, Aug 26 2022 *)
  • PARI
    default(realprecision, 1000); contfrac(sqrt(2)*(exp(sqrt(2))-1)/ (exp(sqrt(2))+1)) \\ Michel Marcus, Oct 11 2016

Formula

a(2*n) = 4*n-3, a(2*n+1) = 8*n-2.
From Colin Barker, Apr 15 2012: (Start)
a(n) = 2*a(n-2) - a(n-4) for n>5.
G.f.: x^2*(1+6*x+3*x^2+2*x^3)/((1-x)^2*(1+x)^2). (End)
a(n) = (2*n-3)*(3-(-1)^n)/2 for n>1, with a(1) = 0. - Wesley Ivan Hurt, Apr 01 2022

A347823 Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 2, 3, 3, 8, 5, 4, 15, 18, 7, 5, 24, 42, 32, 9, 6, 35, 80, 90, 50, 11, 7, 48, 135, 200, 165, 72, 13, 8, 63, 210, 385, 420, 273, 98, 15, 9, 80, 308, 672, 910, 784, 420, 128, 17, 10, 99, 432, 1092, 1764, 1890, 1344, 612, 162, 19, 11, 120, 585, 1680, 3150, 4032, 3570, 2160, 855, 200, 21
Offset: 0

Views

Author

Jules Beauchamp, Jan 23 2022

Keywords

Examples

			Triangle begins:
  1;
  2,  3;
  3,  8,   5;
  4, 15,  18,   7;
  5, 24,  42,  32,   9;
  6, 35,  80,  90,  50,  11;
  7, 48, 135, 200, 165,  72, 13;
  8, 63, 210, 385, 420, 273, 98, 15;
  ...

Links

Crossrefs

Row sums give A053220.
Columns give A000027, A005563, A212343.
Diagonals give A005408, A001105, A059270, A112742.
Cf. A007318, A094727.

Programs

Formula

T(n,k) = A094727(n+1,k)*A007318(n,k).
Row g.f.: (1 + x)^(n-1)*(1 + n + x + 2*n*x). - Stefano Spezia, Jan 23 2022
Showing 1-5 of 5 results.

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