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.

A154614 Triangle read by rows where T(m,n) = m*n + m + n - 1, 1<=n<=m.

Original entry on oeis.org

2, 4, 7, 6, 10, 14, 8, 13, 18, 23, 10, 16, 22, 28, 34, 12, 19, 26, 33, 40, 47, 14, 22, 30, 38, 46, 54, 62, 16, 25, 34, 43, 52, 61, 70, 79, 18, 28, 38, 48, 58, 68, 78, 88, 98, 20, 31, 42, 53, 64, 75, 86, 97, 108, 119, 22, 34, 46, 58, 70, 82, 94, 106, 118, 130, 142
Offset: 1

Views

Author

Vincenzo Librandi, Jan 16 2009

Keywords

Comments

T(m,n)+2 = (n+1)*(m+1) is not prime.
T(m,m)+2 = (m+1)^2.
First column: A005843; second column: A112414; third column: 2*A020742; fourth column: A016885. - Vincenzo Librandi, Nov 17 2012

Examples

			Triangle begins:
2;
4, 7;
6, 10, 14;
8, 13, 18, 23;
10, 16, 22, 28, 34;
12, 19, 26, 33, 40, 47;
14, 22, 30, 38, 46, 54, 62;
16, 25, 34, 43, 52, 61, 70, 79;
18, 28, 38, 48, 58, 68, 78, 88, 98;
20, 31, 42, 53, 64, 75, 86, 97, 108, 119; etc.

Links

Crossrefs

Cf. A040976, A005843, A112414, A020742, A016885.

Programs

  • Magma
    [(n*k + n + k - 1): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 17 2012
  • Mathematica
    t[n_,k_]:=n*k + n + k - 1; Table[t[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 17 2012 *)

A217776 a(n) = n*(n+1) + (n+2)*(n+3) + (n+4)*(n+5) + (n+6)*(n+7).

Original entry on oeis.org

68, 100, 140, 188, 244, 308, 380, 460, 548, 644, 748, 860, 980, 1108, 1244, 1388, 1540, 1700, 1868, 2044, 2228, 2420, 2620, 2828, 3044, 3268, 3500, 3740, 3988, 4244, 4508, 4780, 5060, 5348, 5644, 5948, 6260, 6580, 6908, 7244, 7588, 7940, 8300, 8668, 9044, 9428
Offset: 0

Views

Author

Jon Perry, Mar 24 2013

Keywords

Examples

			a(1) = 1*2 + 3*4 + 5*6 + 7*8 = 2 + 12 + 30 + 56 = 100.

Links

Crossrefs

Cf. A020742, A027690, A051890 (two pairs), A217775 (3 pairs).

Programs

  • GAP
    List([0..50], n-> (2*n+7)^2+19); # G. C. Greubel, Aug 27 2019
  • JavaScript
    for (j=0;j<50;j++) {
a=j*(j+1)+(j+2)*(j+3)+(j+4)*(j+5)+(j+6)*(j+7);
document.write(a+", ");
}
  • Magma
    [(2*n+7)^2+19: n in [0..50]]; // G. C. Greubel, Aug 27 2019
  • Maple
    seq((2*n+7)^2+19, n=0..50); # G. C. Greubel, Aug 27 2019
  • Mathematica
    (2*Range[50] +5)^2 +19 (* G. C. Greubel, Aug 27 2019 *)
Table[4n^2+28n+68,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{68,100,140},50] (* Harvey P. Dale, Jan 15 2020 *)
  • PARI
    a(n)=4*n^2+28*n+68 \\ Charles R Greathouse IV, Jun 17 2017
  • Sage
    [(2*n+7)^2+19 for n in (0..50)] # G. C. Greubel, Aug 27 2019

Formula

From Bruno Berselli, Mar 29 2013: (Start)
G.f.: 4*(17-26*x+11*x^2)/(1-x)^3.
a(n) = 4*n^2 + 28*n + 68.
a(n) = 4*A027690(n+3) = A020742(n)^2 + 19. (End)
E.g.f.: 4*(17 +8*x +x^2)*exp(x). - G. C. Greubel, Aug 27 2019

A192328 Numbers of the form 20*k+7 which are three times a square.

Original entry on oeis.org

27, 147, 507, 867, 1587, 2187, 3267, 4107, 5547, 6627, 8427, 9747, 11907, 13467, 15987, 17787, 20667, 22707, 25947, 28227, 31827, 34347, 38307, 41067, 45387, 48387, 53067, 56307, 61347, 64827, 70227, 73947, 79707, 83667, 89787, 93987
Offset: 1

Views

Author

Bruno Berselli, Jun 28 2011

Keywords

Comments

Text of the theorem in the paper mentioned in References: The necessary and sufficient condition so that a number of the form 20*k+7 is three times a square is that k is of the form 3*h*(5*h+3)+1 or 3*h*(5*h+7)+7.
A119617 gives the values of k.
A080512*120 gives the first differences.

References

  • "Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno) - Mar 1901 - p. 75 (Problem 286 and its generalization, G. Cardoso-Laynes).

Links

Crossrefs

Cf. A134153, A134154.
Cf. A020742, A047218.

Programs

  • Magma
    [m: m in [7..10^5 by 20] | IsSquare(m/3)];
  • Maple
    select(t -> issqr(t/3), [seq(20*i+7,i=1..10000,3)]); # Robert Israel, Apr 28 2023
  • Mathematica
    Select[20 Range[5000] + 7, IntegerQ[Sqrt[#/3]] &] (* or *) LinearRecurrence[{1,2,-2,-1,1}, {27,147,507,867,1587}, 40] (* Harvey P. Dale, Jul 06 2011 *)
CoefficientList[Series[3 (9 + 40 x + 102 x^2 + 40 x^3 + 9 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 19 2013 *)
  • PARI
    for(k=0, 5*10^3, m=20*k+7; if(issquare(m/3), print1(m",")));
  • PARI
    a(n)=my(m=n--\4); 1200*m^2+[360*m+27, 840*m+147, 1560*m+507, 2040*m+867][n%4+1] \\ Charles R Greathouse IV, Jun 29 2011

Formula

G.f.: 3*x*(9 + 40*x + 102*x^2 + 40*x^3 + 9*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = 3*((10*(n-1) + (-1)^(n-1) + 5)/2)^2.
a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
Sum_{i=1..n} a(i) = n*(50*(n-1)*(n+1) + 15*(-1)^(n-1) + 39)/2.
a(n) = 3*A020742(A047218(n))^2.

Extensions

Offset corrected by Mohammed Yaseen, Apr 27 2023

A035329 a(n) = n*(2*n+5)*(2*n+7).

Original entry on oeis.org

0, 63, 198, 429, 780, 1275, 1938, 2793, 3864, 5175, 6750, 8613, 10788, 13299, 16170, 19425, 23088, 27183, 31734, 36765, 42300, 48363, 54978, 62169, 69960, 78375, 87438, 97173, 107604, 118755, 130650, 143313, 156768, 171039, 186150, 202125, 218988, 236763
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • Eric Harold Neville, Jacobian Elliptic Functions, 2nd ed., p. 38.

Links

Crossrefs

Cf. A020742, A033537.

Programs

Formula

From Wesley Ivan Hurt, Oct 05 2020: (Start)
a(n) = 4*n^3 + 24*n^2 + 35*n.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: 3*x*(21-18*x+5*x^2)/(1-x)^4. (End)
From Elmo R. Oliveira, Aug 08 2025: (Start)
E.g.f.: exp(x)*x*(63 + 36*x + 4*x^2).
a(n) = A033537(n)*A020742(n). (End)

Extensions

More terms from Sean A. Irvine, Oct 05 2020
Showing 1-4 of 4 results.

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

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