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.

Previous Showing 51-60 of 65 results. Next

A128090 Denominators in inverse of A128077, numerators = 1.

Original entry on oeis.org

1, -4, 4, -18, -18, 9, -48, -48, -48, 16, -100, -100, -100, -100, 25, -180, -180, -180, -180, -180, 36, -294, -294, -294, -294, -294, -294, 49
Offset: 1

Views

Author

Gary W. Adamson, Feb 14 2007

Keywords

Comments

Row sums of the inverse of A128077: (1/1; -1/4, 1/4; -1/18, -1/18, 1/9; ...) = (1, 0, 0, 0, ...).
Row sums of unsigned terms: (1; 4, 4; 18, 18, 9; ...) = A128091.
A045991 = (0, 0, 4, 18, 48, 100, ...).

Examples

			First few rows of the triangle:
     1
    -4,    4;
   -18,  -18,    9;
   -48,  -48,  -48,   16;
  -100, -100, -100, -100, 25;
  ...

Crossrefs

Cf. A128077, A045991, A128091.

Formula

Denominators in inverse triangular matrix of A128077, i.e., inverse of (1; 1, 4; 1, 2, 9; 1, 2, 3, 16; ...).

A128091 Row sums of unsigned A128090.

Original entry on oeis.org

1, 8, 45, 160, 425, 936, 1813, 3200, 5265, 8200, 12221, 17568, 24505, 33320, 44325, 57856, 74273, 93960, 117325, 144800, 176841, 213928, 256565, 305280, 360625, 423176, 493533, 572320, 660185, 757800, 865861, 985088, 1116225, 1260040, 1417325
Offset: 1

Views

Author

Gary W. Adamson, Feb 14 2007

Keywords

Examples

			a(4) = 160 = sum of row 4 terms of A128090: (48 + 48 + 48 + 16) = 3*A045991(4) + 4^2; where A045991 = (0, 0, 4, 18, 48, 100, ...).

Links

Crossrefs

Cf. A000290, A002522, A128077, A128090, A045991.

Programs

Formula

a(n) = (n-1)*A045991(n) + n^2.
a(n) = n^2*(n^2 - 2*n + 2) = A000290(n)*A002522(n-1). - Philippe Deléham, Mar 16 2014
G.f.: x*(1 + 3*x + 15*x^2 + 5*x^3)/(1-x)^5. - Philippe Deléham, Mar 16 2014
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), a(1) = 1, a(2) = 8, a(3) = 45, a(4) = 160, a(5) = 425. - Philippe Deléham, Mar 16 2014

Extensions

a(10)-a(35) from Philippe Deléham, Mar 16 2014

A133039 a(n) = P(n)^3 - P(n)^2 where P(n) = A000931(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 18, 48, 100, 294, 648, 1584, 3840, 8820, 21168, 49284, 115248, 270400, 628660, 1468548, 3420150, 7960000, 18539400, 43120350, 100328400, 233365440, 542672640, 1262045880, 2934442944, 6822962664, 15863704528, 36881698048, 85746672900, 199347278724, 463445232298
Offset: 0

Views

Author

Omar E. Pol, Nov 02 2007

Keywords

Examples

			a(10)=18 because Padovan(10)=3 and 3^3=27 and 3^2=9 and 27-9=18.

Links

Crossrefs

Cf. A000290, A000578, A045991. Padovan sequence: A000931.

Programs

  • Mathematica
    P[0] := 1; P[1] := 0; P[2] := 0; P[n_] := P[n] = P[n - 2] + P[n - 3]; Table[P[n]^3 - P[n]^2, {n, 0, 50}] (* G. C. Greubel, Oct 02 2017 *)
  • PARI
    x='x+O('x^50); concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(2*x^8*(x^7-x^6+2*x^5+x^2-2*x+2)/((x -1)*(x^3-2*x^2+3*x-1)*(x^3-x^2+2*x-1)*(x^3-x-1)*(x^6+3*x^5+5*x^4 +5*x^3 +5*x^2+3*x+1)))) \\ G. C. Greubel, Oct 02 2017

Formula

a(n) = P(n)^3 - P(n)^2 = A000931(n)^3 - A000931(n)^2.
G.f.: 2*x^8*(x^7-x^6+2*x^5+x^2-2*x+2) / ((x-1) * (x^3-2*x^2+3*x-1) * (x^3-x^2+2*x-1) * (x^3-x-1) * (x^6+3*x^5+5*x^4+5*x^3+5*x^2+3*x+1)). - Colin Barker, Sep 18 2013

Extensions

Incorrect initial zero of the sequence deleted by Colin Barker, Sep 18 2013
Added more terms, Joerg Arndt, Sep 18 2013

A133061 a(n) = 5*p^5 - 3*p^3 - 2*p^2, where p = prime(n).

Original entry on oeis.org

128, 1116, 15200, 82908, 801020, 1849536, 7083968, 12359196, 32144156, 102480896, 143054460, 346565088, 579070880, 734799996, 1146409148, 2090525216, 3573998396, 4222293120, 6749714268, 9020062940, 10364180256, 15383790396, 19693474076, 27918166496, 42933944448, 52547391200
Offset: 1

Views

Author

Omar E. Pol, Nov 05 2007

Keywords

Examples

			a(4)=82908 because the 4th prime is 7, 5*7^5=84035, 3*7^3=1029, 2*7^2=98 and we can write 84035-1029-98=82908.

Links

Crossrefs

Cf. A000290, A000578, A000584, A045991, A133070. Prime numbers: A000040.

Programs

  • Magma
    [5*p^5-3*p^3-2*p^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010
  • Maple
    a:= n-> (p-> (5*p^3-3*p-2)*p^2)(ithprime(n)):
seq(a(n), n=1..26);  # Alois P. Heinz, Sep 23 2024
  • Mathematica
    Table[(Prime[n])^2*(5*Prime[n]^3 - 3*Prime[n] - 2), {n, 1, 50}] (* G. C. Greubel, Oct 09 2017 *)
  • PARI
    for(n=1,25, print1(5*prime(n)^5 - 3*prime(n)^3 - 2*prime(n)^2, ", ")) \\ G. C. Greubel, Oct 09 2017

Formula

a(n) = 5*(p(n))^5 - 3*(p(n))^3 - 2*(p(n))^2, where p(n)=A000040(n).

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A133063 a(n) = 5*p^5 + 3*p^3 - 2*p^2, where p = prime(n).

Original entry on oeis.org

176, 1278, 15950, 84966, 809006, 1862718, 7113446, 12400350, 32217158, 102627230, 143233206, 346869006, 579484406, 735277038, 1147032086, 2091418478, 3575230670, 4223655006, 6751518846, 9022210406, 10366514358, 15386748630, 19696904798, 27922396310, 42939420486, 52553573006
Offset: 1

Views

Author

Omar E. Pol, Nov 05 2007

Keywords

Examples

			a(4)=84966 because the 4th prime is 7, 5*7^5=84035, 3*7^3=1029, 2*7^2=98 and we can write 84035+1029-98=84966.

Links

Crossrefs

Cf. A000290, A000578, A000584, A045991, A133072. Prime numbers: A000040.

Programs

  • Magma
    [5*p^5+3*p^3-2*p^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 15 2010
  • Maple
    a:= n-> (p-> (5*p^3+3*p-2)*p^2)(ithprime(n)):
seq(a(n), n=1..26);  # Alois P. Heinz, Sep 23 2024
  • Mathematica
    Table[(Prime[n])^2*(5*Prime[n]^3 + 3*Prime[n] - 2), {n, 1, 50}] (* G. C. Greubel, Oct 09 2017 *)
  • PARI
    for(n=1,25, print1(5*prime(n)^5 + 3*prime(n)^3 - 2*prime(n)^2, ", ")) \\ G. C. Greubel, Oct 09 2017

Formula

a(n) = 5*(p(n))^5 + 3*(p(n))^3 - 2*(p(n))^2, where p(n)=A000040(n).

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A134630 a(n) = 5*n^5 - 3*n^3 - 2*n^2.

Original entry on oeis.org

0, 0, 128, 1116, 4896, 15200, 38160, 82908, 162176, 292896, 496800, 801020, 1238688, 1849536, 2680496, 3786300, 5230080, 7083968, 9429696, 12359196, 15975200, 20391840, 25735248, 32144156, 39770496, 48780000, 59352800, 71684028, 85984416, 102480896, 121417200, 143054460, 167671808, 195566976, 227056896
Offset: 0

Views

Author

Omar E. Pol, Nov 04 2007

Keywords

Comments

Coefficients and exponents are the first three prime numbers in decreasing order.

Examples

			a(4)=4896 because 4^5=1024, 5*1024=5120, 4^3=64, 3*64=192, 4^2=16, 2*16=32 and we can write 5120-192-32=4896.

Links

Crossrefs

Cf. A000290, A000578, A000584, A045991, A100019, A133070.

Programs

  • Magma
    [5*n^5-3*n^3 -2*n^2: n in [0..50]]; // Vincenzo Librandi, Dec 14 2010
  • Maple
    A134630:=n->5*n^5 - 3*n^3 - 2*n^2; seq(A134630(n), n=0..50); # Wesley Ivan Hurt, May 21 2014
  • Mathematica
    CoefficientList[Series[4 x^2 (32 + 87 x + 30 x^2 + x^3)/(-1 + x)^6, {x, 0, 50}], x] (* Vincenzo Librandi, May 21 2014 *)
Table[5n^5-3n^3-2n^2,{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,0,128,1116,4896,15200},40] (* Harvey P. Dale, Jun 01 2014 *)

Formula

a(n) = 5*n^5 - 3*n^3 - 2*n^2.
G.f.: 4*x^2*(32+87*x+30*x^2+x^3)/(-1+x)^6. - R. J. Mathar, Nov 14 2007
a(0)=0, a(1)=0, a(2)=128, a(3)=1116, a(4)=4896, a(5)=15200, a(n)= 6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jun 01 2014

Extensions

More terms from Vincenzo Librandi, Dec 14 2010

A134632 5*n^5 + 3*n^3 - 2*n^2. Coefficients and exponents are the prime numbers in decreasing order.

Original entry on oeis.org

0, 6, 176, 1278, 5280, 15950, 39456, 84966, 165248, 297270, 502800, 809006, 1249056, 1862718, 2696960, 3806550, 5254656, 7113446, 9464688, 12400350, 16023200, 20447406, 25799136, 32217158, 39853440, 48873750, 59458256, 71802126, 86116128, 102627230, 121579200, 143233206, 167868416, 195782598, 227292720
Offset: 0

Views

Author

Omar E. Pol, Nov 04 2007

Keywords

Examples

			a(4)=5280 because 4^5=1024, 5*1024=5120, 4^3=64, 3*64=192, 4^2=16, 2*16=32 and we can write 5120+192-32=5280.

Links

Crossrefs

Cf. A000290, A000578, A000584, A045991, A100019, A133072.

Programs

Formula

a(n) = 5*n^5 + 3*n^3 - 2*n^2.
G.f.: 2x*(3+70x+156x^2+66x^3+5x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

Extensions

More terms from Vincenzo Librandi, Dec 14 2010

A134633 5*n^5 + 3*n^3 + 2*n^2. Coefficients and exponents are the prime numbers in decreasing order.

Original entry on oeis.org

0, 10, 192, 1314, 5344, 16050, 39600, 85162, 165504, 297594, 503200, 809490, 1249632, 1863394, 2697744, 3807450, 5255680, 7114602, 9465984, 12401794, 16024800, 20449170, 25801072, 32219274, 39855744, 48876250, 59460960, 71805042, 86119264, 102630594, 121582800, 143237050, 167872512, 195786954, 227297344
Offset: 0

Views

Author

Omar E. Pol, Nov 04 2007

Keywords

Examples

			a(4)=5344 because 4^5=1024, 5*1024=5120, 4^3=64, 3*64=192, 4^2=16, 2*16=32 and we can write 5120+192+32=5344.

Links

Crossrefs

Cf. A000290, A000578, A000584, A045991, A100019, A133073.

Programs

  • Magma
    [5*n^5+3*n^3+2*n^2: n in [0..50]]; // Vincenzo Librandi, Dec 14 2010
  • Maple
    A134633:=n->5*n^5 + 3*n^3 + 2*n^2; seq(A134633(n), n=0..50); # Wesley Ivan Hurt, May 21 2014
  • Mathematica
    Table[5n^5+3n^3+2n^2,{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,10,192,1314,5344,16050},40] (* Harvey P. Dale, Apr 25 2012 *)
CoefficientList[Series[2 x (5 + 66 x + 156 x^2 + 70 x^3 + 3x^4)/(1 - x)^6, {x, 0, 50}], x] (* Vincenzo Librandi, May 21 2014 *)

Formula

a(n) = 5*n^5 + 3*n^3 + 2*n^2.
G.f.: 2x*(5+66x+156x^2+70x^3+3x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007
a(0)=0, a(1)=10, a(2)=192, a(3)=1314, a(4)=5344, a(5)=16050, a(n)= 6*a(n-1)- 15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Apr 25 2012

Extensions

More terms from Vincenzo Librandi, Dec 14 2010

A153260 a(n) = n^3 - 3*(n+3)^2.

Original entry on oeis.org

-27, -47, -67, -81, -83, -67, -27, 43, 149, 297, 493, 743, 1053, 1429, 1877, 2403, 3013, 3713, 4509, 5407, 6413, 7533, 8773, 10139, 11637, 13273, 15053, 16983, 19069, 21317, 23733, 26323, 29093, 32049, 35197, 38543, 42093, 45853, 49829, 54027
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

Keywords

Links

Crossrefs

Cf. A045991, A081437, A069778, A153257, A153258, A153259.

Programs

  • Magma
    [n^3-3*(n+3)^2: n in [0..40] ]; // Vincenzo Librandi, Aug 25 2011
  • Mathematica
    a[n_]:=n^3-3*(n+3)^2; a/@ Range[0, 50]
Table[n^3-3(n+3)^2,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{-27,-47,-67,-81},51] (* Harvey P. Dale, Aug 24 2011 *)
  • PARI
    vector(40, n, n--; n^3-3*(n+3)^2) \\ G. C. Greubel, Nov 10 2018

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=-27, a(1)=-47, a(2)=-67, a(3)=-81. - Harvey P. Dale, Aug 24 2011
G.f.: (x*(x*(13*x - 41) + 61) - 27)/(x-1)^4. - Harvey P. Dale, Aug 24 2011
E.g.f.: (-27 - 20*x + x^3)*exp(x). - G. C. Greubel, Nov 10 2018

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Aug 25 2011

A195813 Primes of the form 3^n - 2^n + n^2 - n^3.

Original entry on oeis.org

17, 5857, 18523, 1584103, 4764037, 14312989, 847255040011, 984770884591425105217, 2954312671366461520711, 22528399544934452045357278229190881
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 24 2011

Keywords

Comments

Generated by n = 4,8,9,13,14,15,25,44,45,72,...
This selects primes in the sequence A001047(n)-A045991(n).
Previous Showing 51-60 of 65 results. Next

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

Content is available under The OEIS End-User License Agreement.