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.

A167875 One third of product plus sum of three consecutive nonnegative integers; a(n)=(n+1)(n^2+2n+3)/3.

Original entry on oeis.org

1, 4, 11, 24, 45, 76, 119, 176, 249, 340, 451, 584, 741, 924, 1135, 1376, 1649, 1956, 2299, 2680, 3101, 3564, 4071, 4624, 5225, 5876, 6579, 7336, 8149, 9020, 9951, 10944, 12001, 13124, 14315, 15576, 16909, 18316, 19799, 21360, 23001, 24724, 26531
Offset: 0

Views

Author

Klaus Brockhaus, Nov 14 2009

Keywords

Comments

a(n) = ((n*(n+1)*(n+2))+(n+(n+1)+(n+2)))/3, n >= 0.
Equals A006527 without initial term 0: a(n) = A006527(n+1).
Binomial transform of A167876.
Inverse binomial transform of A080930.
a(n) = A007290(n+2)+n+1.
a(n) = A014820(n)/(n+1) for n > 0.
a(n) = A116731(n+2)-1.
a(n) = A033547(n+1)-n.
a(n) = A054602(n)/3.
a(n) = A086514(n+3)-2.
a(n) = A002061(n+1)+a(n-1) for n > 0.
a(n) = A005894(n)-a(n-1) for n > 0.
First bisection is A057813.
Second differences are in A004277.
a(n) = A177342(n)*(-1)+a(n-1)*5 with n>0. For n=8, a(8)=-A177342(8)+a(7)*5=-631+176*5=249. - Bruno Berselli, May 18 2010

Examples

			a(0) = (0*1*2+0+1+2)/3 = (0+3)/3 = 1.
a(1) = (1*2*3+1+2+3)/3 = (6+6)/3 = 4.
a(6)-4*a(5)+6*a(4)-4*a(3)+a(2) = 119-4*76+6*45-4*24+11 = 0. - _Bruno Berselli_, May 26 2010

Links

Crossrefs

Cf. A001477 (nonnegative integers),
A006527 ((n^3+2*n)/3),
A167876 (1, 3, 4, 2, 0, 0, 0, 0, ...),
A080930,
A007290 (2*C(n, 3)),
A014820 ((1/3)*(n^2+2*n+3)*(n+1)^2),
A116731,
A033547 (n*(n^2+5)/3),
A054602 (Sum_{d|3} phi(d)*n^(3/d)),
A086514 ((n^3-6*n^2+14*n-6)/3),
A002061 (n^2-n+1),
A005894 (centered tetrahedral numbers),
A057813 ((2*n+1)*(4*n^2+4*n+3)/3),
A004277 (1 and the positive even numbers),
A028387 (n+(n+1)^2),
A166941, A166942, A166943.

Programs

  • Magma
    [ (&*s + &+s)/3 where s is [n..n+2]: n in [0..42] ];
  • Mathematica
    Select[Table[(n*(n+1)*(n+2)+n+(n+1)+(n+2))/3,{n,0,5!}],IntegerQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)
(Times@@#+Total[#])/3&/@Partition[Range[0,65],3,1]  (* Harvey P. Dale, Mar 14 2011 *)
  • PARI
    a(n)=(n+1)*(n^2+2*n+3)/3 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (n^3+3*n^2+5*n+3)/3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+2 for n > 3; a(0)=1, a(1)=4, a(2)=11, a(3)=24.
G.f.: (1+x^2)/(1-x)^4.
a(n) = SUM(A109613(k)*A005408(n-k): 0<=k<=n). - Reinhard Zumkeller, Dec 05 2009
a(n)-4*a(n-1)+6*a(n-2)-4*a(n-3)+a(n-4)=0 for n>3. - Bruno Berselli, May 26 2010

A166942 One fifth of product plus sum of five consecutive nonnegative numbers.

Original entry on oeis.org

2, 27, 148, 509, 1350, 3031, 6056, 11097, 19018, 30899, 48060, 72085, 104846, 148527, 205648, 279089, 372114, 488395, 632036, 807597, 1020118, 1275143, 1578744, 1937545, 2358746, 2850147, 3420172, 4077893, 4833054, 5696095
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Oct 24 2009

Keywords

Comments

a(n) = ((n*...*(n+4))+(n+...+(n+4)))/5, n >= 0.
Binomial transform of 2, 25, 96, 144, 96, 24, 0, 0, 0, 0, ....
Partial sums of A062938 where initial term 1 is replaced by 2.

Examples

			a(0) = (0*1*2*3*4 + 0 + 1 + 2 + 3 + 4)/5 = (0 + 10)/5 = 2.
a(1) = (1*2*3*4*5 + 1 + 2 + 3 + 4 + 5)/5 = (120 + 15)/5 = 27.

Links

Crossrefs

Cf. A001477 (nonnegative integers), A062938 (squares of the form n(n+1)(n+2)(n+3)+1), A028387 (n+(n+1)^2), A167875, A166941, A166943.

Programs

  • Magma
    [ (&*s + &+s)/5 where s is [n..n+4]: n in [0..29] ]; // Klaus Brockhaus, Nov 14 2009
  • Mathematica
    Table[((n+4)*(n+3)*(n+2)*(n+1)*n+(n+4)+(n+3)+(n+2)+(n+1)+n)/5, {n,0,100}]
(Total[#]+Times@@#)/5&/@Partition[Range[0,100],5,1]  (* Harvey P. Dale, Mar 05 2011 *)

Formula

a(n) = (n^5 + 10n^4 + 35n^3 + 50n^2 + 29n + 10)/5. - Charles R Greathouse IV, Nov 02 2009
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 24 for n > 4; a(0)=2, a(1)=27, a(2)=148, a(3)=509, a(4)=1350. - Klaus Brockhaus, Nov 14 2009
G.f.: (2+15*x+16*x^2-14*x^3+6*x^4-x^5)/(1-x)^6. - Klaus Brockhaus, Nov 14 2009

Extensions

Edited and offset corrected by Klaus Brockhaus, Nov 14 2009

A166943 One third of product plus sum of six consecutive nonnegative numbers.

Original entry on oeis.org

5, 247, 1689, 6731, 20173, 50415, 110897, 221779, 411861, 720743, 1201225, 1921947, 2970269, 4455391, 6511713, 9302435, 13023397, 17907159, 24227321, 32303083, 42504045, 55255247, 71042449, 90417651, 114004853, 142506055
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Oct 24 2009

Keywords

Comments

a(n) = ((n*...*(n+5))+(n+...+(n+5)))/3, n >= 0.
Binomial transform of 5, 242, 1200, 2400, 2400, 1200, 240, 0, 0, 0, 0, ....

Examples

			a(0) = (0*1*2*3*4*5+0+1+2+3+4+5)/3 = (0+15)/3 = 5.
a(1) = (1*2*3*4*5*6+1+2+3+4+5+6)/3 = (720+21)/3 = 247.

Links

Crossrefs

Cf. A001477 (nonnegative integers), A028387 (n+(n+1)^2), A167875, A166941, A166942.

Programs

  • Magma
    [ (&*s + &+s)/3 where s is [n..n+5]: n in [0..25] ]; // Klaus Brockhaus, Nov 14 2009
  • Mathematica
    lst={};Do[p=(n+5)*(n+4)*(n+3)*(n+2)*(n+1)*n+(n+5)+(n+4)+(n+3)+(n+2)+(n+1)+n;AppendTo[lst,p/3],{n,0,5!}];lst
(Plus@@#+Times@@#)/3&/@Partition[Range[0,30],6,1] (* Harvey P. Dale, Nov 10 2009 *)

Formula

a(n) = (n^6 + 15n^5 + 85n^4 + 225n^3 + 274n^2 + 126n + 15)/3. - Charles R Greathouse IV, Nov 04 2009
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)+240 for n > 5; a(0)=5, a(1)=247, a(2)=1689, a(3)=6731, a(4)=20173, a(5)=50415. - Klaus Brockhaus, Nov 14 2009
G.f.: (5+212*x+65*x^2-80*x^3+55*x^4-20*x^5+3*x^6)/(1-x)^7. - Klaus Brockhaus, Nov 14 2009

Extensions

Edited and offset corrected by Klaus Brockhaus, Nov 14 2009

A167858 A000004 preceded by 3, 14, 36, 36, 12.

Original entry on oeis.org

3, 14, 36, 36, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Klaus Brockhaus, Nov 13 2009

Keywords

Comments

Inverse binomial transform of A166941/2.

Crossrefs

Cf. A000004 (zero sequence), A166941 (product plus sum of four consecutive nonnegative numbers), A166926 (1, 2, 4, 0, 0, 0, 0, ...), A130706 (1, 2, 0, 0, 0, 0, ...), A130779 (1, 1, 2, 0, 0, 0, 0, ...).

Programs

  • PARI
    {concat([3, 14, 36, 36, 12], vector(98))}

Formula

a(0) = 3, a(1) = 14, a(2) = 36, a(3) = 36, a(4) = 12, a(n) = 0 for n > 4.
G.f.: 3+14*x+36*x^2+36*x^3+12*x^4.

A173044 Product plus sum of five consecutive nonnegative numbers.

Original entry on oeis.org

10, 135, 740, 2545, 6750, 15155, 30280, 55485, 95090, 154495, 240300, 360425, 524230, 742635, 1028240, 1395445, 1860570, 2441975, 3160180, 4037985, 5100590, 6375715, 7893720, 9687725, 11793730, 14250735, 17100860, 20389465, 24165270, 28480475, 33390880, 38956005
Offset: 0

Views

Author

Vladimir Joseph Stephan Orlovsky, Nov 21 2010

Keywords

Links

Crossrefs

Cf. A028387, A054602, A166941.

Programs

  • Magma
    [(n+2)*(n^4 +8*n^3 +19*n^2 +12*n +5): n in [0..40]]; // G. C. Greubel, Feb 19 2021
  • Maple
    A173044:= n-> (n+2)*(n^4 +8*n^3 +19*n^2 +12*n +5); seq(A173044(n), n=0..40) # G. C. Greubel, Feb 19 2021
  • Mathematica
    a[n_]:= n*(n+1)*(n+2)*(n+3)*(n+4) + n + (n+1)+(n+2)+(n+3)+(n+4);
Table[a[n],{n,0,5!}]
  • Sage
    [(n+2)*(n^4 +8*n^3 +19*n^2 +12*n +5) for n in (0..40)] # G. C. Greubel, Feb 19 2021

Formula

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4) +n +(n+1) +(n+2) +(n+3) +(n+4).
G.f.: 5*(2 +15*x +16*x^2 -14*x^3 +6*x^4 -x^5)/(1-x)^6. - Colin Barker, Jun 25 2012
a(n) = (n+2)*(n^4 +8*n^3 +19*n^2 +12*n +5) = n^5 +10*n^4 +35*n^3 +50*n^2 +29*n +10. - Bruno Berselli, Jun 25 2012
E.g.f.: (10 +125*x +240*x^2 +120*x^3 +20*x^4 +x^5)*exp(x). - G. C. Greubel, Feb 19 2021

Extensions

Offset corrected by G. C. Greubel, Feb 19 2021
Showing 1-5 of 5 results.

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