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 1-9 of 9 results.

A319866 a(n) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 + 12*11 + ... + (up to the n-th term).

Original entry on oeis.org

2, 2, 6, 14, 20, 44, 52, 100, 110, 190, 202, 322, 336, 504, 520, 744, 762, 1050, 1070, 1430, 1452, 1892, 1916, 2444, 2470, 3094, 3122, 3850, 3880, 4720, 4752, 5712, 5746, 6834, 6870, 8094, 8132, 9500, 9540, 11060, 11102, 12782, 12826, 14674, 14720, 16744
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 29 2018

Keywords

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=2.
The denominators of the generating functions for these sequences are (1 + x)*(1 - x^k)^(k+1). - Georg Fischer and Andrew Howroyd, Mar 07 2020

Examples

			a(1) = 2;
a(2) = 2*1 = 2;
a(3) = 2*1 + 4 = 6;
a(4) = 2*1 + 4*3 = 14;
a(5) = 2*1 + 4*3 + 6 = 20;
a(6) = 2*1 + 4*3 + 6*5 = 44;
a(7) = 2*1 + 4*3 + 6*5 + 8 = 52;
a(8) = 2*1 + 4*3 + 6*5 + 8*7 = 100;
a(9) = 2*1 + 4*3 + 6*5 + 8*7 + 10 = 110;
a(10) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 = 190;
a(11) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 + 12 = 202;
etc.

Links

Crossrefs

For similar sequences, see: A000217 (k=1), this sequence (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).

Programs

  • Maple
    a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,2),n=1..50); # Muniru A Asiru, Sep 30 2018
  • Mathematica
    k:=2; a[n_]:= Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])* Product[n-i-j+k+1, {i,1,j }] , {j,1,k-1} ]  + Sum[(Floor[j/k]-Floor[(j-1)/k])* Product[j-i+1, {i,1,k} ], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)
CoefficientList[Series[2/((-1 + x)^2 (1 + x)^2) + ( 2 (x + 3 x^3))/((-1 + x)^4 (1 + x)^3), {x, 0, 50}], x] (* Stefano Spezia, Sep 30 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{2,2,6,14,20,44,52},60] (* Harvey P. Dale, Mar 01 2025 *)
  • PARI
    Vec(2*x*(1 - x^2 + 4*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^50)) \\ Colin Barker, Sep 30 2018

Formula

G.f.: 2*x/((-1 + x)^2*(1 + x)^2) + 2*(x^2 + 3*x^4)/((-1 + x)^4 (1 + x)^3). - Stefano Spezia, Sep 30 2018
From Colin Barker, Sep 30 2018: (Start)
a(n) = (4*n - 6*n + 3*n^2 + 2*n^3) / 12 for n even.
a(n) = (15 + 4*n + 6*n - 3*n^2 + 2*n^3) / 12 for n odd.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7.
(End)

A319868 a(n) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16*15*14*13 + ... + (up to the n-th term).

Original entry on oeis.org

4, 12, 24, 24, 32, 80, 360, 1704, 1716, 1836, 3024, 13584, 13600, 13824, 16944, 57264, 57284, 57644, 64104, 173544, 173568, 174096, 185688, 428568, 428596, 429324, 448224, 919968, 920000, 920960, 949728, 1783008, 1783044, 1784268, 1825848, 3196728, 3196768
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 29 2018

Keywords

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=4.

Examples

			a(1) = 4;
a(2) = 4*3 = 12;
a(3) = 4*3*2 = 24;
a(4) = 4*3*2*1 = 24;
a(5) = 4*3*2*1 + 8 = 32;
a(6) = 4*3*2*1 + 8*7 = 80;
a(7) = 4*3*2*1 + 8*7*6 = 360;
a(8) = 4*3*2*1 + 8*7*6*5 = 1704;
a(9) = 4*3*2*1 + 8*7*6*5 + 12 = 1716;
a(10) = 4*3*2*1 + 8*7*6*5 + 12*11 = 1836;
a(11) = 4*3*2*1 + 8*7*6*5 + 12*11*10 = 3024;
a(12) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 = 13584;
a(13) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16 = 13600;
a(14) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16*15 = 13824;
a(15) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16*15*14 = 16944;
a(16) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16*15*14*13 = 57264;
a(17) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16*15*14*13 + 20 = 57284;
a(18) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16*15*14*13 + 20*19 = 57644;
a(19) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16*15*14*13 + 20*19*18 = 64104;
a(20) = 4*3*2*1 + 8*7*6*5 + 12*11*10*9 + 16*15*14*13 + 20*19*18*17 = 173544;
etc.

Links

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), this sequence (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).

Programs

  • Maple
    a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,4),n=1..40); # Muniru A Asiru, Sep 30 2018
  • Mathematica
    k:=4; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])*Product[n-i-j+k+1, {i,1,j }], {j,1,k-1} ] + Sum[(Floor[j/k]-Floor[(j-1)/k])*Product[j-i+1, {i,1,k}], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)
LinearRecurrence[{1,0,0,5,-5,0,0,-10,10,0,0,10,-10,0,0,-5,5,0,0,1,-1},{4,12,24,24,32,80,360,1704,1716,1836,3024,13584,13600,13824,16944,57264,57284,57644,64104,173544,173568},60] (* Harvey P. Dale, Jan 29 2020 *)
  • PARI
    Vec(4*x*(1 + 2*x + 3*x^2 - 3*x^4 + 2*x^5 + 55*x^6 + 336*x^7 + 3*x^8 - 10*x^9 - 23*x^10 + 960*x^11 - x^12 + 6*x^13 - 35*x^14 + 240*x^15) / ((1 - x)^6*(1 + x)^5*(1 + x^2)^5) + O(x^40)) \\ Colin Barker, Oct 19 2018

Formula

From Colin Barker, Oct 19 2018: (Start)
G.f.: 4*x*(1 + 2*x + 3*x^2 - 3*x^4 + 2*x^5 + 55*x^6 + 336*x^7 + 3*x^8 - 10*x^9 - 23*x^10 + 960*x^11 - x^12 + 6*x^13 - 35*x^14 + 240*x^15) / ((1 - x)^6*(1 + x)^5*(1 + x^2)^5).
a(n) = a(n-1) + 5*a(n-4) - 5*a(n-5) - 10*a(n-8) + 10*a(n-9) + 10*a(n-12) - 10*a(n-13) - 5*a(n-16) + 5*a(n-17) + a(n-20) - a(n-21) for n>21.
(End)

A319869 a(n) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + ... + (up to the n-th term).

Original entry on oeis.org

5, 20, 60, 120, 120, 130, 210, 840, 5160, 30360, 30375, 30570, 33090, 63120, 390720, 390740, 391100, 397560, 507000, 2251200, 2251225, 2251800, 2265000, 2554800, 8626800, 8626830, 8627670, 8651160, 9284520, 25727520, 25727555, 25728710, 25766790, 26984160
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 29 2018

Keywords

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=5.

Examples

			a(1) = 5;
a(2) = 5*4 = 20;
a(3) = 5*4*3 = 60;
a(4) = 5*4*3*2 = 120;
a(5) = 5*4*3*2*1 = 120;
a(6) = 5*4*3*2*1 + 10 = 130;
a(7) = 5*4*3*2*1 + 10*9 = 210;
a(8) = 5*4*3*2*1 + 10*9*8 = 840;
a(9) = 5*4*3*2*1 + 10*9*8*7 = 5160;
a(10) = 5*4*3*2*1 + 10*9*8*7*6 = 30360;
a(11) = 5*4*3*2*1 + 10*9*8*7*6 + 15 = 30375;
a(12) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14 = 30570;
a(13) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13 = 33090;
a(14) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12 = 63120;
a(15) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 = 390720;
a(16) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20 = 390740;
a(17) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20*19 = 391100;
a(18) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20*19*18 = 397560;
a(19) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20*19*18*17 = 507000;
a(20) = 5*4*3*2*1 + 10*9*8*7*6 + 15*14*13*12*11 + 20*19*18*17*16 = 2251200;
etc.

Links

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), this sequence (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).

Programs

  • Maple
    a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,5),n=1..40); # Muniru A Asiru, Sep 30 2018
  • Mathematica
    k:=5; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])*Product[n-i-j+k+1, {i,1,j }] , {j,1,k-1}] + Sum[(Floor[j/k]-Floor[(j-1)/k])*Product[j-i+1, {i,1,k}], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)

A319870 a(n) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14*13 + ... + (up to the n-th term).

Original entry on oeis.org

6, 30, 120, 360, 720, 720, 732, 852, 2040, 12600, 95760, 666000, 666018, 666306, 670896, 739440, 1694160, 14032080, 14032104, 14032632, 14044224, 14287104, 19132560, 110941200, 110941230, 110942070, 110965560, 111598920, 128041920, 538459200, 538459236
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 30 2018

Keywords

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=6.

Examples

			a(1) = 6;
a(2) = 6*5 = 30;
a(3) = 6*5*4 = 120;
a(4) = 6*5*4*3 = 360;
a(5) = 6*5*4*3*2 = 720;
a(6) = 6*5*4*3*2*1 = 720;
a(7) = 6*5*4*3*2*1 + 12 = 732;
a(8) = 6*5*4*3*2*1 + 12*11 = 852;
a(9) = 6*5*4*3*2*1 + 12*11*10 = 2040;
a(10) = 6*5*4*3*2*1 + 12*11*10*9 = 12600;
a(11) = 6*5*4*3*2*1 + 12*11*10*9*8 = 95760;
a(12) = 6*5*4*3*2*1 + 12*11*10*9*8*7 = 666000;
a(13) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18 = 666018;
a(14) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17 = 666306;
a(15) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16 = 670896;
a(16) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15 = 739440;
a(17) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14 = 1694160;
a(18) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14*13 = 14032080;
a(19) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14*13 + 24 = 14032104;
a(20) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14*13 + 24*23 = 14032632;
etc.

Links

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), this sequence (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).

Programs

  • Maple
    a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,6),n=1..35); # Muniru A Asiru, Sep 30 2018
  • Mathematica
    k:=6; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])*Product[n-i-j+k+1, {i,1,j }] , {j,1,k-1} ]  + Sum[(Floor[j/k]-Floor[(j-1)/k])*Product[j-i+1, {i,1,k}], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)

A319871 a(n) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + ... + (up to the n-th term).

Original entry on oeis.org

7, 42, 210, 840, 2520, 5040, 5040, 5054, 5222, 7224, 29064, 245280, 2167200, 17302320, 17302341, 17302740, 17310300, 17445960, 19744200, 56372400, 603353520, 603353548, 603354276, 603373176, 603844920, 615147120, 874606320, 6570915120, 6570915155, 6570916310
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 30 2018

Keywords

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=7.

Examples

			a(1) = 7;
a(2) = 7*6 = 42;
a(3) = 7*6*5 = 210;
a(4) = 7*6*5*4 = 840;
a(5) = 7*6*5*4*3 = 2520;
a(6) = 7*6*5*4*3*2 = 5040;
a(7) = 7*6*5*4*3*2*1 = 5040;
a(8) = 7*6*5*4*3*2*1 + 14 = 5054;
a(9) = 7*6*5*4*3*2*1 + 14*13 = 5222;
a(10) = 7*6*5*4*3*2*1 + 14*13*12 = 7224;
a(11) = 7*6*5*4*3*2*1 + 14*13*12*11 = 29064;
a(12) = 7*6*5*4*3*2*1 + 14*13*12*11*10 = 245280;
a(13) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9 = 2167200;
a(14) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 = 17302320;
a(15) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21 = 17302341;
a(16) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20 = 17302740;
a(17) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20*19 = 17310300;
a(18) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20*19*18 = 17445960;
a(19) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20*19*18*17 = 19744200;
a(20) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20*19*18*17*16 = 56372400;
etc.

Links

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), this sequence (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).

Programs

  • Maple
    a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,7),n=1..30); # Muniru A Asiru, Sep 30 2018
  • Mathematica
    k:=7; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])*Product[n-i-j+k+1, {i,1,j }] , {j,1,k-1} ] + Sum[(Floor[j/k]-Floor[(j-1)/k])*Product[j-i+1, {i,1,k}], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)

A319872 a(n) = 8*7*6*5*4*3*2*1 + 16*15*14*12*11*10*9 + ... + (up to the n-th term).

Original entry on oeis.org

8, 56, 336, 1680, 6720, 20160, 40320, 40320, 40336, 40560, 43680, 84000, 564480, 5806080, 57697920, 518958720, 518958744, 518959272, 518970864, 519213744, 524059200, 615867840, 2263322880, 30173149440, 30173149472, 30173150432, 30173179200, 30174012480
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 30 2018

Keywords

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=8.

Examples

			a(1) = 8;
a(2) = 8*7 = 56;
a(3) = 8*7*6 = 336;
a(4) = 8*7*6*5 = 1680;
a(5) = 8*7*6*5*4 = 6720;
a(6) = 8*7*6*5*4*3 = 20160;
a(7) = 8*7*6*5*4*3*2 = 40320;
a(8) = 8*7*6*5*4*3*2*1 = 40320;
a(9) = 8*7*6*5*4*3*2*1 + 16 = 40336;
a(10) = 8*7*6*5*4*3*2*1 + 16*15 = 40560;
a(11) = 8*7*6*5*4*3*2*1 + 16*15*14 = 43680;
a(12) = 8*7*6*5*4*3*2*1 + 16*15*14*13 = 84000;
a(13) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12 = 564480;
a(14) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11 = 5806080;
a(15) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10 = 57697920;
a(16) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 = 518958720;
a(17) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24 = 518958744;
a(18) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23 = 518959272;
a(19) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23*22 = 518970864;
a(20) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23*22*21 = 519213744;
etc.

Links

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), this sequence (k=8), A319873 (k=9), A319874 (k=10).

Programs

  • Maple
    a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,8),n=1..30); # Muniru A Asiru, Sep 30 2018
  • Mathematica
    k:=8; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])* Product[n-i-j+k+1, {i,1,j }], {j,1,k-1} ]  + Sum[(Floor[j/k]-Floor[(j-1)/k])* Product[j-i+1, {i,1,k} ], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)

A319873 a(n) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 + ... + (up to the n-th term).

Original entry on oeis.org

9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880, 362898, 363186, 367776, 436320, 1391040, 13728960, 160755840, 1764685440, 17643588480, 17643588507, 17643589182, 17643606030, 17644009680, 17653276080, 17856715680, 22119259680, 107157012480
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 30 2018

Keywords

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=9.

Examples

			a(1) = 9;
a(2) = 9*8 = 72;
a(3) = 9*8*7 = 504;
a(4) = 9*8*7*6 = 3024;
a(5) = 9*8*7*6*5 = 15120;
a(6) = 9*8*7*6*5*4 = 60480;
a(7) = 9*8*7*6*5*4*3 = 181440;
a(8) = 9*8*7*6*5*4*3*2 = 362880;
a(9) = 9*8*7*6*5*4*3*2*1 = 362880;
a(10) = 9*8*7*6*5*4*3*2*1 + 18 = 362898;
a(11) = 9*8*7*6*5*4*3*2*1 + 18*17 = 363186;
a(12) = 9*8*7*6*5*4*3*2*1 + 18*17*16 = 367776
a(13) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15 = 436320;
a(14) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14 = 1391040;
a(15) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13 = 13728960;
a(16) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12 = 160755840;
a(17) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11 = 1764685440;
a(18) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 = 17643588480;
a(19) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 + 27 = 17643588507;
etc.

Links

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), this sequence (k=9), A319874 (k=10).

Programs

  • Maple
    a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,9),n=1..30); # Muniru A Asiru, Sep 30 2018
  • Mathematica
    k:=9; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])* Product[n-i-j+k+1, {i,1,j }], {j,1,k-1} ]  + Sum[(Floor[j/k]-Floor[(j-1)/k])* Product[j-i+1, {i,1,k} ], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)

A319874 a(n) = 10*9*8*7*6*5*4*3*2*1 + 20*19*18*17*16*15*14*13*12*11 + ... + (up to the n-th term).

Original entry on oeis.org

10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800, 3628820, 3629180, 3635640, 3745080, 5489280, 31536000, 394329600, 5082739200, 60952953600, 670446201600, 670446201630, 670446202470, 670446225960, 670446859320, 670463302320, 670873719600
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 30 2018

Keywords

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=10.

Examples

			a(1) = 10;
a(2) = 10*9 = 90;
a(3) = 10*9*8 = 720;
a(4) = 10*9*8*7 = 5040;
a(5) = 10*9*8*7*6 = 30240;
a(6) = 10*9*8*7*6*5 = 151200;
a(7) = 10*9*8*7*6*5*4 = 604800;
a(8) = 10*9*8*7*6*5*4*3 = 1814400;
a(9) = 10*9*8*7*6*5*4*3*2 = 3628800;
a(10) = 10*9*8*7*6*5*4*3*2*1 = 3628800;
a(11) = 10*9*8*7*6*5*4*3*2*1 + 20 = 3628820;
a(12) = 10*9*8*7*6*5*4*3*2*1 + 20*19 = 3629180;
a(13) = 10*9*8*7*6*5*4*3*2*1 + 20*19*18 = 3635640;
a(14) = 10*9*8*7*6*5*4*3*2*1 + 20*19*18*17 = 3745080;
a(15) = 10*9*8*7*6*5*4*3*2*1 + 20*19*18*17*16 = 5489280;
a(16) = 10*9*8*7*6*5*4*3*2*1 + 20*19*18*17*16*15 = 31536000;
a(17) = 10*9*8*7*6*5*4*3*2*1 + 20*19*18*17*16*15*14 = 394329600;
a(18) = 10*9*8*7*6*5*4*3*2*1 + 20*19*18*17*16*15*14*13 = 5082739200;
a(19) = 10*9*8*7*6*5*4*3*2*1 + 20*19*18*17*16*15*14*13*12 = 60952953600;
etc.

Links

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), this sequence (k=10).

Programs

  • Maple
    a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,10),n=1..25); # Muniru A Asiru, Sep 30 2018
  • Mathematica
    k:=10; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])* Product[n-i-j+k+1, {i,1,j }], {j,1,k-1} ] + Sum[(Floor[j/k]-Floor[(j-1)/k])* Product[j-i+1, {i,1,k} ], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)

A268685 a(n) = 3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4.

Original entry on oeis.org

6, 126, 630, 1950, 4680, 9576, 17556, 29700, 47250, 71610, 104346, 147186, 202020, 270900, 356040, 459816, 584766, 733590, 909150, 1114470, 1352736, 1627296, 1941660, 2299500, 2704650, 3161106, 3673026, 4244730, 4880700, 5585580, 6364176, 7221456, 8162550
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 11 2016

Keywords

Comments

a(n) is the total volume of the family of (n+1) rectangular prisms, where the k-th prism has dimensions (3k) X (3k-1) X (3k-2). - Wesley Ivan Hurt, Oct 02 2018

Examples

			a(0) = 1*2*3 = 6;
a(1) = 1*2*3 + 4*5*6 = 126;
a(2) = 1*2*3 + 4*5*6 + 7*8*9 = 630;
a(3) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 = 1950;
a(4) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 = 4680;
a(5) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 = 9576, etc.

Links

Crossrefs

Cf. A000027, A054776, A116689, A135036.
Trisection of A319014 and A319867.

Programs

  • Magma
    [3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4: n in [0..40]]; // Vincenzo Librandi, Feb 11 2016
  • Mathematica
    Table[3 (n + 1) (n + 2) (3 n + 1) ((3 n + 4)/4), {n, 0, 32}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {6, 126, 630, 1950, 4680}, 32]
CoefficientList[Series[6 (10 x^2 + 16 x + 1) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 11 2016 *)
  • PARI
    a(n) = 3*(n+1)*(n+2)*(3*n+1)*(3*n+4)/4 \\ Felix Fröhlich, Jun 07 2016

Formula

G.f.: -6*(10*x^2 + 16*x + 1)/(x - 1)^5.
a(n) = Sum_{k = 0..n} (3*k + 1)(3*k + 2)(3*k + 3).
Sum {n>=0} 1/a(n) = 2*(sqrt(3)*Pi + 9*log(3) - 14)/15 = 0.1771878254287521...
a(n) mod 6 = 0.
a(n) = 6*A116689(n+1). - R. J. Mathar, Jun 07 2016
E.g.f.: 3*exp(x)*(8 + 160*x +256*x^2 + 96*x^3 + 9*x^4)/4. - Stefano Spezia, Apr 18 2023
Sum_{n>=0} (-1)^n/a(n) = 28/15 - 8*Pi/(15*sqrt(3)) - 16*log(2)/15. - Amiram Eldar, Apr 30 2023
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