A319014 -id:A319014 - cp's OEIS frontend

A228958 a(n) = 12 + 34 + 56 + 78 + 910 + 1112 + 13*14 + ... + (up to n).

1, 2, 5, 14, 19, 44, 51, 100, 109, 190, 201, 322, 335, 504, 519, 744, 761, 1050, 1069, 1430, 1451, 1892, 1915, 2444, 2469, 3094, 3121, 3850, 3879, 4720, 4751, 5712, 5745, 6834, 6869, 8094, 8131, 9500, 9539, 11060, 11101, 12782, 12825, 14674, 14719, 16744

Offset: 1

Robert Pfister, Sep 09 2013

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=2. - Wesley Ivan Hurt, Sep 10 2018

a(2n) is the total area of the family of n rectangles, where the k-th rectangle has dimensions (2k) X (2k-1). - Wesley Ivan Hurt, Oct 01 2018

			1                            =   1
1*2                          =   2
1*2 + 3                      =   5
1*2 + 3*4                    =  14
1*2 + 3*4 + 5                =  19
1*2 + 3*4 + 5*6              =  44
1*2 + 3*4 + 5*6 + 7          =  51
1*2 + 3*4 + 5*6 + 7*8        = 100
1*2 + 3*4 + 5*6 + 7*8 + 9    = 109
1*2 + 3*4 + 5*6 + 7*8 + 9*10 = 190
...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A093361, (k=1) A000217, (k=2) this sequence, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.

[(1/12)*(2*n^3+4*n+3/2+(3*n^2-6*n-3/2)*(-1)^n): n in [1..50]]; // Vincenzo Librandi, Sep 11 2018

a[n_?OddQ] := (2*n^3-3*n^2+10*n+3)/12; a[n_?EvenQ] := n*(n+2)*(2*n-1)/12; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Sep 10 2013 *)
CoefficientList[Series[x(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3), {x, 0, 40}], x] (* Stefano Spezia, Sep 23 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,2,5,14,19,44,51},50] (* Harvey P. Dale, Mar 11 2023 *)

Vec( x*(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3) + O(x^66) ) \\ Joerg Arndt, Sep 17 2013

a(n) = (1/12)*(2*n^3+4*n+3/2+(3*n^2-6*n-3/2)*(-1)^n). [based on Alcover program]
G.f.: x*(x^5 - x^4 + 6*x^3 + x + 1)/((x-1)^4*(x+1)^3). [Joerg Arndt, Sep 13 2013]
E.g.f.: (x*(9 + 9*x + 2*x^2)*cosh(x) + (3 + 3*x + 3*x^2 + 2*x^3)*sinh(x))/12. - Stefano Spezia, Apr 18 2023

Definition corrected by Ivan Panchenko, Dec 02 2013

A319205 a(n) = 1234 + 5678 + 9101112 + 13141516 + 171819*20 + ... + (up to n).

Original entry on oeis.org

1, 2, 6, 24, 29, 54, 234, 1704, 1713, 1794, 2694, 13584, 13597, 13766, 16314, 57264, 57281, 57570, 63078, 173544, 173565, 174006, 184170, 428568, 428593, 429218, 446118, 919968, 919997, 920838, 946938, 1783008, 1783041, 1784130, 1822278, 3196728, 3196765

Offset: 1

Wesley Ivan Hurt, Sep 13 2018

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=4.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4 + 5 = 29;
a(6) = 1*2*3*4 + 5*6 = 54;
a(7) = 1*2*3*4 + 5*6*7 = 234;
a(8) = 1*2*3*4 + 5*6*7*8 = 1704;
a(9) = 1*2*3*4 + 5*6*7*8 + 9 = 1713;
a(10) = 1*2*3*4 + 5*6*7*8 + 9*10 = 1794;
a(11) = 1*2*3*4 + 5*6*7*8 + 9*10*11 = 2694;
a(12) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 = 13584;
a(13) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13 = 13597;
a(14) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14 = 13766;
a(15) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15 = 16314;
a(16) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16 = 57264;
a(17) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16 + 17 = 57281;
a(18) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16 + 17*18 = 57570;
a(19) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16 + 17*18*19 = 63078;
etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,5,-5,0,0,-10,10,0,0,10,-10,0,0,-5,5,0,0,1,-1).

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) this sequence, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.

a[n_]:=Sum[(4*i)!/(4*i-4)!, {i, 1, Floor[n/4] }] + Sum[(1-Sign[Mod[n-j,4]])*Product[n-i+1, {i, 1, j}], {j, 1, 3}] ; Array[a, 40] (* Stefano Spezia, Sep 17 2018 *)

Vec(x*(1 + x + 4*x^2 + 18*x^3 + 20*x^5 + 160*x^6 + 1380*x^7 - 6*x^8 - 34*x^9 + 40*x^10 + 3720*x^11 + 8*x^12 + 4*x^13 - 192*x^14 + 1020*x^15 - 3*x^16 + 9*x^17 - 12*x^18 + 6*x^19) / ((1 - x)^6*(1 + x)^5*(1 + x^2)^5) + O(x^40)) \\ Colin Barker, Sep 14 2018

a(n) = Sum_{i=1..floor(n/4)} (4*i)!/(4*i-4)! + Sum_{j=1..3} (1-sign((n-j) mod 4)) * (Product_{i=1..j} n-i+1).
From Colin Barker, Sep 14 2018: (Start)
G.f.: x*(1 + x + 4*x^2 + 18*x^3 + 20*x^5 + 160*x^6 + 1380*x^7 - 6*x^8 - 34*x^9 + 40*x^10 + 3720*x^11 + 8*x^12 + 4*x^13 - 192*x^14 + 1020*x^15 - 3*x^16 + 9*x^17 - 12*x^18 + 6*x^19) / ((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)

A319206 a(n) = 12345 + 678910 + 1112131415 + ... + (up to n).

Original entry on oeis.org

1, 2, 6, 24, 120, 126, 162, 456, 3144, 30360, 30371, 30492, 32076, 54384, 390720, 390736, 390992, 395616, 483744, 2251200, 2251221, 2251662, 2261826, 2506224, 8626800, 8626826, 8627502, 8646456, 9196824, 25727520, 25727551, 25728512, 25760256, 26840544

Offset: 1

Wesley Ivan Hurt, Sep 13 2018

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=5.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5 + 6 = 126;
a(7) = 1*2*3*4*5 + 6*7 = 162;
a(8) = 1*2*3*4*5 + 6*7*8 = 456;
a(9) = 1*2*3*4*5 + 6*7*8*9 = 3144;
a(10) = 1*2*3*4*5 + 6*7*8*9*10 = 30360;
a(11) = 1*2*3*4*5 + 6*7*8*9*10 + 11 = 30371;
a(12) = 1*2*3*4*5 + 6*7*8*9*10 + 11*12 = 30492; etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,6,-6,0,0,0,-15,15,0,0,0,20,-20,0,0,0,-15,15,0,0,0,6,-6,0,0,0,-1,1).

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) this sequence, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.

a[n_]:=Sum[(5*i)!/(5*i-5)!, {i, 1, Floor[n/5] }] + Sum[(1-Sign[Mod[n-j, 5]])*Product[n-i+1, {i, 1, j}], {j, 1, 4}] ; Array[a, 34] (* Stefano Spezia, Apr 18 2023 *)

Vec(x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 30*x^6 + 270*x^7 + 2580*x^8 + 26640*x^9 - 10*x^10 - 80*x^11 - 120*x^12 + 6450*x^13 + 174480*x^14 + 20*x^15 + 50*x^16 - 550*x^17 - 5760*x^18 + 155760*x^19 - 15*x^20 + 15*x^21 + 360*x^22 - 3240*x^23 + 18000*x^24 + 4*x^25 - 16*x^26 + 36*x^27 - 48*x^28 + 24*x^29) / ((1 - x)^7*(1 + x + x^2 + x^3 + x^4)^6) + O(x^40)) \\ Colin Barker, Sep 14 2018

a(n) = Sum_{i=1..floor(n/5)} (5*i)!/(5*i-5)! + Sum_{j=1..4} (1-sign((n-j) mod 5)) * (Product_{i=1..j} n-i+1).
From Colin Barker, Sep 14 2018: (Start)
G.f.: x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 30*x^6 + 270*x^7 + 2580*x^8 + 26640*x^9 - 10*x^10 - 80*x^11 - 120*x^12 + 6450*x^13 + 174480*x^14 + 20*x^15 + 50*x^16 - 550*x^17 - 5760*x^18 + 155760*x^19 - 15*x^20 + 15*x^21 + 360*x^22 - 3240*x^23 + 18000*x^24 + 4*x^25 - 16*x^26 + 36*x^27 - 48*x^28 + 24*x^29) / ((1 - x)^7*(1 + x + x^2 + x^3 + x^4)^6).
a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - 15*a(n-10) + 15*a(n-11) + 20*a(n-15) - 20*a(n-16) - 15*a(n-20) + 15*a(n-21) + 6*a(n-25) - 6*a(n-26) - a(n-30) + a(n-31) for n>31.
(End)

A319207 a(n) = 123456 + 789101112 + 1314151617*18 + ... + (up to n).

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 727, 776, 1224, 5760, 56160, 666000, 666013, 666182, 668730, 709680, 1408560, 14032080, 14032099, 14032460, 14040060, 14207640, 18069960, 110941200, 110941225, 110941850, 110958750, 111432600, 125191800, 538459200, 538459231, 538460192

Offset: 1

Wesley Ivan Hurt, Sep 13 2018

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=6.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5*6 = 720;
a(7) = 1*2*3*4*5*6 + 7 = 727;
a(8) = 1*2*3*4*5*6 + 7*8 = 776;
a(9) = 1*2*3*4*5*6 + 7*8*9 = 1224;
a(10) = 1*2*3*4*5*6 + 7*8*9*10 = 5760;
a(11) = 1*2*3*4*5*6 + 7*8*9*10*11 = 56160;
a(12) = 1*2*3*4*5*6 + 7*8*9*10*11*12 = 666000;
a(13) = 1*2*3*4*5*6 + 7*8*9*10*11*12 + 13 = 666013;
a(14) = 1*2*3*4*5*6 + 7*8*9*10*11*12 + 13*14 = 666182; etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,7,-7,0,0,0,0,-21,21,0,0,0,0,35,-35,0,0,0,0,-35,35,0,0,0,0,21,-21,0,0,0,0,-7,7,0,0,0,0,1,-1).

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) this sequence, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.

a[n_]:=Sum[(6*i)!/(6*i-6)!, {i, 1, Floor[n/6] }] + Sum[(1-Sign[Mod[n-j,6]])*Product[n-i+1, {i, 1, j}], {j, 1, 5}] ; Array[a, 40] (* Stefano Spezia, Sep 17 2018 *)

Vec(x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 42*x^7 + 420*x^8 + 4410*x^9 + 49728*x^10 + 605640*x^11 - 15*x^12 - 153*x^13 - 504*x^14 + 9576*x^15 + 348096*x^16 + 8367240*x^17 + 40*x^18 + 172*x^19 - 968*x^20 - 24444*x^21 + 25200*x^22 + 17292240*x^23 - 45*x^24 - 33*x^25 + 1668*x^26 + 2610*x^27 - 361200*x^28 + 6939240*x^29 + 24*x^30 - 54*x^31 - 540*x^32 + 7650*x^33 - 61680*x^34 + 387240*x^35 - 5*x^36 + 25*x^37 - 80*x^38 + 180*x^39 - 240*x^40 + 120*x^41) / ((1 - x)^8*(1 + x)^7*(1 - x + x^2)^7*(1 + x + x^2)^7) + O(x^40)) \\ Colin Barker, Sep 14 2018

a(n) = Sum_{i=1..floor(n/6)} (6*i)!/(6*i-6)! + Sum_{j=1..5} (1-sign((n-j) mod 6)) * (Product_{i=1..j} n-i+1).
G.f.: x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 42*x^7 + 420*x^8 + 4410*x^9 + 49728*x^10 + 605640*x^11 - 15*x^12 - 153*x^13 - 504*x^14 + 9576*x^15 + 348096*x^16 + 8367240*x^17 + 40*x^18 + 172*x^19 - 968*x^20 - 24444*x^21 + 25200*x^22 + 17292240*x^23 - 45*x^24 - 33*x^25 + 1668*x^26 + 2610*x^27 - 361200*x^28 + 6939240*x^29 + 24*x^30 - 54*x^31 - 540*x^32 + 7650*x^33 - 61680*x^34 + 387240*x^35 - 5*x^36 + 25*x^37 - 80*x^38 + 180*x^39 - 240*x^40 + 120*x^41) / ((1 - x)^8*(1 + x)^7*(1 - x + x^2)^7*(1 + x + x^2)^7). - Colin Barker, Sep 14 2018
a(n) = a(n-1) + 7*a(n-6) - 7*a(n-7) - 21*a(n-12) + 21*a(n-13) + 35*a(n-18) - 35*a(n-19) - 35*a(n-24) + 35*a(n-25) + 21*a(n-30) - 21*a(n-31) - 7*a(n-36) + 7*a(n-37) + a(n-42) - a(n-43). - Wesley Ivan Hurt, Jun 20 2024

A319208 a(n) = 1234567 + 891011121314 + 15161718192021 + ... + (up to n).

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 5048, 5112, 5760, 12960, 100080, 1240560, 17302320, 17302335, 17302560, 17306400, 17375760, 18697680, 45209520, 603353520, 603353542, 603354026, 603365664, 603657120, 611247120, 816480720, 6570915120, 6570915149, 6570915990

Offset: 1

Wesley Ivan Hurt, Sep 13 2018

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=7.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5*6 = 720;
a(7) = 1*2*3*4*5*6*7 = 5040;
a(8) = 1*2*3*4*5*6*7 + 8 = 5048;
a(9) = 1*2*3*4*5*6*7 + 8*9 = 5112;
a(10) = 1*2*3*4*5*6*7 + 8*9*10 = 5760;
a(11) = 1*2*3*4*5*6*7 + 8*9*10*11 = 12960;
a(12) = 1*2*3*4*5*6*7 + 8*9*10*11*12 = 100080;
a(13) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13 = 1240560;
a(14) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 = 17302320;
a(15) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15 = 17302335;
a(16) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15*16 = 17302560; etc.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) this sequence, (k=8) A319209, (k=9) A319211, (k=10) A319212.

Table[Total[Times@@@Partition[Range[n],UpTo[7]]],{n,30}] (* Harvey P. Dale, Aug 02 2020 *)

a(n) = Sum_{i=1..floor(n/7)} (7*i)!/(7*i-7)! + Sum_{j=1..6} (1-sign((n-j) mod 7)) * (Product_{i=1..j} n-i+1).

A319209 a(n) = 12345678 + 910111213141516 + 17181920212223*24 + ... + (up to n).

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 40329, 40410, 41310, 52200, 194760, 2202480, 32472720, 518958720, 518958737, 518959026, 518964534, 519075000, 521400600, 572680080, 1754550000, 30173149440, 30173149465, 30173150090, 30173166990, 30173640840, 30187400040

Offset: 1

Wesley Ivan Hurt, Sep 13 2018

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=8.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5*6 = 720;
a(7) = 1*2*3*4*5*6*7 = 5040;
a(8) = 1*2*3*4*5*6*7*8 = 40320;
a(9) = 1*2*3*4*5*6*7*8 + 9 = 40329;
a(10) = 1*2*3*4*5*6*7*8 + 9*10 = 40410;
a(11) = 1*2*3*4*5*6*7*8 + 9*10*11 = 41310;
a(12) = 1*2*3*4*5*6*7*8 + 9*10*11*12 = 52200;
a(13) = 1*2*3*4*5*6*7*8 + 9*10*11*12*13 = 194760;
a(14) = 1*2*3*4*5*6*7*8 + 9*10*11*12*13*14 = 2202480;
a(15) = 1*2*3*4*5*6*7*8 + 9*10*11*12*13*14*15 = 32472720;
a(16) = 1*2*3*4*5*6*7*8 + 9*10*11*12*13*14*15*16 = 518958720;
a(17) = 1*2*3*4*5*6*7*8 + 9*10*11*12*13*14*15*16 + 17 = 518958737;
a(18) = 1*2*3*4*5*6*7*8 + 9*10*11*12*13*14*15*16 + 17*18 = 518959026; etc.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) this sequence, (k=9) A319211, (k=10) A319212.

a[n_]:=Sum[(8*i)!/(8*i-8)!, {i, 1, Floor[n/8] }] + Sum[(1-Sign[Mod[n-j, 8]])*Product[n-i+1, {i, 1, j}], {j, 1, 7}] ; Array[a, 29] (* Stefano Spezia, Apr 18 2023 *)

a(n) = Sum_{i=1..floor(n/8)} (8*i)!/(8*i-8)! + Sum_{j=1..7} (1-sign((n-j) mod 8)) * (Product_{i=1..j} n-i+1).

A319211 a(n) = 123456789 + 101112131415161718 + 192021222324252627 + ... + (up to n).

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 362890, 362990, 364200, 380040, 603120, 3966480, 58020480, 980542080, 17643588480, 17643588499, 17643588860, 17643596460, 17643764040, 17647626360, 17740497600, 20066316480, 80634516480, 1718398644480

Offset: 1

Wesley Ivan Hurt, Sep 13 2018

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=9.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5*6 = 720;
a(7) = 1*2*3*4*5*6*7 = 5040;
a(8) = 1*2*3*4*5*6*7*8 = 40320;
a(9) = 1*2*3*4*5*6*7*8*9 = 362880;
a(10) = 1*2*3*4*5*6*7*8*9 + 10 = 362890;
a(11) = 1*2*3*4*5*6*7*8*9 + 10*11 = 362990;
a(12) = 1*2*3*4*5*6*7*8*9 + 10*11*12 = 364200;
a(13) = 1*2*3*4*5*6*7*8*9 + 10*11*12*13 = 380040;
a(14) = 1*2*3*4*5*6*7*8*9 + 10*11*12*13*14 = 603120;
a(15) = 1*2*3*4*5*6*7*8*9 + 10*11*12*13*14*15 = 3966480;
a(16) = 1*2*3*4*5*6*7*8*9 + 10*11*12*13*14*15*16 = 58020480;
a(17) = 1*2*3*4*5*6*7*8*9 + 10*11*12*13*14*15*16*17 = 980542080;
a(18) = 1*2*3*4*5*6*7*8*9 + 10*11*12*13*14*15*16*17*18 = 17643588480;
a(19) = 1*2*3*4*5*6*7*8*9 + 10*11*12*13*14*15*16*17*18 + 19 = 17643588499; etc.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) this sequence, (k=10) A319212.

a[n_]:=Sum[(9*i)!/(9*i-9)!, {i, 1, Floor[n/9] }] + Sum[(1-Sign[Mod[n-j, 9]])*Product[n-i+1, {i, 1, j}], {j, 1, 8}] ; Array[a, 27] (* Stefano Spezia, Apr 18 2023 *)
Table[Total[Times@@@Partition[Range[n],UpTo[9]]],{n,30}] (* Harvey P. Dale, Dec 04 2024 *)

a(n) = Sum_{i=1..floor(n/9)} (9*i)!/(9*i-9)! + Sum_{j=1..8} (1-sign((n-j) mod 9)) * (Product_{i=1..j} n-i+1).

A319212 a(n) = 123456789*10 + 11121314151617181920 + 21222324252627282930 + ... + (up to n).

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 3628811, 3628932, 3630516, 3652824, 3989160, 9394560, 101646720, 1767951360, 33525757440, 670446201600, 670446201621, 670446202062, 670446212226, 670446456624, 670452577200, 670611967200, 674921872800

Offset: 1

Wesley Ivan Hurt, Sep 13 2018

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=10.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5*6 = 720;
a(7) = 1*2*3*4*5*6*7 = 5040;
a(8) = 1*2*3*4*5*6*7*8 = 40320;
a(9) = 1*2*3*4*5*6*7*8*9 = 362880;
a(10) = 1*2*3*4*5*6*7*8*9*10 = 3628800;
a(11) = 1*2*3*4*5*6*7*8*9*10 + 11 = 3628811;
a(12) = 1*2*3*4*5*6*7*8*9*10 + 11*12 = 3628932;
a(13) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13 = 3630516;
a(14) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14 = 3652824;
a(15) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15 = 3989160;
a(16) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16 = 9394560;
a(17) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17 = 101646720;
a(18) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17*18 = 1767951360;
a(19) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17*18*19 = 33525757440;
a(20) = 1*2*3*4*5*6*7*8*9*10 + 11*12*13*14*15*16*17*18*19*20 = 670446201600;
etc.

Georg Fischer, Table of n, a(n) for n = 1..1000 (first 739 terms from Colin Barker)

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) this sequence.

a[n_]:=Sum[(10*i)!/(10*i-10)!, {i, 1, Floor[n/10] }] + Sum[(1-Sign[Mod[n-j,10]])*Product[n-i+1, {i, 1, j}], {j, 1, 9}] ; Array[a, 40] (* or *)
CoefficientList[Series[x (1 + x + 4 x^2 + 18 x^3 + 96 x^4 + 600 x^5 + 4320 x^6 + 35280 x^7 + 322560 x^8 + 3265920 x^9 + 110 x^11 + 1540 x^12 + 22110 x^13 + 335280 x^14 + 5398800 x^15 + 92204640 x^16 + 1665916560 x^17 + 31754257920 x^18 + 636884519040 x^19 - 45 x^20 - 835 x^21 - 7040 x^22 + 2426160 x^24 + 99963600 x^25 + 3295369440 x^26 + 102515711760 x^27 + 3159608094720 x^28 + 98387160157440 x^29 + 240 x^30 + 2600 x^31 + 6400 x^32 - 384120 x^33 - 11000880 x^34 - 92637600 x^35 + 8150963040 x^36 + 682266206160 x^37 + 38076411985920 x^38 + 1874796686864640 x^39 - 630 x^40 - 4270 x^41 + 22120 x^42 + 1067820 x^43 + 8250000 x^44 - 525742800 x^45 - 23300782560 x^46 + 150285587760 x^47 + 93849442283520 x^48 + 9232053795296640 x^49 + 1008 x^50 + 3668 x^51 - 67928 x^52 - 1130796 x^53 + 15384048 x^54 + 861484800 x^55 - 7313090400 x^56 - 1717130091600 x^57 + 1723567507200 x^58 + 14964584346835200 x^59 - 1050 x^60 - 910 x^61 + 81760 x^62 + 291240 x^63 - 27736080 x^64 - 136792800 x^65 + 29138931360 x^66 - 117003292560 x^67 - 93887882161920 x^68 + 8480246509848960 x^69 + 720 x^70 - 1240 x^71 - 49760 x^72 + 371400 x^73 + 13094640 x^74 - 362037600 x^75 - 4579273440 x^76 + 749464032240 x^77 - 39511261278720 x^78 + 1564662885730560 x^79 - 315 x^80 + 1325 x^81 + 13300 x^82 - 301230 x^83 + 553680 x^84 + 130188600 x^85 - 5213255040 x^86 + 144639915840 x^87 - 3405078673920 x^88 + 72262987695360 x^89 + 80 x^90 - 530 x^91 + 180 x^92 + 60030 x^93 - 1288800 x^94 + 20098800 x^95 - 270829440 x^96 + 3295434240 x^97 - 36561611520 x^98 + 368739423360 x^99 - 9 x^100 + 81 x^101 - 576 x^102 + 3528 x^103 - 18144 x^104 + 75600 x^105 - 241920 x^106 + 544320 x^107 - 725760 x^108 + 362880 x^109)/((1 - x)^12 (1 + x)^11 (1 + x^2 + x^4 + x^6 + x^8)^11), {x, 0, 40}], x] (* Stefano Spezia, Sep 17 2018 *)

a(n) = Sum_{i=1..floor(n/10)} (10*i)!/(10*i-10)! + Sum_{j=1..9} (1-sign((n-j) mod 10)) * (Product_{i=1..j} n-i+1).
From Stefano Spezia, Sep 17 2018: (Start)
G.f.: x*(1 + x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 4320*x^6 + 35280*x^7 + 322560*x^8 + 3265920*x^9 + 110*x^11 + 1540*x^12 + 22110*x^13 + 335280*x^14 + 5398800*x^15 + 92204640*x^16 + 1665916560*x^17 + 31754257920*x^18 + 636884519040*x^19 - 45*x^20 - 835*x^21 - 7040*x^22 + 2426160*x^24 + 99963600*x^25 + 3295369440*x^26 + 102515711760*x^27 + 3159608094720*x^28 + 98387160157440*x^29 + 240*x^30 + 2600*x^31 + 6400*x^32 - 384120*x^33 - 11000880*x^34 - 92637600*x^35 + 8150963040*x^36 + 682266206160*x^37 + 38076411985920*x^38 + 1874796686864640*x^39 - 630*x^40 - 4270*x^41 + 22120*x^42 + 1067820*x^43 + 8250000*x^44 - 525742800*x^45 - 23300782560*x^46 + 150285587760*x^47 + 93849442283520*x^48 + 9232053795296640*x^49 + 1008*x^50 + 3668*x^51 - 67928*x^52 - 1130796*x^53 + 15384048*x^54 + 861484800*x^55 - 7313090400*x^56 - 1717130091600*x^57 + 1723567507200*x^58 + 14964584346835200*x^59 - 1050*x^60 - 910*x^61 + 81760*x^62 + 291240*x^63 - 27736080*x^64 - 136792800*x^65 + 29138931360*x^66 - 117003292560*x^67 - 93887882161920*x^68 + 8480246509848960*x^69 + 720*x^70 - 1240*x^71 - 49760*x^72 + 371400*x^73 + 13094640*x^74 - 362037600*x^75 - 4579273440*x^76 + 749464032240*x^77 - 39511261278720*x^78 + 1564662885730560*x^79 - 315*x^80 + 1325*x^81 + 13300*x^82 - 301230*x^83 + 553680*x^84 + 130188600*x^85 - 5213255040*x^86 + 144639915840*x^87 - 3405078673920*x^88 + 72262987695360*x^89 + 80*x^90 - 530*x^91 + 180*x^92 + 60030*x^93 - 1288800*x^94 + 20098800*x^95 - 270829440*x^96 + 3295434240*x^97 - 36561611520*x^98 + 368739423360*x^99 - 9*x^100 + 81*x^101 - 576*x^102 + 3528*x^103 - 18144*x^104 + 75600*x^105 - 241920*x^106 + 544320*x^107 - 725760*x^108 + 362880*x^109)/((1 - x)^12*(1 + x)^11*(1 + x^2 + x^4 + x^6 + x^8)^11).
(End)

A319543 a(n) = 123 - 456 + 789 - 101112 + 131415 - ... + (up to n).

Original entry on oeis.org

1, 2, 6, 2, -14, -114, -107, -58, 390, 380, 280, -930, -917, -748, 1800, 1784, 1528, -3096, -3077, -2716, 4884, 4862, 4378, -7260, -7235, -6610, 10290, 10262, 9478, -14070, -14039, -13078, 18666, 18632, 17476, -24174, -24137, -22768, 30660, 30620, 29020

Offset: 1

Wesley Ivan Hurt, Sep 22 2018

In general, for alternating sequences that multiply the first k natural numbers, and subtract/add the products of the next k natural numbers (preserving the order of operations up to n), we have a(n) = (-1)^floor(n/k) * Sum_{i=1..k-1} (1-sign((n-i) mod k)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/k)+1) * (1-sign(i mod k)) * (Product_{j=1..k} (i-j+1)). Here k=3.

An alternating version of A319014.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3 - 4 = 2;
a(5) = 1*2*3 - 4*5 = -14;
a(6) = 1*2*3 - 4*5*6 = -114;
a(7) = 1*2*3 - 4*5*6 + 7 = -107;
a(8) = 1*2*3 - 4*5*6 + 7*8 = -58;
a(9) = 1*2*3 - 4*5*6 + 7*8*9 = 390;
a(10) = 1*2*3 - 4*5*6 + 7*8*9 - 10 = 380;
a(11) = 1*2*3 - 4*5*6 + 7*8*9 - 10*11 = 280;
a(12) = 1*2*3 - 4*5*6 + 7*8*9 - 10*11*12 = -930;
a(13) = 1*2*3 - 4*5*6 + 7*8*9 - 10*11*12 + 13 = -917;
a(14) = 1*2*3 - 4*5*6 + 7*8*9 - 10*11*12 + 13*14 = -748;
a(15) = 1*2*3 - 4*5*6 + 7*8*9 - 10*11*12 + 13*14*15 = 1800; etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,-4,4,0,-6,6,0,-4,4,0,-1,1).

For similar sequences, see: A001057 (k=1), A319373 (k=2), this sequence (k=3), A319544 (k=4), A319545 (k=5), A319546 (k=6), A319547 (k=7), A319549 (k=8), A319550 (k=9), A319551 (k=10).

Cf. A319014.

seq(coeff(series((x*(1+x+4*x^2-12*x^4-84*x^5-3*x^6-9*x^7+72*x^8-2*x^9+4*x^10-2*x^11))/((1-x)*(1+x)^4*(1-x+x^2)^4),x,n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 01 2018

LinearRecurrence[{1, 0, -4, 4, 0, -6, 6, 0, -4, 4, 0, -1, 1},{1, 2, 6, 2, -14, -114, -107, -58, 390, 380, 280, -930, -917}, 40] (* Stefano Spezia, Sep 23 2018 *)

Vec(x*(1 + x + 4*x^2 - 12*x^4 - 84*x^5 - 3*x^6 - 9*x^7 + 72*x^8 - 2*x^9 + 4*x^10 - 2*x^11) / ((1 - x)*(1 + x)^4*(1 - x + x^2)^4) + O(x^40)) \\ Colin Barker, Sep 23 2018

a(n) = (-1)^floor(n/3) * Sum_{i=1..2} (1-sign((n-i) mod 3)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/3)+1) * (1-sign(i mod 3)) * (Product_{j=1..3} (i-j+1)).
From Colin Barker, Sep 23 2018: (Start)
G.f.: x*(1 + x + 4*x^2 - 12*x^4 - 84*x^5 - 3*x^6 - 9*x^7 + 72*x^8 - 2*x^9 + 4*x^10 - 2*x^11) / ((1 - x)*(1 + x)^4*(1 - x + x^2)^4).
a(n) = a(n-1) - 4*a(n-3) + 4*a(n-4) - 6*a(n-6) + 6*a(n-7) - 4*a(n-9) + 4*a(n-10) - a(n-12) + a(n-13) for n>13.
(End)

A305189 a(n) = 12 + 3 + 45 + 6 + 78 + 9 + 1011 + 12 + ... + (up to n).

Original entry on oeis.org

1, 2, 5, 9, 25, 31, 38, 87, 96, 106, 206, 218, 231, 400, 415, 431, 687, 705, 724, 1085, 1106, 1128, 1612, 1636, 1661, 2286, 2313, 2341, 3125, 3155, 3186, 4147, 4180, 4214, 5370, 5406, 5443, 6812, 6851, 6891, 8491, 8533, 8576, 10425, 10470, 10516, 12632

Offset: 1

Wesley Ivan Hurt, Sep 15 2018

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2 + 3 = 5;
a(4) = 1*2 + 3 + 4 = 9;
a(5) = 1*2 + 3 + 4*5 = 25;
a(6) = 1*2 + 3 + 4*5 + 6 = 31;
a(7) = 1*2 + 3 + 4*5 + 6 + 7 = 38;
a(8) = 1*2 + 3 + 4*5 + 6 + 7*8 = 87;
a(9) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 = 96;
a(10) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10 = 106;
a(11) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 = 206;
a(12) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 = 218; etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A093361, A228958, A319014.

seq(coeff(series((x*(1+x+3*x^2+x^3+13*x^4-3*x^5-2*x^6+4*x^7))/((1-x)^4*(1+x+x^2)^3),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Sep 16 2018

Table[3*Floor[n/3]*(Floor[n/3] + 1)/2 + Floor[(n + 1)/3]*(3*Floor[(n + 1)/3]^2 - 1) + n*(Floor[(n - 1)/3] - Floor[(n - 2)/3]), {n, 50}]
LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1 }, {1, 2, 5, 9, 25, 31, 38, 87, 96, 106}, 50] (* Stefano Spezia, Sep 16 2018 *)

Vec(x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Sep 16 2018

a(n) = 3*floor(n/3)*(floor(n/3) + 1)/2 + floor((n+1)/3)*(3*floor((n+1)/3)^2 - 1) + n*(floor((n-1)/3) - floor((n-2)/3)).
From Colin Barker, Sep 16 2018: (Start)
G.f.: x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.
(End)

cp's OEIS Frontend

A228958 a(n) = 1*2 + 3*4 + 5*6 + 7*8 + 9*10 + 11*12 + 13*14 + ... + (up to n).

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A319205 a(n) = 1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16 + 17*18*19*20 + ... + (up to n).

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A319206 a(n) = 1*2*3*4*5 + 6*7*8*9*10 + 11*12*13*14*15 + ... + (up to n).

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A319207 a(n) = 1*2*3*4*5*6 + 7*8*9*10*11*12 + 13*14*15*16*17*18 + ... + (up to n).

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A319208 a(n) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15*16*17*18*19*20*21 + ... + (up to n).

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A319209 a(n) = 1*2*3*4*5*6*7*8 + 9*10*11*12*13*14*15*16 + 17*18*19*20*21*22*23*24 + ... + (up to n).

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A319211 a(n) = 1*2*3*4*5*6*7*8*9 + 10*11*12*13*14*15*16*17*18 + 19*20*21*22*23*24*25*26*27 + ... + (up to n).

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A228958 a(n) = 12 + 34 + 56 + 78 + 910 + 1112 + 13*14 + ... + (up to n).

A319205 a(n) = 1234 + 5678 + 9101112 + 13141516 + 171819*20 + ... + (up to n).

A319206 a(n) = 12345 + 678910 + 1112131415 + ... + (up to n).

A319207 a(n) = 123456 + 789101112 + 1314151617*18 + ... + (up to n).

A319208 a(n) = 1234567 + 891011121314 + 15161718192021 + ... + (up to n).

A319209 a(n) = 12345678 + 910111213141516 + 17181920212223*24 + ... + (up to n).

A319211 a(n) = 123456789 + 101112131415161718 + 192021222324252627 + ... + (up to n).

A319212 a(n) = 123456789*10 + 11121314151617181920 + 21222324252627282930 + ... + (up to n).

A319543 a(n) = 123 - 456 + 789 - 101112 + 131415 - ... + (up to n).

A305189 a(n) = 12 + 3 + 45 + 6 + 78 + 9 + 1011 + 12 + ... + (up to n).