A228958 -id:A228958 - cp's OEIS frontend

A319014 a(n) = 123 + 456 + 789 + 101112 + 131415 + 161718 + ... + (up to n).

1, 2, 6, 10, 26, 126, 133, 182, 630, 640, 740, 1950, 1963, 2132, 4680, 4696, 4952, 9576, 9595, 9956, 17556, 17578, 18062, 29700, 29725, 30350, 47250, 47278, 48062, 71610, 71641, 72602, 104346, 104380, 105536, 147186, 147223, 148592, 202020, 202060, 203660

Offset: 1

Wesley Ivan Hurt, Sep 07 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=3.

			a(1)  = 1;
a(2)  = 1*2 = 2;
a(3)  = 1*2*3 = 6;
a(4)  = 1*2*3 + 4 = 10;
a(5)  = 1*2*3 + 4*5 = 26;
a(6)  = 1*2*3 + 4*5*6 = 126;
a(7)  = 1*2*3 + 4*5*6 + 7 = 133;
a(8)  = 1*2*3 + 4*5*6 + 7*8 = 182;
a(9)  = 1*2*3 + 4*5*6 + 7*8*9 = 630;
a(10) = 1*2*3 + 4*5*6 + 7*8*9 + 10 = 640;
a(11) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11 = 740;
a(12) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 = 1950;
a(13) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13 = 1963;
a(14) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14 = 2132;
a(15) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 = 4680;
a(16) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16 = 4696;
a(17) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17 = 4952;
a(18) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 = 9576;
a(19) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 + 19 = 9595;
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).

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) this sequence, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) A319208, (k=8) A319209, (k=9) A319211, (k=10) A319212.

Cf. A049347, A061347, A268685 (trisection).

CoefficientList[Series[(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)^5*(1 + x + x^2)^4), {x, 0, 50}], x] (* after Colin Barker *)

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)^5*(1 + x + x^2)^4) + O(x^50)) \\ Colin Barker, Sep 08 2018

a(n) = Sum_{i=1..floor(n/3)} (3*i)!/(3*i-3)! + Sum_{j=1..2} (1-sign((n-j) mod 3)) * (Product_{i=1..j} n-i+1).
From Colin Barker, Sep 08 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)^5*(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)
a(3*k) = 3*k*(k+1)*(3*k-2)*(3*k+1)/4, a(3*k+1) = a(3*k) + 3*k + 1, a(3*k+2) = a(3*k) + (3*k+2)*(3*k+1). - Giovanni Resta, Sep 08 2018
a(n) = (3*n^4 - 6*n^3 + 9*n^2 + 6*n - 8 - 2*(3*n^3 - 6*n^2 - 6*n + 2)*A061347(n-3) + 6*(n^3 - 6*n^2 + 6*n + 2)*A049347(n-2))/36. - Stefano Spezia, Apr 23 2023

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)

A319373 a(n) = 12 - 34 + 56 - 78 + 910 - 1112 + 13*14 - ... + (up to n).

Original entry on oeis.org

1, 2, -1, -10, -5, 20, 13, -36, -27, 54, 43, -78, -65, 104, 89, -136, -119, 170, 151, -210, -189, 252, 229, -300, -275, 350, 323, -406, -377, 464, 433, -528, -495, 594, 559, -666, -629, 740, 701, -820, -779, 902, 859, -990, -945, 1080, 1033, -1176, -1127

Offset: 1

Wesley Ivan Hurt, Sep 17 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=2.

An alternating version of A228958.

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2 - 3 = -1;
a(4) = 1*2 - 3*4 = -10;
a(5) = 1*2 - 3*4 + 5 = -5;
a(6) = 1*2 - 3*4 + 5*6 = 20;
a(7) = 1*2 - 3*4 + 5*6 - 7 = 13;
a(8) = 1*2 - 3*4 + 5*6 - 7*8 = -36;
a(9) = 1*2 - 3*4 + 5*6 - 7*8 + 9 = -27;
a(10) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 = 54;
a(11) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11 = 43;
a(12) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 = -78;
a(13) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13 = -65;
a(14) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13*14 = 104;
a(15) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13*14 - 15 = 89; etc.

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, A228958, A305189.

For similar sequences, see: A001057 (k=1), this sequence (k=2), A319543 (k=3), A319544 (k=4), A319545 (k=5), A319546 (k=6), A319547 (k=7), A319549 (k=8), A319550 (k=9), A319551 (k=10).

Table[(Cos[n Pi/2] (1 - n - n^2) + Sin[n Pi/2] (1 + 3 n - n^2) - 1)/2, {n, 50}]
a[n_] := (-1)^Floor[n/2] Sum[(1 - Sign[Mod[n - i, 2]]) Product[n - j + 1, {j, 1, i}], {i, 1, 1}] + Sum[(-1)^(Floor[i/2] + 1) (1 - Sign[Mod[i, 2]]) Product[i - j + 1, {j, 1, 2}], {i, 1, n}]; Array[a, 30] (* Stefano Spezia, Sep 23 2018 *)

Vec(x*(1 + x - 6*x^3 - x^4 + x^5) / ((1 - x)*(1 + x^2)^3) + O(x^50)) \\ Colin Barker, Sep 18 2018

a(n) = (cos(n*Pi/2)*(1-n-n^2) + sin(n*Pi/2)*(1+3*n-n^2) - 1)/2.
From Colin Barker, Sep 18 2018: (Start)
G.f.: x*(1 + x - 6*x^3 - x^4 + x^5) / ((1 - x)*(1 + x^2)^3).
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)
a(n) = (-1 + (-1)^((n-1)*n/2))/2 + (-2 + (-1)^n)*(-1)^(n*(n+1)/2)*n/2 - (-1)^((n-1)*n/2)*n^2/2. - Bruno Berselli, Sep 25 2018

A227364 a(n) = 1 + 23 + 456 + 78910 + ... + ...*n (see Example lines).

Original entry on oeis.org

0, 1, 3, 7, 11, 27, 127, 134, 183, 631, 5167, 5178, 5299, 6883, 29191, 365527, 365543, 365799, 370423, 458551, 2226007, 39435607, 39435629, 39436113, 39447751, 39739207, 47329207, 252562807, 6006997207, 6006997236, 6006998077, 6007024177, 6007860247, 6035477527, 6975328087

Offset: 0

Alex Ratushnyak, Jul 07 2013

nonn

			a(5) = 1 + 2*3 + 4*5 = 27;
a(6) = 1 + 2*3 + 4*5*6 = 127;
a(7) = 1 + 2*3 + 4*5*6 + 7 = 134.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A227363, A227365, A227366, A227367, A228958.

for n in range(55):
  sum = 0
  i = k = 1
  while i<=n:
    product = 1
    for x in range(k):
      product *= i
      i += 1
      if i>n: break
    sum += product
    k += 1
  print(str(sum), end=',')

cp's OEIS Frontend

A319014 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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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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A319014 a(n) = 123 + 456 + 789 + 101112 + 131415 + 161718 + ... + (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).

A319373 a(n) = 12 - 34 + 56 - 78 + 910 - 1112 + 13*14 - ... + (up to n).

A227364 a(n) = 1 + 23 + 456 + 78910 + ... + ...*n (see Example lines).