A045991 -id:A045991 - cp's OEIS frontend

A163283 Triangle read by rows in which row n lists n+1 terms, starting with n^3 and ending with n^4, such that the difference between successive terms is equal to n^3 - n^2.

0, 1, 1, 8, 12, 16, 27, 45, 63, 81, 64, 112, 160, 208, 256, 125, 225, 325, 425, 525, 625, 216, 396, 576, 756, 936, 1116, 1296, 343, 637, 931, 1225, 1519, 1813, 2107, 2401, 512, 960, 1408, 1856, 2304, 2752, 3200, 3648, 4096, 729, 1377, 2025, 2673, 3321, 3969

Offset: 0

Omar E. Pol, Jul 24 2009

The first term of row n is A000578(n) and the last term of row n is A000583(n).

			Triangle begins:
0;
1,    1;
8,    12,   16;
27,   45,   63,   81;
64,   112,  160,  208,  256;
125,  225,  325,  425,  525,  625;
216,  396,  576,  756,  936,  1116, 1296;
343,  637,  931,  1225, 1519, 1813, 2107, 2401;
512,  960,  1408, 1856, 2304, 2752, 3200, 3648, 4096;
729,  1377, 2025, 2673, 3321, 3969, 4617, 5265, 5913, 6561;
1000, 1900, 2800, 3700, 4600, 5500, 6400, 7300, 8200, 9100, 10000;
...

G. C. Greubel, Table of n, a(n) for the first 50 rows

Row sums: A099903.

Cf. A000578, A000583, A045991, A159797, A162611, A162614, A163282, A163284, A163285.

Table[n^3 + k*(n^3 - n^2), {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 13 2016 *)

A163283(n, k)=n^3 +k*(n^3 -n^2) \\ G. C. Greubel, Dec 13 2016

T(n, k) = n^3 + k*(n^3 - n^2), for 0 <= k <= n, n >= 0. - G. C. Greubel, Dec 13 2016

Edited by Omar E. Pol, Jul 25 2009

A059409 a(n) = 4^n * (2^n - 1).

Original entry on oeis.org

0, 4, 48, 448, 3840, 31744, 258048, 2080768, 16711680, 133955584, 1072693248, 8585740288, 68702699520, 549688705024, 4397778075648, 35183298347008, 281470681743360, 2251782633816064, 18014329790005248, 144114913197948928, 1152920405095219200

Offset: 0

N. J. A. Sloane, Alford Arnold, Jan 30 2001

Jordan's totient functions are described more fully in A059379 and A059380; for example, J_1(n) is Euler's totient function and J_2(n) the Moebius transform of squares.

			(4,48,448,3840,...) = (8,64,512,4096,...) - (2,12,56,240,...) - (1,3,7,15,...) - (1,1,1,1,...)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Cf. A059379, A059380, A016152.
Cf. A024023, A000225, A000012.
Cf. A065442.

List([0..100], n->4^n * (2^n - 1)); # Muniru A Asiru, Jan 29 2018

[4^n*(2^n - 1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011

seq(4^n * (2^n - 1), n=0..20); # Muniru A Asiru, Jan 29 2018

Table[4^n*(2^n - 1), {n,0,30}] (* G. C. Greubel, Jan 29 2018 *)
LinearRecurrence[{12,-32},{0,4},20] (* Harvey P. Dale, Oct 14 2019 *)

a(n) = { 4^n*(2^n - 1) } \\ Harry J. Smith, Jun 26 2009

Equals J_n(8) (see A059379).
J_n(8) = 8^n - A024023(n) - A000225(n) - A000012(n).
a(n) = 4*A016152(n).
G.f.: 4*x / ( (8*x-1)*(4*x-1) ). - R. J. Mathar, Nov 23 2018
Sum_{n>0} 1/a(n) = E - 4/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022
a(n) = A291779(A008585(n)) = A045991(A000079(n)). - Mathew Englander, Feb 08 2024

A240930 a(n) = n^7 - n^6.

Original entry on oeis.org

0, 0, 64, 1458, 12288, 62500, 233280, 705894, 1835008, 4251528, 9000000, 17715610, 32845824, 57921708, 97883968, 159468750, 251658240, 386201104, 578207808, 846825858, 1216000000, 1715322420, 2380977984, 3256789558, 4395368448, 5859375000, 7722894400, 10072932714

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 7-digit positive integers in base n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001014, A001015.

Cf. A002378, A045991, A085537, A085538, A085539, A240931, A240932, A240933.

[n^7-n^6 : n in [0..30]]; // Wesley Ivan Hurt, Aug 03 2014

A240930:=n->n^7-n^6: seq(A240930(n), n=0..30); # Wesley Ivan Hurt, Aug 03 2014

Table[n^7 - n^6, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 03 2014 *)
CoefficientList[Series[2 (32*x^2 + 473*x^3 + 1208*x^4 + 718*x^5 + 88*x^6 + x^7)/(x - 1)^8, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)

vector(100, n, (n-1)^7 - (n-1)^6) \\ Derek Orr, Aug 03 2014

a(n) = n^6*(n-1) = n^7 - n^6.
a(n) = A001015(n) - A001014(n).
G.f.: 2*(32*x^2 + 473*x^3 + 1208*x^4 + 718*x^5 + 88*x^6 + x^7)/(x - 1)^8. - Wesley Ivan Hurt, Aug 03 2014
Recurrence: a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Aug 03 2014
Sum_{n>=2} 1/a(n) = 6 - Sum_{k=2..6} zeta(k). - Amiram Eldar, Jul 05 2020

A240931 a(n) = n^8 - n^7.

Original entry on oeis.org

0, 0, 128, 4374, 49152, 312500, 1399680, 4941258, 14680064, 38263752, 90000000, 194871710, 394149888, 752982204, 1370375552, 2392031250, 4026531840, 6565418768, 10407740544, 16089691302, 24320000000, 36021770820, 52381515648, 74906159834, 105488842752, 146484375000

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 8-digit positive integers in base n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A001015, A001016.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240932, A240933.

[n^8-n^7 : n in [0..30]]; // Wesley Ivan Hurt, Aug 09 2014

A240931:=n->n^8-n^7: seq(A240931(n), n=0..30); # Wesley Ivan Hurt, Aug 09 2014

Table[n^8 - n^7, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 09 2014 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,128,4374,49152,312500,1399680,4941258,14680064},30] (* Harvey P. Dale, Apr 29 2016 *)

vector(100, n, (n-1)^8 - (n-1)^7) \\ Derek Orr, Aug 03 2014

concat([0,0], Vec(-2*x^2*(x^6+183*x^5+2682*x^4+8422*x^3+7197*x^2+1611*x+64) / (x-1)^9 + O(x^100))) \\ Colin Barker, Aug 08 2014

a(n) = n^7*(n-1) = n^8 - n^7.
a(n) = A001016(n) - A001015(n).
G.f.: -2*x^2*(x^6+183*x^5+2682*x^4+8422*x^3+7197*x^2+1611*x+64) / (x-1)^9. - Colin Barker, Aug 08 2014
Sum_{n>=2} 1/a(n) = 7 - Sum_{k=2..7} zeta(k). - Amiram Eldar, Jul 05 2020

A240932 a(n) = n^9 - n^8.

Original entry on oeis.org

0, 0, 256, 13122, 196608, 1562500, 8398080, 34588806, 117440512, 344373768, 900000000, 2143588810, 4729798656, 9788768652, 19185257728, 35880468750, 64424509440, 111612119056, 187339329792, 305704134738, 486400000000, 756457187220, 1152393344256, 1722841676182

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 9-digit positive integers in base n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001016, A001017.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240933.

[n^9-n^8: n in [0..40]]; // Vincenzo Librandi, Aug 15 2016

Table[n^9 - n^8, {n, 0, 40}] (* Vincenzo Librandi, Aug 15 2016 *)

vector(100, n, (n-1)^9 - (n-1)^8) \\ Derek Orr, Aug 03 2014

a(n) = n^8*(n-1) = n^9 - n^8.
a(n) = A001017(n) - A001016(n).
G.f.: 2*x^2*(x^7+374*x^6+9327*x^5+49780*x^4+78095*x^3+38454*x^2+5281*x+128) / (x-1)^10. - Colin Barker, Aug 08 2014
Sum_{n>=2} 1/a(n) = 8 - Sum_{k=2..8} zeta(k). - Amiram Eldar, Jul 05 2020

A240933 a(n) = n^10 - n^9.

Original entry on oeis.org

0, 0, 512, 39366, 786432, 7812500, 50388480, 242121642, 939524096, 3099363912, 9000000000, 23579476910, 56757583872, 127253992476, 268593608192, 538207031250, 1030792151040, 1897406023952, 3372107936256, 5808378560022, 9728000000000, 15885600931620, 25352653573632

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 10-digit positive integers in base n.

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A001017, A008454.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240932.

[n^10-n^9 : n in [0..30]]; // Wesley Ivan Hurt, Aug 03 2014

A240933:=n->n^10-n^9: seq(A240933(n), n=0..30); # Wesley Ivan Hurt, Aug 03 2014

Table[n^10 - n^9, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 03 2014 *)
CoefficientList[Series[2 (256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,0,512,39366,786432,7812500,50388480,242121642,939524096,3099363912,9000000000},40] (* Harvey P. Dale, Oct 19 2022 *)

vector(100, n, (n-1)^10 - (n-1)^9) \\ Derek Orr, Aug 03 2014

a(n) = n^9*(n-1) = n^10 - n^9.
a(n) = A008454(n) - A001017(n). - Michel Marcus, Aug 03 2014
G.f.: 2*(256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11. - Wesley Ivan Hurt, Aug 03 2014
Recurrence: a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Wesley Ivan Hurt, Aug 03 2014
Sum_{n>=2} 1/a(n) = 9 - Sum_{k=2..9} zeta(k). - Amiram Eldar, Jul 05 2020

A067389 a(n) = 3n^3 + 2n^2 + n.

Original entry on oeis.org

0, 6, 34, 102, 228, 430, 726, 1134, 1672, 2358, 3210, 4246, 5484, 6942, 8638, 10590, 12816, 15334, 18162, 21318, 24820, 28686, 32934, 37582, 42648, 48150, 54106, 60534, 67452, 74878, 82830, 91326, 100384, 110022, 120258, 131110, 142596

Offset: 0

George E. Antoniou, Jan 21 2002

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

[3*n^3 + 2*n^2 + n: n in [0..60]]; // Vincenzo Librandi, May 08 2011

a:=n->n+2*n^2+3*n^3: seq(a(n), n=0..36); # Zerinvary Lajos, Oct 05 2007

Table[3*n^3+2*n^2+n,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, May 07 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,6,34,102},40] (* Harvey P. Dale, Oct 01 2019 *)

a(n) = n*A056109(n) = A045991(n+1)+A033431(n). - Henry Bottomley, Jan 25 2002
From Chai Wah Wu, Apr 25 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
G.f.: 2*x*(x^2 + 5*x + 3)/(x - 1)^4. (End)

More terms from Henry Bottomley, Jan 25 2002

A114364 a(n) = n*(n+1)^2.

Original entry on oeis.org

4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, 2028, 2548, 3150, 3840, 4624, 5508, 6498, 7600, 8820, 10164, 11638, 13248, 15000, 16900, 18954, 21168, 23548, 26100, 28830, 31744, 34848, 38148, 41650, 45360, 49284, 53428, 57798, 62400

Offset: 1

Cino Hilliard, Feb 09 2006

Former name was "Numbers k such that k*x^3 + x + 1 is not prime."

Theorem: y = k*x^3 + x + 1 is not prime for k = 4, 18, 48, ..., n*(n+1)^2. Proof: n*(n+1)^2*x^3 + x + 1 = ((n+1)*x + 1)*((n^2+n)*x^2 - n*x + 1). Thus (n+1)*x + 1 divides y. This could possibly be used as a pre-test for compositeness. This sequence is the same as beginning with the third term of A045991.

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

Cf. A045991.

Equals twice A006002.

seq(2*binomial(n,2)*n, n=2..40); # Zerinvary Lajos, Apr 25 2007

CoefficientList[Series[(2 (2 + x))/(-1 + x)^4, {x, 0, 38}], x] (* or *)
Array[# (# + 1)^2 &, 39] (* Michael De Vlieger, Feb 03 2019 *)

g2(n) = for(x=1,n,y=x*(x+1)^2;print1(y","))

a(n) = n*(n+1)^2.
G.f.: 2 * (2 + x)/(-1 + x)^4. - Michael De Vlieger, Feb 03 2019
From Amiram Eldar, Jan 02 2021: (Start)
Sum_{n>=1} 1/a(n) = 2 - Pi^2/6.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 +2*log(2) - 2. (End)
E.g.f.: exp(x)*x*(4 + 5*x + x^2). - Stefano Spezia, May 20 2021

Name changed by Jon E. Schoenfield, Feb 03 2019

A212969 Number of (w,x,y) with all terms in {0,...,n} and w != x and x > range(w,x,y).

Original entry on oeis.org

0, 0, 2, 10, 26, 56, 100, 166, 252, 368, 510, 690, 902, 1160, 1456, 1806, 2200, 2656, 3162, 3738, 4370, 5080, 5852, 6710, 7636, 8656, 9750, 10946, 12222, 13608, 15080, 16670, 18352, 20160, 22066, 24106, 26250, 28536, 30932, 33478, 36140

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A212959, A045991, A212970.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w != x > (Max[w, x, y] - Min[w, x, y]),
  s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212969 *)
m/2 (* integers *)

a(n) = (n-1)*(2*n*(7*n-2) - 3*(-1)^n + 3)/24.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = 2*(x^2)*(1 + 3*x + 2*x^2 + x^3) and g(x) = ((1-x)^4)*(1+x)^2.
a(n) = A045991(n) - A212970(n-1). - Ayoub Saber Rguez, Mar 31 2023

A212970 Number of (w,x,y) with all terms in {0,...,n} and w != x and x < range(w,x,y).

Original entry on oeis.org

0, 2, 8, 22, 44, 80, 128, 196, 280, 390, 520, 682, 868, 1092, 1344, 1640, 1968, 2346, 2760, 3230, 3740, 4312, 4928, 5612, 6344, 7150, 8008, 8946, 9940, 11020, 12160, 13392, 14688, 16082, 17544, 19110, 20748, 22496, 24320, 26260, 28280

Offset: 0

Clark Kimberling, Jun 02 2012

For a guide to related sequences, see A212959.

Twice the partial sums of A210977. - J. M. Bergot, Aug 10 2013

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A088003, A210977, A212959.

Cf. A045991, A212969.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w != x < (Max[w, x, y] - Min[w, x, y]),
   s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212970 *)
m/2 (* essentially A088003 *)

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = 2*x*(1 + 2*x + 2*x^2) and g(x) = ((1-x)^4)(1+x)^2.
a(n) = 2 * A088003(n) for n>0.
From Ayoub Saber Rguez, Mar 31 2023: (Start)
a(n) + A212969(n+1) = A045991(n+1).
a(n) = (10*n^3 + 24*n^2 + 8*n + (6*n)*(n mod 2))/24. (End)

Typo in name corrected by Ayoub Saber Rguez, Mar 31 2023

cp's OEIS Frontend

A163283 Triangle read by rows in which row n lists n+1 terms, starting with n^3 and ending with n^4, such that the difference between successive terms is equal to n^3 - n^2.

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A059409 a(n) = 4^n * (2^n - 1).

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A240930 a(n) = n^7 - n^6.

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A240931 a(n) = n^8 - n^7.

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A240932 a(n) = n^9 - n^8.

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A240933 a(n) = n^10 - n^9.

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A067389 a(n) = 3n^3 + 2n^2 + n.