A017017 -id:A017017 - cp's OEIS frontend

A017020 a(n) = (7*n + 3)^4.

81, 10000, 83521, 331776, 923521, 2085136, 4100625, 7311616, 12117361, 18974736, 28398241, 40960000, 57289761, 78074896, 104060401, 136048896, 174900625, 221533456, 276922881, 342102016, 418161601

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A017017, A017018, A017019, A017021 - A017028.

[(7*n+3)^4: n in [0..35]]; // Vincenzo Librandi, Jul 14 2011

(7*Range[0,20]+3)^4 (* or *) LinearRecurrence[{5,-10,10,-5,1},{81,10000,83521,331776,923521},30] (* Harvey P. Dale, Oct 29 2019 *)

[(7*n+3)^4 for n in range(41)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (81 + 9595*x + 34331*x^2 + 13361*x^3 + 256*x^4)/(1 - x)^5.
E.g.f.: (81 + 9919*x + 31801*x^2 + 18522*x^3 + 2401*x^4)*exp(x). (End)

A017021 a(n) = (7*n + 3)^5.

Original entry on oeis.org

243, 100000, 1419857, 7962624, 28629151, 79235168, 184528125, 380204032, 714924299, 1252332576, 2073071593, 3276800000, 4984209207, 7339040224, 10510100501, 14693280768, 20113571875, 27027081632, 35723051649, 46525874176

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A017017 - A017020, A017022 - A017028.

[(7*n+3)^5: n in [0..35]]; // Vincenzo Librandi, Jul 14 2011

(7*Range[0,20]+3)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{243,100000,1419857,7962624,28629151,79235168},20] (* Harvey P. Dale, Aug 27 2017 *)

[(7*n+3)^5 for n in range(41)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (243 + 98542*x + 823502*x^2 + 938622*x^3 + 154907*x^4 + 1024*x^5)/(1-x)^6.
E.g.f.: (243 + 99757*x + 610050*x^2 + 667135*x^3 + 204085*x^4 + 16807*x^5)*exp(x). (End)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Jun 26 2025

A017022 a(n) = (7*n + 3)^6.

Original entry on oeis.org

729, 1000000, 24137569, 191102976, 887503681, 3010936384, 8303765625, 19770609664, 42180533641, 82653950016, 151334226289, 262144000000, 433626201009, 689869781056, 1061520150601, 1586874322944

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A017017 - A017021, A017023 - A017028.

[(7*n+3)^6: n in [0..35]]; // Vincenzo Librandi, Jul 14 2011

(7*Range[0,40] +3)^6 (* G. C. Greubel, Oct 17 2023 *)

[(7*n+3)^6 for n in range(41)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (729 + 994897*x + 17152878*x^2 + 43114478*x^3 + 21697313*x^4 + 1742889*x^5 + 4096*x^6)/(1-x)^7.
E.g.f.: (729 + 999271*x + 11069149*x^2 + 20281590*x^3 + 10996580*x^4 + 2067261*x^5 + 117649*x^6)*exp(x). (End)

A017023 a(n) = (7*n + 3)^7.

Original entry on oeis.org

2187, 10000000, 410338673, 4586471424, 27512614111, 114415582592, 373669453125, 1028071702528, 2488651484819, 5455160701056, 11047398519097, 20971520000000, 37725479487783, 64847759419264

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A017017 - A017022, A017024 - A017028.

[(7*n+3)^7: n in [0..30]]; // Vincenzo Librandi, Jul 14 2011

(7*Range[0,30]+3)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28, 8,-1}, {2187,10000000,410338673,4586471424,27512614111,114415582592, 373669453125,1028071702528}, 30] (* Harvey P. Dale, Jul 07 2020 *)

[(7*n+3)^7 for n in range(41)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (2187 + 9982504*x + 330399909*x^2 + 1583639568*x^3 + 1750478653*x^4 + 456781416*x^5 + 19356099*x^6 + 16384*x^7)/(1-x)^8.
E.g.f.: (2187 + 9997813*x + 195170430*x^2 + 564242203*x^3 + 482865110*x^4 + 155531978*x^5 + 19765032*x^6 + 823543*x^7)*exp(x). (End)

A168331 a(n) = (5 + 14n + 7(-1)^n)/4.

Original entry on oeis.org

3, 10, 10, 17, 17, 24, 24, 31, 31, 38, 38, 45, 45, 52, 52, 59, 59, 66, 66, 73, 73, 80, 80, 87, 87, 94, 94, 101, 101, 108, 108, 115, 115, 122, 122, 129, 129, 136, 136, 143, 143, 150, 150, 157, 157, 164, 164, 171, 171, 178, 178, 185, 185, 192, 192, 199, 199, 206, 206

Offset: 1

Vincenzo Librandi, Nov 23 2009

Essentially the same as A168376.

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

Cf. A017017, A168376.

[5/4+7*n/2+7*(-1)^n/4: n in [1..70]]; // Vincenzo Librandi, Sep 17 2013

A168331:=n->(5+14*n+7*(-1)^n)/4: seq(A168331(n), n=1..100); # Wesley Ivan Hurt, May 03 2017

Table[5/4 + 7n/2 + 7 (-1)^n/4, {n,60}] (* or *) LinearRecurrence[{1, 1, -1}, {3, 10, 10}, 60] (* Harvey P. Dale, Oct 24 2011 *)
CoefficientList[Series[- (- 3 - 7 x + 3 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 17 2013 *)

a(n) = 7*n - a(n-1) - 1, with n > 1, a(1)=3.
G.f.: x*(3+7*x-3*x^2) / ((1+x)*(x-1)^2). - R. J. Mathar, Jan 05 2011
a(1)=3, a(2)=10, a(3)=10; for n>3, a(n) = a(n-1)+a(n-2)-a(n-3). - Harvey P. Dale, Oct 24 2011
E.g.f.: (1/4)*(7 - 12*exp(x) + (5 + 14*x)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 18 2016

New definition by R. J. Mathar, Jan 05 2011

A177071 a(n) = (7n + 3)(7*n + 4).

Original entry on oeis.org

12, 110, 306, 600, 992, 1482, 2070, 2756, 3540, 4422, 5402, 6480, 7656, 8930, 10302, 11772, 13340, 15006, 16770, 18632, 20592, 22650, 24806, 27060, 29412, 31862, 34410, 37056, 39800, 42642, 45582, 48620, 51756, 54990, 58322, 61752, 65280, 68906, 72630, 76452

Offset: 0

Vincenzo Librandi, May 31 2010

Cf. Zumkeller's contribution in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2, therefore a(n) = 49*A002061(n+1) - 37. - Bruno Berselli, Aug 24 2010

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

Cf. A002061, A017017, A017029, A061792, A177059.

Table[(7n+3)(7n+4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{12,110,306},40] (* Harvey P. Dale, Oct 09 2011 *)

a(n)=2*binomial(7*n+4,2) \\ Charles R Greathouse IV, Jan 11 2012

a(n) = 98*n + a(n-1) with n > 0, a(0)=12.
From Harvey P. Dale, Oct 09 2011: (Start)
a(0)=12, a(1)=110, a(2)=306, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -((2*(x+6)*(6*x+1))/(x-1)^3). (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017017(n)*A017029(n).
Sum_{n>=0} 1/a(n) = tan(Pi/14)*Pi/7.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/14)*cos(sqrt(5)*Pi/14).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/14)*cosh(sqrt(3)*Pi/14). (End)
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: exp(x)*(12 + 49*x*(2 + x)).
a(n) = 2*A061792(n). (End)

Edited by N. J. A. Sloane, Jun 22 2010

A017019 a(n) = (7*n + 3)^3.

Original entry on oeis.org

27, 1000, 4913, 13824, 29791, 54872, 91125, 140608, 205379, 287496, 389017, 512000, 658503, 830584, 1030301, 1259712, 1520875, 1815848, 2146689, 2515456, 2924207, 3375000, 3869893, 4410944, 5000211

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A017017, A017018, A017020 - A017028.

[(7*n+3)^3: n in [0..35]]; // Vincenzo Librandi, Jul 14 2011

(7*Range[0, 40] +3)^3 (* G. C. Greubel, Oct 17 2023 *)

[(7*n+3)^3 for n in range(41)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (27 + 892*x + 1075*x^2 + 64*x^3)/(1 - x)^4.
E.g.f.: (27 + 973*x + 1470*x^2 + 343*x^3)*exp(x). (End)

A017024 a(n) = (7*n + 3)^8.

Original entry on oeis.org

6561, 100000000, 6975757441, 110075314176, 852891037441, 4347792138496, 16815125390625, 53459728531456, 146830437604321, 360040606269696, 806460091894081, 1677721600000000, 3282116715437121, 6095689385410816

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A017017 - A017023, A017025 - A017028.

[(7*n+3)^8: n in [0..30]]; // Vincenzo Librandi, Jul 14 2011

(7*Range[0,40] +3)^8 (* G. C. Greubel, Oct 17 2023 *)

[(7*n+3)^8 for n in range(41)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (6561 +99940951*x +6075993637*x^2 +50892946083*x^3 +104941304419*x^4 +61119660133*x^5 +9093089943*x^6 +213769057*x^7 +65536*x^8)/(1-x)^9.
E.g.f.: (6561 +99993439*x +3387882001*x^2 +14908005882*x^3 +18918513831*x^4 +9290270934*x^5 +1978150286*x^6 +181179460*x^7 +5764801*x^8)*exp(x). (End)

A137185 Lucky numbers (A000959) which are congruent to 3 mod 7.

Original entry on oeis.org

3, 31, 73, 87, 115, 129, 171, 241, 283, 297, 339, 367, 409, 451, 535, 577, 591, 619, 717, 745, 787, 801, 885, 927, 997, 1011, 1039, 1053, 1095, 1123, 1179, 1249, 1263, 1291, 1389, 1417, 1459, 1473, 1501, 1543, 1585, 1599, 1641, 1711, 1767, 1809, 1879, 1893, 1921

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000959 and A017017.

A195029 a(n) = n(14n + 13) + 3.

Original entry on oeis.org

3, 30, 85, 168, 279, 418, 585, 780, 1003, 1254, 1533, 1840, 2175, 2538, 2929, 3348, 3795, 4270, 4773, 5304, 5863, 6450, 7065, 7708, 8379, 9078, 9805, 10560, 11343, 12154, 12993, 13860, 14755, 15678, 16629, 17608, 18615, 19650, 20713, 21804, 22923, 24070, 25245

Offset: 0

Omar E. Pol, Sep 07 2011

Sequence found by reading the line from 3, in the direction 3, 30, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the semi-diagonal parallel to A195024 and also parallel to A195028 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

56*a(n) + 1 is a perfect square. - Bruno Berselli, Feb 14 2017

Matthew House, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A022264.
Cf. A144555, A152760, A195019, A195020, A195021, A195023, A195024, A195025, A195026, A195027, A195028, A195320.
Cf. A005408, A017017, A185019, A193053, A198017.

Table[n (14 n + 13) + 3, {n, 0, 40}] (* Bruno Berselli, Feb 14 2017 *)
LinearRecurrence[{3,-3,1},{3,30,85},50] (* Harvey P. Dale, May 03 2018 *)

a(n)=n*(14*n+13)+3 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 14*n^2 + 13*n + 3 = A195028(n) + 3 = (2*n + 1)*(7*n + 3).
From Colin Barker, Apr 09 2012: (Start)
G.f.: (3 + 21*x + 4*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Dec 29 2024: (Start)
E.g.f.: exp(x)*(3 + 27*x + 14*x^2).
a(n) = A005408(n)*A017017(n) = A022264(2*n+1). (End)

Edited by Bruno Berselli, Feb 14 2017

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A017020 a(n) = (7*n + 3)^4.

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A017021 a(n) = (7*n + 3)^5.

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A017022 a(n) = (7*n + 3)^6.

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A017023 a(n) = (7*n + 3)^7.

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A168331 a(n) = (5 + 14*n + 7*(-1)^n)/4.

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A177071 a(n) = (7*n + 3)*(7*n + 4).

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A017019 a(n) = (7*n + 3)^3.

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A168331 a(n) = (5 + 14n + 7(-1)^n)/4.

A177071 a(n) = (7n + 3)(7*n + 4).

A195029 a(n) = n(14n + 13) + 3.