A185019 -id:A185019 - cp's OEIS frontend

A013656 a(n) = n(9n-2).

0, 7, 32, 75, 136, 215, 312, 427, 560, 711, 880, 1067, 1272, 1495, 1736, 1995, 2272, 2567, 2880, 3211, 3560, 3927, 4312, 4715, 5136, 5575, 6032, 6507, 7000, 7511, 8040, 8587, 9152, 9735, 10336, 10955, 11592, 12247, 12920, 13611, 14320, 15047, 15792, 16555

Offset: 0

David W. Wilson

For n>0, numbers such that sqrt(a(n)) has the continued fraction {k;[1,1,1,2k]}, where the part in [] is repeated and k is of the form 3m+2 (A016789). - Bruno Berselli, May 30 2013

For n >= 1, the continued fraction expansion of sqrt(4*a(n)) is [6n-1; {3, 3n-1, 3, 12n-2}]. - Magus K. Chu, Sep 18 2022

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A010701, A017257, A185019.

Cf. A016789, A061039, A144454.

[n*(9*n-2): n in [0..60]]; // G. C. Greubel, Mar 11 2022

Table[n(9n-2),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,32},50] (* Harvey P. Dale, Jul 07 2012 *)

a(n)=n*(9*n-2) \\ Charles R Greathouse IV, Jun 17 2017

a(n+1) = A144454(9*n+7) = A061039(27*n+21). - Paul Curtz, Nov 05 2008
a(n) = a(n-1) + 18*n - 11 with n>0, a(0)=0. - Vincenzo Librandi, Nov 22 2010
a(0)=0, a(1)=7, a(2)=32, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 07 2012
From G. C. Greubel, Mar 11 2022: (Start)
G.f.: x*(7 - 11*x)/(1-x)^3.
E.g.f.: x*(7 + 9*x)*exp(x). (End)
Sum_{n>=1} 1/a(n) = -(psi(7/9)+gamma)/2 = (A354640-A001620)/2 = 0.22000753... - R. J. Mathar, Apr 22 2024

A147296 a(n) = n(9n+2).

Original entry on oeis.org

0, 11, 40, 87, 152, 235, 336, 455, 592, 747, 920, 1111, 1320, 1547, 1792, 2055, 2336, 2635, 2952, 3287, 3640, 4011, 4400, 4807, 5232, 5675, 6136, 6615, 7112, 7627, 8160, 8711, 9280, 9867, 10472, 11095, 11736, 12395, 13072, 13767, 14480, 15211, 15960

Offset: 0

Paul Curtz, Nov 05 2008

For n >= 1, the continued fraction expansion of sqrt(4*a(n)) is [6n; {1, 1, 1, 3n-1, 1, 1, 1, 12n}]. - Magus K. Chu, Sep 17 2022

Reply to V. Librandi, A147296 (SeqFan list). - _M. F. Hasler_, Mar 01 2009
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Equals first 9-fold decimation of A144454.

Cf. A010701, A017173, A147296, A185019.

Table[n(9n+2),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,40},50] (* Harvey P. Dale, Dec 19 2014 *)

A147296(n) = n*(9*n + 2) \\ M. F. Hasler, Mar 01 2009

a(n) = n*(9*n + 2), as conjectured by V. Librandi. - M. F. Hasler, Mar 01 2009
G.f.: x*(11+7*x)/(1-x)^3. - Jaume Oliver Lafont, Aug 30 2009
a(n) = floor((3*n + 1/3)^2). - Reinhard Zumkeller, Apr 14 2010
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: exp(x)*x*(11 + 9*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

More terms from M. F. Hasler, Mar 01 2009

A195026 a(n) = 7n(2*n + 1).

Original entry on oeis.org

0, 21, 70, 147, 252, 385, 546, 735, 952, 1197, 1470, 1771, 2100, 2457, 2842, 3255, 3696, 4165, 4662, 5187, 5740, 6321, 6930, 7567, 8232, 8925, 9646, 10395, 11172, 11977, 12810, 13671, 14560, 15477, 16422, 17395, 18396, 19425, 20482, 21567, 22680, 23821, 24990

Offset: 0

Omar E. Pol, Oct 13 2011

Sequence found by reading the line from 0, in the direction 0, 21, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. Semi-diagonal opposite to A195320 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

Sum of the numbers from 6*n to 8*n. - Wesley Ivan Hurt, Dec 23 2015

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

Cf. A014105, A144555, A152760, A195019, A195020, A195021, A195023, A195024, A195025, A195320.

Cf. A185019, A193053, A198017.

[14*n^2 +7*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

A195026:=n->7*n*(2*n+1): seq(A195026(n), n=0..50); # Wesley Ivan Hurt, Dec 23 2015

Table[7*n*(2*n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Dec 23 2015 *)
LinearRecurrence[{3,-3,1},{0,21,70},50] (* Harvey P. Dale, Apr 26 2017 *)

a(n)=7*n*(2*n+1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 14*n^2 + 7*n.
a(n) = 7*A014105(n). - Bruno Berselli, Oct 13 2011
From Colin Barker, Apr 09 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: 7*x*(3+x)/(1-x)^3. (End)
a(n) = Sum_{i=6*n..8*n} i. - Wesley Ivan Hurt, Dec 23 2015
E.g.f.: 7*exp(x)*x*(3 + 2*x). - Elmo R. Oliveira, Dec 29 2024

A195030 a(n) = (n-2)(14n-39) for n > 2, otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 34, 93, 180, 295, 438, 609, 808, 1035, 1290, 1573, 1884, 2223, 2590, 2985, 3408, 3859, 4338, 4845, 5380, 5943, 6534, 7153, 7800, 8475, 9178, 9909, 10668, 11455, 12270, 13113, 13984, 14883, 15810, 16765, 17748, 18759, 19798, 20865, 21960, 23083

Offset: 0

Omar E. Pol, Oct 18 2011

Union of [1, 2] and A195021.

Sequence found by reading the line from 0, in the direction 0, 1,..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-axis of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195030
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A195019, A195020, A195021, A185019, A193053, A198017.

[0,1,2] cat[(n-2)*(14*n-39): n in [3..50]]; // Vincenzo Librandi, Jul 10 2012

Join[{0,1,2},Table[(n-2)*(14*n-39),{n,3,50}]] (* Vincenzo Librandi, Jul 10 2012 *)

a(n)=if(n,([0,1,0; 0,0,1; 1,-3,3]^n*[0;1;2])[1,1],0) \\ Charles R Greathouse IV, Oct 16 2015

G.f.: x*(1-x+30*x^3-2*x^4)/(1-x)^3. - Bruno Berselli, Oct 18 2011

Both sequence (based on A195021) and definition suggested by Bruno Berselli, Oct 18 2011

A195027 a(n) = 2n(7*n + 5).

Original entry on oeis.org

0, 24, 76, 156, 264, 400, 564, 756, 976, 1224, 1500, 1804, 2136, 2496, 2884, 3300, 3744, 4216, 4716, 5244, 5800, 6384, 6996, 7636, 8304, 9000, 9724, 10476, 11256, 12064, 12900, 13764, 14656, 15576, 16524, 17500, 18504, 19536, 20596, 21684, 22800, 23944, 25116, 26316

Offset: 0

Omar E. Pol, Oct 13 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. Semi-axis opposite to A195023 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

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

Cf. A144555, A152760, A179986, A195019, A195020, A195021, A195023, A195024, A195025, A195026, A195320.

Cf. A185019, A193053, A198017.

[14*n^2 +10*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[2n(7n+5),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,24,76},50] (* Harvey P. Dale, Jul 24 2012 *)

a(n)=2*n*(7*n+5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 14*n^2 + 10*n.
a(n) = 4*A179986(n). - Bruno Berselli, Oct 13 2011
G.f.: 4*x*(6+x)/(1-x)^3. - Colin Barker, Jan 09 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=24, a(2)=76. - Harvey P. Dale, Jul 24 2012
E.g.f.: 2*exp(x)*x*(12 + 7*x). - Elmo R. Oliveira, Dec 30 2024

A195028 a(n) = n(14n + 13).

Original entry on oeis.org

0, 27, 82, 165, 276, 415, 582, 777, 1000, 1251, 1530, 1837, 2172, 2535, 2926, 3345, 3792, 4267, 4770, 5301, 5860, 6447, 7062, 7705, 8376, 9075, 9802, 10557, 11340, 12151, 12990, 13857, 14752, 15675, 16626, 17605, 18612, 19647, 20710, 21801, 22920, 24067, 25242

Offset: 0

Omar E. Pol, Oct 13 2011

Sequence found by reading the line from 0, in the direction 0, 27, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. Numbers opposite to the semi-diagonal A195024 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

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

Cf. A144555, A152760, A185019, A193053, A195019, A195020, A195021, A195023, A195024, A195025, A195026, A195027, A195320, A198017.

[14*n^2 +13*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

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

a(n) = 14*n^2 + 13*n.
G.f.: x*(27+x)/(1-x)^3. - Colin Barker, Jan 09 2012
From Elmo R. Oliveira, Dec 30 2024: (Start)
E.g.f.: exp(x)*x*(27 + 14*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Name suggested by Bruno Berselli, Oct 13 2011

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

cp's OEIS Frontend

A013656 a(n) = n*(9*n-2).

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Magma

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A147296 a(n) = n*(9*n+2).

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A195026 a(n) = 7*n*(2*n + 1).

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A195030 a(n) = (n-2)*(14*n-39) for n > 2, otherwise a(n) = n.

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Magma

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A195027 a(n) = 2*n*(7*n + 5).

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Magma

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A195028 a(n) = n*(14*n + 13).

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Magma

PARI

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Extensions

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

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A013656 a(n) = n(9n-2).

A147296 a(n) = n(9n+2).

A195026 a(n) = 7n(2*n + 1).

A195030 a(n) = (n-2)(14n-39) for n > 2, otherwise a(n) = n.

A195027 a(n) = 2n(7*n + 5).

A195028 a(n) = n(14n + 13).

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