A110449 -id:A110449 - cp's OEIS frontend

A022271 a(n) = n(13n + 1)/2.

0, 7, 27, 60, 106, 165, 237, 322, 420, 531, 655, 792, 942, 1105, 1281, 1470, 1672, 1887, 2115, 2356, 2610, 2877, 3157, 3450, 3756, 4075, 4407, 4752, 5110, 5481, 5865, 6262, 6672, 7095, 7531, 7980, 8442, 8917, 9405, 9906, 10420, 10947, 11487, 12040

Offset: 0

N. J. A. Sloane

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. A022270, A110449.

Cf. similar sequences listed in A022289.

[n*(13*n + 1)/2: n in [0..45]]; // Vincenzo Librandi, Mar 31 2015

Table[n (13 n + 1)/2, {n, 0, 40}] (* Vincenzo Librandi, Mar 31 2015 *)
LinearRecurrence[{3,-3,1},{0,7,27},50] (* Harvey P. Dale, Jul 03 2022 *)

a(n)=n*(13*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A110449(n, 6) for n>5.
a(n) = 13*n + a(n-1) - 6 with n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
G.f.: x*(7+6*x)/(1-x)^3. - Vincenzo Librandi, Mar 31 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2 - Vincenzo Librandi, Mar 31 2015
a(n) = A022270(-n). - Bruno Berselli, Mar 31 2015
a(n) = A000217(7*n) - A000217(6*n). - Bruno Berselli, Oct 13 2016
E.g.f.: (x/2)*(13*x + 14)*exp(x). - G. C. Greubel, Aug 23 2017

More terms from Vincenzo Librandi, Mar 31 2015

A085786 a(n) = n(2n^2 + n + 1)/2.

Original entry on oeis.org

2, 11, 33, 74, 140, 237, 371, 548, 774, 1055, 1397, 1806, 2288, 2849, 3495, 4232, 5066, 6003, 7049, 8210, 9492, 10901, 12443, 14124, 15950, 17927, 20061, 22358, 24824, 27465, 30287, 33296, 36498, 39899, 43505, 47322, 51356, 55613, 60099, 64820

Offset: 1

Jon Perry, Jul 23 2003

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

Cf. A000217 [t(n)], A000096 [t(n)+n], A005449 [t(n)+n^2].

a(n) = A110449(n, n).

[n*(2*n^2 + n + 1)/2: n in [1..40]]; // Vincenzo Librandi, Aug 14 2017

CoefficientList[Series[(x + 1) (x + 2) / (x - 1)^4, {x, 0, 40}], x] (* or *) LinearRecurrence[{4, -6, 4, -1}, {2, 11, 33, 74}, 40] (* Vincenzo Librandi, Aug 14 2017 *)

t(n)=n*(n+1)/2;
vector(40, i, t(i)+i^3)

a(n) = A000217(n) + n^3.
From Colin Barker, Jan 20 2014: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(x+1)*(x+2) / (x-1)^4. (End)
E.g.f.: (x/2)*(4 + 7*x + 2*x^2)*exp(x). - G. C. Greubel, Aug 24 2017

Name changed by Wesley Ivan Hurt, Apr 30 2022

A022273 a(n) = n(15n + 1)/2.

Original entry on oeis.org

0, 8, 31, 69, 122, 190, 273, 371, 484, 612, 755, 913, 1086, 1274, 1477, 1695, 1928, 2176, 2439, 2717, 3010, 3318, 3641, 3979, 4332, 4700, 5083, 5481, 5894, 6322, 6765, 7223, 7696, 8184, 8687, 9205, 9738, 10286, 10849, 11427, 12020, 12628, 13251, 13889

Offset: 0

N. J. A. Sloane

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. A022272, A064761, A110449.

Cf. similar sequences listed in A022289.

[n*(15*n + 1)/2: n in [0..45]]; // Vincenzo Librandi, Mar 31 2015

Table[n (15 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Mar 12 2015 *)
CoefficientList[Series[x (8 + 7 x) / (1 - x)^3, {x, 0, 40}], x]; (* Vincenzo Librandi, Mar 31 2015 *)

a(n)=n*(15*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A110449(n, 7) for n>6.
a(n) = 15*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
G.f.: x*(8+7*x)/(1-x)^3. - Vincenzo Librandi, Mar 31 2015
a(n) = 3*a(n-1) - 3*a(n-2) - a(n-3) for n>2. - Vincenzo Librandi, Mar 31 2015
a(n) = A022272(-n). - Bruno Berselli, Mar 31 2015
a(n) + a(-n) = A064761(n). - Bruno Berselli, Mar 31 2015
a(n) = A000217(8*n) - A000217(7*n). - Bruno Berselli, Oct 13 2016
E.g.f.: (x/2)*(15*x + 16)*exp(x). - G. C. Greubel, Aug 23 2017

More terms from Vincenzo Librandi, Mar 31 2015

A110451 a(n) = n(4n^2 + 2*n + 1).

Original entry on oeis.org

0, 7, 42, 129, 292, 555, 942, 1477, 2184, 3087, 4210, 5577, 7212, 9139, 11382, 13965, 16912, 20247, 23994, 28177, 32820, 37947, 43582, 49749, 56472, 63775, 71682, 80217, 89404, 99267, 109830, 121117, 133152, 145959, 159562, 173985, 189252

Offset: 0

Reinhard Zumkeller, Jul 21 2005

a(n) = A110449(2*n,n), central terms in triangle A110449.

2*a(n) is the sum of the consecutive integers from A000384(n)+1 to A000384(n+1)-1. - Bruno Berselli, Jun 27 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000384, A110449.

List([0..40],n->n*(4*n^2+2*n+1)); # Muniru A Asiru, Jun 27 2018

[n*(4*n^2+2*n+1): n in [0..40]]; // Vincenzo Librandi, Dec 26 2010

seq(n*(4*n^2+2*n+1),n=0..40); # Muniru A Asiru, Jun 27 2018

Table[n*(4*n^2 + 2*n + 1), {n, 0, 50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,7,42,129}, 50] (* G. C. Greubel, Aug 24 2017 *)

x='x+O('x^50); Vec((7*x + 14*x^2 + 3*x^3)/(1 - x)^4) \\ G. C. Greubel, Aug 24 2017

From G. C. Greubel, Aug 24 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-1) + 4*a(n-2) - a(n-4).
G.f.: (7*x + 14*x^2 + 3*x^3)/(1 - x)^4.
E.g.f.: x*(7 + 14*x + 4*x^2)*exp(x). (End)

cp's OEIS Frontend

A022271 a(n) = n*(13*n + 1)/2.

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A085786 a(n) = n*(2*n^2 + n + 1)/2.

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A022273 a(n) = n*(15*n + 1)/2.

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A110451 a(n) = n*(4*n^2 + 2*n + 1).

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A022271 a(n) = n(13n + 1)/2.

A085786 a(n) = n(2n^2 + n + 1)/2.

A022273 a(n) = n(15n + 1)/2.

A110451 a(n) = n(4n^2 + 2*n + 1).