A152773 -id:A152773 - cp's OEIS frontend

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

0, 3, 36, 99, 192, 315, 468, 651, 864, 1107, 1380, 1683, 2016, 2379, 2772, 3195, 3648, 4131, 4644, 5187, 5760, 6363, 6996, 7659, 8352, 9075, 9828, 10611, 11424, 12267, 13140, 14043, 14976, 15939, 16932, 17955, 19008, 20091, 21204

Offset: 0

Omar E. Pol, Dec 26 2008

This sequence is related to A172117 by 3*A172117(n) = n*a(n) - Sum_{i=0..n-1} a(i) and this is the case d=10 in the identity n*(3*n*(d*n - d + 2)/2) - Sum_{k=0..n-1} 3*k*(d*k - d + 2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/2. - Bruno Berselli, Aug 26 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051624, A152965.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153875, A172117.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=30: see Comments lines of A226492.

Table[3n(5n-4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,3,36},40] (* Harvey P. Dale, Jun 18 2014 *)
3*PolygonalNumber[12,Range[0,60]] (* Harvey P. Dale, May 13 2022 *)

a(n)=3*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 15*n^2 - 12*n = A051624(n)*3.
a(n) = 30*n + a(n-1) - 27 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 9*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=36. - Harvey P. Dale, Jun 18 2014
E.g.f.: 3*x*(1 + 5*x)*exp(x). - G. C. Greubel, Aug 21 2016
a(n) = (4*n-2)^2 - (n-2)^2. In general, if P(k,n) is the k-th n-gonal number, then (2*k+1)*P(8*k+4,n) = ((3*k+1)*n-2*k)^2 - (k*n-2*k)^2. - Charlie Marion, Jul 29 2021

A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3n(9*n-7)/2.

Original entry on oeis.org

0, 3, 33, 90, 174, 285, 423, 588, 780, 999, 1245, 1518, 1818, 2145, 2499, 2880, 3288, 3723, 4185, 4674, 5190, 5733, 6303, 6900, 7524, 8175, 8853, 9558, 10290, 11049, 11835, 12648, 13488, 14355, 15249, 16170, 17118, 18093, 19095

Offset: 0

Omar E. Pol, Jan 02 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682, A152995.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=27: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,27}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Table[3*n*(9*n-7)/2, {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,3,33},25] (* G. C. Greubel, Aug 28 2016 *)

a(n)=3*n*(9*n-7)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (27*n^2 - 21*n)/2 = A051682(n)*3.
a(n) = 27*n + a(n-1) - 24, with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 8*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From G. C. Greubel, Aug 28 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*x*(2 + 9*x)*exp(x). (End)

A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3n(11*n - 9)/2.

Original entry on oeis.org

0, 3, 39, 108, 210, 345, 513, 714, 948, 1215, 1515, 1848, 2214, 2613, 3045, 3510, 4008, 4539, 5103, 5700, 6330, 6993, 7689, 8418, 9180, 9975, 10803, 11664, 12558, 13485, 14445, 15438, 16464, 17523, 18615, 19740, 20898, 22089

Offset: 0

Omar E. Pol, Jan 03 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051865, A152997.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=33: see Comments lines of A226492.

Table[(33*n^2 - 27*n)/2, {n,0,25}] (* G. C. Greubel, Aug 31 2016 *)

a(n)=3*n*(11*n-9)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (33*n^2 - 27*n)/2 = A051865(n)*3.
a(n) = a(n-1) + 33*n - 30, with n>0, a(0)=0. - Vincenzo Librandi, Dec 14 2010
G.f.: 3*x*(1 + 10*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From G. C. Greubel, Aug 31 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*x*(2 + 11*x)*exp(x). (End)

A153784 4 times heptagonal numbers: a(n) = 2n(5*n-3).

Original entry on oeis.org

0, 4, 28, 72, 136, 220, 324, 448, 592, 756, 940, 1144, 1368, 1612, 1876, 2160, 2464, 2788, 3132, 3496, 3880, 4284, 4708, 5152, 5616, 6100, 6604, 7128, 7672, 8236, 8820, 9424, 10048, 10692, 11356, 12040, 12744, 13468, 14212, 14976, 15760, 16564, 17388, 18232, 19096

Offset: 0

Omar E. Pol, Jan 02 2009

Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012

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

Cf. A000566, A085787, A087348, A135706, A152745, A152773.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,4,6!,20}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Table[2n(5n-3),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,4,28},50] (* Harvey P. Dale, Mar 19 2015 *)

a(n)=2*n*(5*n-3) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 10*n^2 - 6*n = 4*A000566(n) = 2*A135706(n).
a(n) = 20*n + a(n-1) - 16 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(n) = A087348(n) - 1, n >= 1. - Omar E. Pol, Jul 18 2012
a(0)=0, a(1)=4, a(2)=28, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 19 2015
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 4*x*(1 + 4*x)/(1 - x)^3.
E.g.f.: 2*exp(x)*x*(2 + 5*x).
a(n) = A152745(n) - n. (End)

A153786 6 times heptagonal numbers: a(n) = 3n(5*n-3).

Original entry on oeis.org

0, 6, 42, 108, 204, 330, 486, 672, 888, 1134, 1410, 1716, 2052, 2418, 2814, 3240, 3696, 4182, 4698, 5244, 5820, 6426, 7062, 7728, 8424, 9150, 9906, 10692, 11508, 12354, 13230, 14136, 15072, 16038, 17034, 18060, 19116, 20202, 21318

Offset: 0

Omar E. Pol, Jan 02 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566, A135706, A152773, A152777, A153785.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,6,8!,30}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 03 2009 *)
Table[ 3*n*(5*n-3), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,6,42}, 25] (* G. C. Greubel, Aug 28 2016 *)

a(n) = 3*n*(5*n-3); \\ Michel Marcus, Aug 28 2016

a(n) = 15*n^2 - 9*n = A000566(n)*6 = A135706(n)*3 = A152773(n)*2.
a(n) = 30*n + a(n-1) - 24 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From G. C. Greubel, Aug 28 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 6*x*(1 + 4*x)/(1 - x)^3.
E.g.f.: 3*x*(2 + 5*x)*exp(x). (End)

cp's OEIS Frontend

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3*n*(5*n-4).

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A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3*n*(9*n-7)/2.

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A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3*n*(11*n - 9)/2.

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A153784 4 times heptagonal numbers: a(n) = 2*n*(5*n-3).

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A153786 6 times heptagonal numbers: a(n) = 3*n*(5*n-3).

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A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3n(9*n-7)/2.

A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3n(11*n - 9)/2.

A153784 4 times heptagonal numbers: a(n) = 2n(5*n-3).

A153786 6 times heptagonal numbers: a(n) = 3n(5*n-3).