A152965 -id:A152965 - cp's OEIS frontend

A226488 a(n) = n(13n - 9)/2.

0, 2, 17, 45, 86, 140, 207, 287, 380, 486, 605, 737, 882, 1040, 1211, 1395, 1592, 1802, 2025, 2261, 2510, 2772, 3047, 3335, 3636, 3950, 4277, 4617, 4970, 5336, 5715, 6107, 6512, 6930, 7361, 7805, 8262, 8732, 9215, 9711, 10220, 10742, 11277, 11825, 12386, 12960

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th octagonal number and n-th 9-gonal (nonagonal) number.

Sum of reciprocals of a(n), for n>0: 0.629618994194109711163742089971688...

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

Cf. A000567, A001106, A153080 (first differences).

Cf. numbers of the form n*(n*k-k+4)/2 listed in A005843 (k=0), A000096 (k=1), A002378 (k=2), A005449 (k=3), A001105 (k=4), A005476 (k=5), A049450 (k=6), A218471 (k=7), A002939 (k=8), A062708 (k=9), A135706 (k=10), A180223 (k=11), A139267 (n=12), this sequence (k=13), A139268 (k=14), A226489 (k=15), A139271 (k=16), A180232 (k=17), A152995 (k=18), A226490 (k=19), A152965 (k=20), A226491 (k=21), A152997 (k=22).

List([0..50], n-> n*(13*n-9)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(13*n-9)/2: n in [0..50]];
```

I:=[0,2,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2) +Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A226488:=n->n*(13*n - 9)/2; seq(A226488(n), n=0..50); # Wesley Ivan Hurt, Feb 25 2014

Table[n(13n-9)/2, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {0, 2, 17}, 50] (* Harvey P. Dale, Jun 19 2013 *)
CoefficientList[Series[x(2+11x)/(1-x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)

a(n)=n*(13*n-9)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(13*n-9)/2 for n in (0..50)] # G. C. Greubel, Aug 30 2019

G.f.: x*(2+11*x)/(1-x)^3.
a(n) + a(-n) = A152742(n).
a(0)=0, a(1)=2, a(2)=17; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 19 2013
E.g.f.: x*(4 + 13*x)*exp(x)/2. - G. C. Greubel, Aug 30 2019
a(n) = A000567(n) + A001106(n). - Michel Marcus, Aug 31 2019

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

Original entry on oeis.org

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

A195015 Main axis of the square spiral whose edges have length A195013 and whose vertices are the numbers A195014.

Original entry on oeis.org

0, 2, 12, 24, 44, 66, 96, 128, 168, 210, 260, 312, 372, 434, 504, 576, 656, 738, 828, 920, 1020, 1122, 1232, 1344, 1464, 1586, 1716, 1848, 1988, 2130, 2280, 2432, 2592, 2754, 2924, 3096, 3276, 3458, 3648, 3840, 4040, 4242, 4452, 4664, 4884

Offset: 0

Omar E. Pol, Sep 26 2011

Sequence found by reading the line from 0, in the direction 0, 2, ..., and the same line from 0, in the direction 0, 12, ..., in the square spiral mentioned above. Axis perpendicular to A195016 in the same spiral.

Also four times A005475 and positives A152965 interleaved.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000566, A005475, A005476, A028895, A033571, A152965, A153126, A195013, A195014, A195016.

[(2*n*(5*n+2)+3*(-1)^n-3)/4: n in [0..50]]; // Vincenzo Librandi, Oct 28 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 2, 12, 24}, 50] (* Paolo Xausa, Feb 09 2024 *)

From Bruno Berselli, Oct 14 2011: (Start)
G.f.: 2*x*(1+4*x)/((1+x)*(1-x)^3).
a(n) = (2*n*(5*n+2) + 3*(-1)^n-3)/4.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) + a(n-1) = A135706(n). (End)

A195016 a(n) = (n(5n+7)-(-1)^n+1)/2.

Original entry on oeis.org

0, 7, 17, 34, 54, 81, 111, 148, 188, 235, 285, 342, 402, 469, 539, 616, 696, 783, 873, 970, 1070, 1177, 1287, 1404, 1524, 1651, 1781, 1918, 2058, 2205, 2355, 2512, 2672, 2839, 3009, 3186, 3366, 3553, 3743, 3940, 4140, 4347, 4557, 4774, 4994

Offset: 0

Omar E. Pol, Sep 26 2011

Sequence found by reading the line from 0, in the direction 0, 7,..., and the same line from 0, in the direction 0, 17,..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. Axis perpendicular to the main axis A195015 in the same spiral.

Also sequence found by reading the line from 0, in the direction 0, 7,..., and the same line from 0, in the direction 0, 17,..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This line is parallel to A153126 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000566, A005476, A028895, A033571, A085787, A152965, A153126, A195013, A195014, A195015.

&cat[[n*t,(n+1)*t] where t is 10*n+7: n in [0..22]]; // Bruno Berselli, Oct 14 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 7, 17, 34}, 50] (* Paolo Xausa, Feb 09 2024 *)

n*(10*n-3), if n >= 1, and (2*n+1)*(5*n+1)-1, if n >= 0, interleaved.

G.f.: x*(7+3*x)/((1+x)*(1-x)^3). - Bruno Berselli, Oct 14 2011

Concise definition by Bruno Berselli, Oct 14 2011

cp's OEIS Frontend

A226488 a(n) = n*(13*n - 9)/2.

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

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A195015 Main axis of the square spiral whose edges have length A195013 and whose vertices are the numbers A195014.

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Author

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Programs

Magma

Mathematica

Formula

A195016 a(n) = (n*(5*n+7)-(-1)^n+1)/2.

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Author

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Programs

Magma

Mathematica

Formula

Extensions

A226488 a(n) = n(13n - 9)/2.

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

A195016 a(n) = (n(5n+7)-(-1)^n+1)/2.