A194454 -id:A194454 - cp's OEIS frontend

A049453 Second pentagonal numbers with even index: a(n) = n(6n+1).

0, 7, 26, 57, 100, 155, 222, 301, 392, 495, 610, 737, 876, 1027, 1190, 1365, 1552, 1751, 1962, 2185, 2420, 2667, 2926, 3197, 3480, 3775, 4082, 4401, 4732, 5075, 5430, 5797, 6176, 6567, 6970, 7385, 7812, 8251, 8702, 9165, 9640, 10127, 10626, 11137, 11660, 12195

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Number of edges in the join of the complete tripartite graph of order 3n and the cycle graph of order n, K_n,n,n * C_n. - Roberto E. Martinez II, Jan 07 2002
Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
First bisection of A036498. - Bruno Berselli, Nov 25 2012

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. A001318, A005449, A033568, A033570, A036498, A049452, A185019, A194454.

Cf. A254963 (comment).

seq(binomial(6*n+1,2)/3, n=0..42); # Zerinvary Lajos, Jan 21 2007

s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 12}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[n*(6*n + 1), {n,0,50}] (* G. C. Greubel, Jun 07 2017 *)

x='x+O('x^50); concat([0], Vec(x*(7+5*x)/(1-x)^3)) \\ G. C. Greubel, Jun 07 2017

G.f.: x*(7+5*x)/(1-x)^3.
a(n) = 12*n + a(n-1) - 5 with n > 0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Jun 07 2017
From Amiram Eldar, Feb 18 2022: (Start)
Sum_{n>=1} 1/a(n) = 6 - sqrt(3)*Pi/2 - 2*log(2) - 3*log(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi + log(2) + sqrt(3)*log(2 + sqrt(3)) - 6. (End)
E.g.f.: exp(x)*x*(7 + 6*x). - Elmo R. Oliveira, Dec 12 2024

A172073 a(n) = (4n^3 + n^2 - 3n)/2.

Original entry on oeis.org

0, 1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, 3510, 4459, 5565, 6840, 8296, 9945, 11799, 13870, 16170, 18711, 21505, 24564, 27900, 31525, 35451, 39690, 44254, 49155, 54405, 60016, 66000, 72369, 79135, 86310, 93906, 101935, 110409, 119340, 128740

Offset: 0

Vincenzo Librandi, Jan 25 2010

14-gonal (or tetradecagonal) pyramidal numbers generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=6.
In fact, the sequence is related to A000567 by a(n) = n*A000567(n) - Sum_{i=0..n-1} A000567(i) and this is the case d=6 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Nov 29 2010
Except for the initial 0, this is the principal diagonal of the convolution array A213761. - Clark Kimberling, Jul 04 2012
Starting (1, 15, 54, ...), this is the binomial transform of (1, 14, 25, 12, 0, 0, 0, ...). - Gary W. Adamson, Jul 29 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - Bruno Berselli, Feb 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index to sequences related to pyramidal numbers.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A237616.

Cf. A000567, A051866, A194454.

List([0..40], n-> n*(n+1)*(4*n-3)/2); # G. C. Greubel, Aug 30 2019

[(4*n^3+n^2-3*n)/2: n in [0..50]]; // Vincenzo Librandi, Jan 01 2014

seq(n*(n+1)*(4*n-3)/2, n=0..40); # G. C. Greubel, Aug 30 2019

f[n_]:= n(n+1)(4n-3)/2; Array[f, 40, 0]
LinearRecurrence[{4,-6,4,-1},{0,1,15,54},40] (* Harvey P. Dale, Jan 29 2013 *)
CoefficientList[Series[x (1+11x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)

a(n)=(4*n^3+n^2-3*n)/2 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(4*n-3)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

a(n) = n*(n+1)*(4*n-3)/2.
From Bruno Berselli, Dec 15 2010: (Start)
G.f.: x*(1+11*x)/(1-x)^4.
a(n) = Sum_{i=0..n} A051866(i). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=15, a(3)=54. - Harvey P. Dale, Jan 29 2013
a(n) = Sum_{i=0..n-1} (n-i)*(12*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi/21 + 8*log(2)/7 - 2/7.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/21 + 8*sqrt(2)*log(sqrt(2)+2)/21 - (20 + 4*sqrt(2))*log(2)/21 + 2/7. (End)
E.g.f.: exp(x)*x*(2 + 13*x + 4*x^2)/2. - Elmo R. Oliveira, Aug 04 2025

Edited by Bruno Berselli, Dec 14 2010

A154106 a(n) = 12n^2 + 22n + 11.

Original entry on oeis.org

11, 45, 103, 185, 291, 421, 575, 753, 955, 1181, 1431, 1705, 2003, 2325, 2671, 3041, 3435, 3853, 4295, 4761, 5251, 5765, 6303, 6865, 7451, 8061, 8695, 9353, 10035, 10741, 11471, 12225, 13003, 13805, 14631, 15481, 16355, 17253, 18175, 19121

Offset: 0

Klaus Brockhaus, Jan 04 2009

Sequence found by reading the line from 11, in the direction 11, 45,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

			a(3) = 12*3^2 + 22*3 + 11 = 185 = 2*3*29 + 11 = 2*3*A016969(4) + 11.
a(4) = a(3) +24*4 +10 = 185 +96 +10 = 291.

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

Cf. A016969 (6n+5), A153286, A185918, A194454.

Magma
```
[ 12*n^2+22*n+11: n in [0..39] ];
```

Table[12n^2+22n+11,{n,0,50}]  (* Harvey P. Dale, Mar 16 2011 *)
LinearRecurrence[{3,-3,1},{11,45,103}, 25] (* G. C. Greubel, Sep 02 2016 *)

a(n)=12*n^2+22*n+11 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: (1 +x)*(11 +x)/(1-x)^3.
a(n) = 2*n*A016969(n+1) + 11.
a(0) = 11; for n > 0, a(n) = a(n-1) + 24*n + 10.
a(n) = 2 + A185918(n+1). - Omar E. Pol, Jul 18 2012
E.g.f.: (11 + 34*x + 12*x^2)*exp(x). - G. C. Greubel, Sep 02 2016

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A049453 Second pentagonal numbers with even index: a(n) = n*(6*n+1).

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A172073 a(n) = (4*n^3 + n^2 - 3*n)/2.

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A154106 a(n) = 12*n^2 + 22*n + 11.

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A049453 Second pentagonal numbers with even index: a(n) = n(6n+1).

A172073 a(n) = (4n^3 + n^2 - 3n)/2.

A154106 a(n) = 12n^2 + 22n + 11.