A237616 -id:A237616 - cp's OEIS frontend

A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n(n+1)(10*n-7)/6.

0, 1, 13, 46, 110, 215, 371, 588, 876, 1245, 1705, 2266, 2938, 3731, 4655, 5720, 6936, 8313, 9861, 11590, 13510, 15631, 17963, 20516, 23300, 26325, 29601, 33138, 36946, 41035, 45415, 50096, 55088, 60401, 66045, 72030, 78366, 85063, 92131, 99580, 107420, 115661

Offset: 0

N. J. A. Sloane, R. K. Guy

Binomial transform of [1, 12, 21, 10, 0, 0, 0, ...] = (1, 13, 46, 110, ...). - Gary W. Adamson, Nov 28 2007

This sequence is related to A000566 by a(n) = n*A000566(n) - Sum_{i=0..n-1} A000566(i) and this is the case d=5 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, Oct 18 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, 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), 2008.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A093645 ((10, 1) Pascal, column m=3). Partial sums of A051624.
Cf. A000566.
See similar sequences listed in A237616.

List([0..45], n-> n*(n+1)*(10*n-7)/6); # G. C. Greubel, Aug 30 2019

[ n eq 1 select 0 else Self(n-1)+(n-1)*(5*n-9): n in [1..45] ]; // Klaus Brockhaus, Nov 20 2008

A007587:=n->n*(n+1)*(10*n-7)/6: seq(A007587(n), n=0..50); # Wesley Ivan Hurt, Oct 23 2014

CoefficientList[Series[x(1+9x)/(1-x)^4, {x,0,45}], x] (* Vincenzo Librandi, Jun 20 2013 *)
Table[n(n+1)(10n-7)/6,{n,0,50}] (* Harvey P. Dale, Nov 13 2013 *)

a(n)=if(n,([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;1;13;46])[1,1],0) \\ Charles R Greathouse IV, Oct 07 2015

vector(45, n, n*(n-1)*(10*n-17)/6) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(10*n-7)/6 for n in (0..45)] # G. C. Greubel, Aug 30 2019

a(n) = (10*n-7)*binomial(n+1, 2)/3.
G.f.: x*(1+9*x)/(1-x)^4.
a(n) = Sum_{k=0..n} k*(5*k-4). - Klaus Brockhaus, Nov 20 2008
a(n) = Sum_{i=0..n-1} (n-i)*(10*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 23 2014
E.g.f.: exp(x)*x*(6 + 33*x + 10*x^2)/6. - Elmo R. Oliveira, Aug 04 2025

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

A172078 a(n) = n(16n^2 + 3*n - 13)/6.

Original entry on oeis.org

0, 1, 19, 70, 170, 335, 581, 924, 1380, 1965, 2695, 3586, 4654, 5915, 7385, 9080, 11016, 13209, 15675, 18430, 21490, 24871, 28589, 32660, 37100, 41925, 47151, 52794, 58870, 65395, 72385, 79856, 87824, 96305, 105315, 114870, 124986, 135679

Offset: 0

Vincenzo Librandi, Jan 25 2010

Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=8.
In fact, the sequence is related to A001107 by a(n) = n*A001107(n) - Sum_{k=0..n-1} A001107(k), and this is the case d=8 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, Dec 14 2010
Inverse binomial transform of this sequence: 0, 1, 17, 16, 0, 0 (0 continued). - Bruno Berselli, Dec 14 2010
Principal diagonal of the convolution array A213835. - Clark Kimberling, Jul 04 2012

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
B. 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. A001107.

Cf. similar sequences listed in A237616.

List([0..40], n-> n*(n+1)*(16*n-13)/6); # G. C. Greubel, Aug 30 2019

[n*(n+1)*(16*n-13)/6: n in [0..40]]; // G. C. Greubel, Aug 30 2019

A172078:=n->(16*n^3+3*n^2-13*n)/6: seq(A172078(n), n=0..50); # Wesley Ivan Hurt, Jan 21 2017

LinearRecurrence[{4,-6,4,-1}, {0,1,19,70}, 50] (* Vincenzo Librandi, Mar 01 2012 *)
Table[n (16n^2+3n-13)/6,{n,0,40}] (* Harvey P. Dale, Aug 14 2023 *)

a(n)=n*(16*n^2+3*n-13)/6 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(16*n-13)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

a(n) = n*(n+1)*(16*n-13)/6.
G.f.: x*(1+15*x)/(1-x)^4. - Bruno Berselli, Dec 14 2010
a(n) = Sum_{i=0..n-1} (n-i)*(16*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(6 +51*x +16*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019

A050441 Partial sums of A051865.

Original entry on oeis.org

0, 1, 14, 50, 120, 235, 406, 644, 960, 1365, 1870, 2486, 3224, 4095, 5110, 6280, 7616, 9129, 10830, 12730, 14840, 17171, 19734, 22540, 25600, 28925, 32526, 36414, 40600, 45095, 49910, 55056, 60544, 66385, 72590, 79170, 86136, 93499, 101270

Offset: 0

Barry E. Williams, Dec 23 1999

This sequence is related to A180223 by 2*a(n) = n*A180223(n) - Sum_{i=0..n-1} A180223(i). Also, 13-gonal (or tridecagonal) pyramidal numbers. - Bruno Berselli, Dec 14 2010

			After 0, the sequence is provided by the row sums of the triangle (see above, fourth formula):
  1;
  2, 12;
  3, 24, 23;
  4, 36, 46, 34;
  5, 48, 69, 68, 45; ... - _Vincenzo Librandi_, Feb 12 2014

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189-196.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

Bruno Berselli, 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. A051865, A180223.

Similar sequences are listed in A237616.

List([0..40], n-> n*(n+1)*(11*n-8)/6); # G. C. Greubel, Aug 30 2019

I:=[0,1,14,50]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4) : n in [1..50]]; // Vincenzo Librandi, Feb 12 2014

seq(n*(n+1)*(11*n-8)/6, n=0..40); # G. C. Greubel, Aug 30 2019

Accumulate[Table[n (11n-9)/2,{n,0,40}]] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,1,14,50},40] (* Harvey P. Dale, Nov 14 2011 *)
CoefficientList[Series[x (1 + 10 x)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)

a(n)=n*(n+1)*(11*n-8)/6 \\ Charles R Greathouse IV, Oct 07 2015

[n*(n+1)*(11*n-8)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

a(n) = n*(n+1)*(11*n-8)/6.
G.f.: x*(1+10*x)/(1-x)^4. - Bruno Berselli, Aug 19 2010
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Bruno Berselli, Aug 19 2010
a(n) = Sum_{i=0..n-1} (n-i)*(11*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: exp(x)*x*(6 + 36*x + 11*x^2)/6. - Stefano Spezia, May 04 2022

A172076 a(n) = n(n+1)(14*n-11)/6.

Original entry on oeis.org

0, 1, 17, 62, 150, 295, 511, 812, 1212, 1725, 2365, 3146, 4082, 5187, 6475, 7960, 9656, 11577, 13737, 16150, 18830, 21791, 25047, 28612, 32500, 36725, 41301, 46242, 51562, 57275, 63395, 69936, 76912, 84337, 92225, 100590, 109446, 118807, 128687

Offset: 0

Vincenzo Librandi, Jan 25 2010

Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=7.
From Bruno Berselli, Dec 14 2010: (Start)
In fact, the sequence is related to A001106 by a(n) = n*A001106(n) - Sum_{k=0..n-1} A001106(k) and this is the case d=7 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.
Also 16-gonal (or hexadecagonal) pyramidal numbers.
Inverse binomial transform of this sequence: 0, 1, 15, 14, 0, 0 (0 continued). (End)

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

Vincenzo Librandi, 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), 2008.
Index to sequences related to pyramidal numbers.
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.

List([0..40], n-> n*(n+1)*(14*n-11)/6); # G. C. Greubel, Aug 30 2019

[n*(n+1)*(14*n-11)/6: n in [0..40]] // G. C. Greubel, Aug 30 2019

A172076:=n->n*(n+1)*(14*n-11)/6; seq(A172076(n), n=0..50); # Wesley Ivan Hurt, Feb 26 2014

LinearRecurrence[{4,-6,4,-1}, {0, 1, 17, 62}, 50] (* Vincenzo Librandi, Mar 01 2012 *)

vector(40, n, n*(n-1)*(14*n-25)/6) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(14*n-11)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1+13*x)/(1-x)^4. - Bruno Berselli, Dec 15 2010
a(n) = Sum_{i=0..n} A051868(i). - Bruno Berselli, Dec 15 2010
a(n) = Sum_{i=0..n-1} (n-i)*(14*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(6 + 45*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019

A172082 a(n) = n(n+1)(6*n-5)/2.

Original entry on oeis.org

0, 1, 21, 78, 190, 375, 651, 1036, 1548, 2205, 3025, 4026, 5226, 6643, 8295, 10200, 12376, 14841, 17613, 20710, 24150, 27951, 32131, 36708, 41700, 47125, 53001, 59346, 66178, 73515, 81375, 89776, 98736, 108273, 118405, 129150, 140526

Offset: 0

Vincenzo Librandi, Jan 25 2010

Generated by formula: n*(n+1)*(2*d*n-2*d+3)/6 with d=9.

This sequence is related to A051682 by a(n) = n*A051682(n) - Sum_{i=0..n-1} A051682(i); in fact this is the case d=9 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n -2*d + 3)/6. - Bruno Berselli, Apr 16 2012

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. A051682.

Cf. similar sequences listed in A237616.

List([0..40], n-> n*(n+1)*(6*n-5)/2); # G. C. Greubel, Aug 30 2019

[(18*n^3+3*n^2-15*n)/6: n in [0..40]]; // Vincenzo Librandi, Jan 02 2014

seq(n*(n+1)*(6*n-5)/2, n=0..40); # G. C. Greubel, Aug 30 2019

Table[(18n^3+3n^2-15n)/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,21,78}, 40] (* Harvey P. Dale, Jun 29 2011 *)
CoefficientList[Series[x*(1+17*x)/(1-x)^4, {x,0,40}], x] (* Vincenzo Librandi, Jan 02 2014 *)

vector(40, n, n*(n-1)*(6*n-11)/2) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(6*n-5)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

a(0)=0, a(1)=1, a(2)=21, a(3)=78; for n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jun 29 2011
G.f.: x*(1+17*x)/(1-x)^4. - Harvey P. Dale, Jun 29 2011
a(n) = Sum_{i=0..n-1} (n-i)*(18*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(2 + 19*x + 6*x^2)*exp(x)/2. - G. C. Greubel, Aug 30 2019
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(3*sqrt(3)*Pi + 9*log(3) + 12*log(2) - 5)/55.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(6*Pi + 6*sqrt(3)*log(sqrt(3)+2) - 16*log(2) + 5)/55. (End)

A172117 a(n) = n(n+1)(20*n-17)/6.

Original entry on oeis.org

0, 1, 23, 86, 210, 415, 721, 1148, 1716, 2445, 3355, 4466, 5798, 7371, 9205, 11320, 13736, 16473, 19551, 22990, 26810, 31031, 35673, 40756, 46300, 52325, 58851, 65898, 73486, 81635, 90365, 99696, 109648, 120241, 131495, 143430, 156066

Offset: 0

Vincenzo Librandi, Jan 26 2010

Generated by the formula n*(n+1)*(2*d*n-2*d+3)/6 for d=10.
This sequence is related to A051624 by a(n) = n*A051624(n) - Sum_{i=0..n-1} A051624(i) = n*(n+1)*(20*n-17)/2; in fact, this is the case d=10 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Aug 26 2010
Also, a(n) = n*A190816(n) - Sum_{i=0..n-1} A190816(i) for n>0. - Bruno Berselli, Dec 18 2013
Starting with offset 1, the sequence is the binomial transform of (1, 22, 41, 20, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 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 entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051624.

Cf. similar sequences listed in A237616.

[n*(n+1)*(20*n-17)/6: n in [0..50]]; // Vincenzo Librandi, Aug 01 2015

Table[(20n^3+3n^2-17n)/6,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,1,23,86},40] (* Harvey P. Dale, May 15 2011 *)

a(n)=n*(20*n^2+3*n-17)/6 \\ Charles R Greathouse IV, Jan 11 2012

[sum( (-1)^j*(20-j)*binomial(n+2-j, 3-j) for j in (0..1)) for n in (0..50)] # G. C. Greubel, Apr 15 2022

G.f.: x*(1+19*x)/(1-x)^4. - Bruno Berselli, Aug 26 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Harvey P. Dale, May 15 2011
a(n) = Sum_{i=0..n-1} (n-i)*(20*i+1), with a(0)=0. - Bruno Berselli, Feb 11 2014
E.g.f.: (1/6)*x*(6 + 63*x + 20*x^2)*exp(x). - G. C. Greubel, Apr 15 2022

A256716 a(n) = n(n+1)(22*n-19)/6.

Original entry on oeis.org

0, 1, 25, 94, 230, 455, 791, 1260, 1884, 2685, 3685, 4906, 6370, 8099, 10115, 12440, 15096, 18105, 21489, 25270, 29470, 34111, 39215, 44804, 50900, 57525, 64701, 72450, 80794, 89755, 99355, 109616, 120560, 132209, 144585, 157710, 171606, 186295, 201799, 218140

Offset: 0

Bruno Berselli, Apr 09 2015

This sequence is related to the tridecagonal numbers (A051865) by a(n) = n*A051865(n) - Sum_{i=0..n-1} A051865(i).

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (22nd row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A051876.
Cf. similar sequences listed in A237616.
Cf. A051865.

Magma
```
[n*(n+1)*(22*n-19)/6: n in [0..40]];
```

Table[n (n + 1) (22 n - 19)/6, {n, 0, 40}]

vector(40, n, n--; n*(n+1)*(22*n-19)/6)

Sage
```
[n*(n+1)*(22*n-19)/6 for n in (0..40)]
```

G.f.: x*(1 + 21*x)/(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with n>3, a(0)=0, a(1)=1, a(2)=25, a(3)=94.
a(n) = Sum_{i=0..n-1} (n-i)*(22*i+1) for n>0.
E.g.f.: exp(x)*x*(6 + 69*x + 22*x^2)/6. - Elmo R. Oliveira, Aug 04 2025

A177890 15-gonal (or pentadecagonal) pyramidal numbers: a(n) = n(n+1)(13*n-10)/6.

Original entry on oeis.org

0, 1, 16, 58, 140, 275, 476, 756, 1128, 1605, 2200, 2926, 3796, 4823, 6020, 7400, 8976, 10761, 12768, 15010, 17500, 20251, 23276, 26588, 30200, 34125, 38376, 42966, 47908, 53215, 58900, 64976, 71456, 78353, 85680, 93450, 101676, 110371, 119548, 129220

Offset: 0

Bruno Berselli, Dec 14 2010

Also a(n) = (15-m)*A000292(n-1) + n*(n+1)*((m-2)*n - (m-5))/6 being n*(n+1)*((m-2)*n - (m-5))/6 a m-gonal pyramidal number (1 < m < 15). For m=6, a(n) = 9*A000292(n-1) + A002412(n).

Inverse binomial transform of this sequence: 0, 1, 14, 13, 0, 0 (0 continued).

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (thirteenth row of the table).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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. A000292, A002412, A051867.

List([0..40], n-> n*(n+1)*(13*n-10)/6); # G. C. Greubel, Aug 30 2019

I:=[0,1,16,58]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2) +4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 04 2012

[n*(n+1)*(13*n-10)/6: n in [0..40]]; // G. C. Greubel, Aug 30 2019

seq(n*(n+1)*(13*n-10)/6, n=0..40); # G. C. Greubel, Aug 30 2019

CoefficientList[Series[x*(1+12*x)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 04 2012 *)
Table[n*(n-1)*(13*n-23)/6, {n,40}] (* G. C. Greubel, Aug 30 2019 *)
LinearRecurrence[{4,-6,4,-1},{0,1,16,58},40] (* Harvey P. Dale, Dec 21 2022 *)

vector(40, n, n*(n-1)*(13*n-23)/6) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(13*n-10)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1+12*x)/(1-x)^4.
a(n) = Sum_{i=0..n} A051867(i).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 04 2012
a(n) = Sum_{i=0..n-1} (n-i)*(13*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(6 + 42*x + 13*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019

A237617 a(n) = n(n + 1)(17*n - 14)/6.

Original entry on oeis.org

0, 1, 20, 74, 180, 355, 616, 980, 1464, 2085, 2860, 3806, 4940, 6279, 7840, 9640, 11696, 14025, 16644, 19570, 22820, 26411, 30360, 34684, 39400, 44525, 50076, 56070, 62524, 69455, 76880, 84816, 93280, 102289, 111860, 122010, 132756, 144115, 156104, 168740

Offset: 0

Bruno Berselli, Feb 11 2014

Also 19-gonal (or nonadecagonal) pyramidal numbers.

This sequence is related to A180232 by 2*a(n) = n*A180232(n) - Sum_{i=0..n-1} A180232(i).

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  18;
   3,  36,  35;
   4,  54,  70,  52;
   5,  72, 105, 104,  69;
   6,  90, 140, 156, 138,  86;
   7, 108, 175, 208, 207, 172, 103;
   8, 126, 210, 260, 276, 258, 206, 120;
   9, 144, 245, 312, 345, 344, 309, 240, 137;
  10, 162, 280, 364, 414, 430, 412, 360, 274, 154; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 17*r - 16 and T(r,c) = T(r-1,c) + T(r,r) = (r-c+1) * T(r,r), with r>=c>0.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (seventeenth row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pyramidal Number.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051871, A180232.

Cf. similar sequences listed in A237616.

List([0..40], n-> n*(n+1)*(17*n-14)/6); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(n+1)*(17*n-14)/6: n in [0..40]];
```

I:=[0,1,20,74]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2) +4*Self(n-3)-Self(n-4) : n in [1..40]]; // Vincenzo Librandi, Feb 12 2014

seq(n*(n+1)*(17*n-14)/6, n=0..40); # G. C. Greubel, Aug 30 2019

Table[n(n+1)(17*n-14)/6, {n, 0, 40}]
CoefficientList[Series[x(1+16x)/(1-x)^4, {x,0,40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,1,20,74},40] (* Harvey P. Dale, Aug 04 2021 *)

vector(40, n, n*(n-1)*(17*n-31)/6) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(17*n-14)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1 + 16*x)/(1 - x)^4.
For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(17*i+1); see the generalization in A237616 (Formula field).
E.g.f.: x*(6 + 54*x + 17*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019

cp's OEIS Frontend

A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n*(n+1)*(10*n-7)/6.

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

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A172078 a(n) = n*(16*n^2 + 3*n - 13)/6.

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A050441 Partial sums of A051865.

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A172076 a(n) = n*(n+1)*(14*n-11)/6.

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A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n(n+1)(10*n-7)/6.

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

A172078 a(n) = n(16n^2 + 3*n - 13)/6.

A172076 a(n) = n(n+1)(14*n-11)/6.

A172082 a(n) = n(n+1)(6*n-5)/2.

A172117 a(n) = n(n+1)(20*n-17)/6.

A256716 a(n) = n(n+1)(22*n-19)/6.

A177890 15-gonal (or pentadecagonal) pyramidal numbers: a(n) = n(n+1)(13*n-10)/6.

A237617 a(n) = n(n + 1)(17*n - 14)/6.