A007587 -id:A007587 - cp's OEIS frontend

A051799 Partial sums of A007587.

1, 14, 60, 170, 385, 756, 1344, 2220, 3465, 5170, 7436, 10374, 14105, 18760, 24480, 31416, 39729, 49590, 61180, 74690, 90321, 108284, 128800, 152100, 178425, 208026, 241164, 278110, 319145, 364560, 414656, 469744, 530145, 596190

Offset: 0

Barry E. Williams, Dec 11 1999

4-dimensional pyramidal number, composed of consecutive 3-dimensional slices; each of which is a 3-dimensional 12-gonal (or dodecagonal) pyramidal number; which in turn is composed of consecutive 2-dimensional slices 12-gonal numbers. - Jonathan Vos Post, Mar 17 2006

Convolution of A000027 with A051624 (excluding 0). - Bruno Berselli, Dec 07 2012

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A007587, A001622, A051624.
Cf. A093645 ((10, 1) Pascal, column m=4).
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

/* A000027 convolved with A051624 (excluding 0): */ A051624:=func; [&+[(n-i+1)*A051624(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Accumulate[Table[n(n+1)(10n-7)/6,{n,0,50}]] (* Harvey P. Dale, Nov 13 2013 *)

a(n) = C(n+3, 3)*(5*n+2)/2 = (n+1)*(n+2)*(n+3)*(5*n+2)/12.
G.f.: (1+9*x)/(1-x)^5.
From Amiram Eldar, Feb 11 2022: (Start)
Sum_{n>=0} 1/a(n) = (125*log(5) + 10*sqrt(5*(5-2*sqrt(5)))*Pi - 50*sqrt(5)*log(phi) - 84)/104, where phi is the golden ratio (A001622).
Sum_{n>=0} (-1)^n/a(n) = (50*sqrt(5)*log(phi) + 5*sqrt(50-10*sqrt(5))*Pi - 256*log(2) + 90)/52. (End)

A051624 12-gonal (or dodecagonal) numbers: a(n) = n(5n-4).

Original entry on oeis.org

0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 793, 924, 1065, 1216, 1377, 1548, 1729, 1920, 2121, 2332, 2553, 2784, 3025, 3276, 3537, 3808, 4089, 4380, 4681, 4992, 5313, 5644, 5985, 6336, 6697, 7068, 7449, 7840, 8241, 8652

Offset: 0

Barry E. Williams

Zero followed by partial sums of A017281. - Klaus Brockhaus, Nov 20 2008
Sequence found by reading the line from 0, in the direction 0, 12, ... and the parallel line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized 12-gonal numbers A195162. - Omar E. Pol, Jul 18 2012
This is also a star hexagonal number: a(n) = A000384(n) + 6*A000217(n-1). - Luciano Ancora, Mar 30 2015
Starting with offset 1, this is the binomial transform of (1, 11, 10, 0, 0, 0, ...). - Gary W. Adamson, Aug 01 2015
a(n+1) is the sum of the odd numbers from 4n+1 to 6n+1. - Wesley Ivan Hurt, Dec 14 2015
For n >= 2, a(n) is the number of intersection points of all unit circles centered on the inner lattice points of an (n+1) X (n+1) square grid. - Wesley Ivan Hurt, Dec 08 2020
The final digit of a(n) equals the final digit of n, A010879(n). - Enrique Pérez Herrero, Nov 13 2022
a(n-1) is the maximum second Zagreb index of maximal 2-degenerate graphs with n vertices.  (The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.) - Allan Bickle, Apr 16 2024

			The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

T. D. Noe, Table of n, a(n) for n = 0..1000
John Elias, Illustration: compass configuration , Illustration: cross configuration.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Wikipedia, Dodecagonal number
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A007587.
Cf. A093645 ((10, 1) Pascal, column m=2). Partial sums of A017281.
Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).
Cf. A372027 (second Zagreb index of MOPs).

[ n eq 1 select 0 else Self(n-1)+10*(n-2)+1: n in [1..43] ]; // Klaus Brockhaus, Nov 20 2008

RecurrenceTable[{a[0]==0, a[1]==1, a[2]==12, a[n]== 3*a[n-1] - 3*a[n-2] + a[n-3]}, a, {n, 30}] (* G. C. Greubel, Jul 31 2015 *)
Table[n*(5*n - 4), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

a(n)=(5*n-4)*n \\ Charles R Greathouse IV, Jun 16 2011

G.f.: x*(1+9*x)/(1-x)^3.
a(n) = Sum_{k=0..n-1} 10*k+1. - Klaus Brockhaus, Nov 20 2008
a(n) = 10*n + a(n-1) - 9 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010
a(n) = A131242(10n). - Philippe Deléham, Mar 27 2013
a(10*a(n) + 46*n + 1) = a(10*a(n) + 46*n) + a(10*n+1). - Vladimir Shevelev, Jan 24 2014
E.g.f.: x*(5*x + 1) * exp(x). - G. C. Greubel, Jul 31 2015
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=12. - G. C. Greubel, Jul 31 2015
Sum_{n>=1} 1/a(n) = sqrt(1 + 2/sqrt(5))*Pi/8 + 5*log(5)/16 + sqrt(5)*log((1 + sqrt(5))/2)/8 = 1.177956057922663858735173968... . - Vaclav Kotesovec, Apr 27 2016
a(n) + 4*(n-1)^2 = (3*n-2)^2. Let P(k,n) be the n-th k-gonal number. Then, in general, P(4k,n) + (k-1)^2*(n-1)^2 = (k*n-k+1)^2. - Charlie Marion, Feb 04 2020
Product_{n>=2} (1 - 1/a(n)) = 5/6. - Amiram Eldar, Jan 21 2021
a(n) = (3*n-2)^2 - (2*n-2)^2. In general, if we let P(k,n) = the n-th k-gonal number, then P(4k,n) = (k*n-(k-1))^2 - ((k-1)*n-(k-1))^2. - Charlie Marion, Nov 11 2021

A093645 (10,1) Pascal triangle.

Original entry on oeis.org

1, 10, 1, 10, 11, 1, 10, 21, 12, 1, 10, 31, 33, 13, 1, 10, 41, 64, 46, 14, 1, 10, 51, 105, 110, 60, 15, 1, 10, 61, 156, 215, 170, 75, 16, 1, 10, 71, 217, 371, 385, 245, 91, 17, 1, 10, 81, 288, 588, 756, 630, 336, 108, 18, 1, 10, 91, 369, 876, 1344, 1386, 966, 444, 126, 19, 1

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(10;n,m) gives in the columns m >= 1 the figurate numbers based on A017281, including the 12-gonal numbers A051624 (see the W. Lang link).
This is the tenth member, d=10, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-5 and A093644 for d=1..9.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m is G(z,x) = (1+9*z)/(1-(1+x)*z).
The SW-NE diagonals give A022100(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k, k), n >= 1, with n=0 value 9. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

			Triangle begins
   1;
  10,  1;
  10, 11,  1;
  10, 21, 12,  1;
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch. 5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Lang, First 10 rows and array of figurate numbers .

Row sums: 1 for n=0 and A005015(n-1), n >= 1, alternating row sums are 1 for n=0, 9 for n=2 and 0 otherwise.

The column sequences give for m=1..9: A017281, A051624 (12-gonal), A007587, A051799, A051880, A050406, A052254, A056125, A093646.

a093645 n k = a093645_tabl !! n !! k
a093645_row n = a093645_tabl !! n
a093645_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [10, 1]
-- Reinhard Zumkeller, Aug 31 2014

t[0, 0] = 1; t[n_, k_] := Binomial[n, k] + 9*Binomial[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Philippe Deléham *)

a(n, m) = F(10;n-m, m) for 0 <= m <= n, else 0, with F(10;0, 0)=1, F(10;n, 0)=10 if n >= 1 and F(10;n, m):=(10*n+m)*binomial(n+m-1, m-1)/m if m >= 1.
Recursion: a(n, m)=0 if m > n, a(0, 0)=1; a(n, 0)=10 if n >= 1; a(n, m) = a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+9*x)/(1-x)^(m+1), m >= 0.
T(n, k) = C(n, k) + 9*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(10 + 21*x + 12*x^2/2! + x^3/3!) = 10 + 31*x + 64*x^2/2! + 110*x^3/3! + 170*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A237616 a(n) = n(n + 1)(5*n - 4)/2.

Original entry on oeis.org

0, 1, 18, 66, 160, 315, 546, 868, 1296, 1845, 2530, 3366, 4368, 5551, 6930, 8520, 10336, 12393, 14706, 17290, 20160, 23331, 26818, 30636, 34800, 39325, 44226, 49518, 55216, 61335, 67890, 74896, 82368, 90321, 98770, 107730, 117216, 127243, 137826, 148980, 160720

Offset: 0

Bruno Berselli, Feb 10 2014

Also 17-gonal (or heptadecagonal) pyramidal numbers.

This sequence is related to A226489 by 2*a(n) = n*A226489(n) - Sum_{i=0..n-1} A226489(i).

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  16;
   3,  32,  31;
   4,  48,  62,  46;
   5,  64,  93,  92,  61;
   6,  80, 124, 138, 122,  76;
   7,  96, 155, 184, 183, 152,  91;
   8, 112, 186, 230, 244, 228, 182, 106;
   9, 128, 217, 276, 305, 304, 273, 212, 121;
  10, 144, 248, 322, 366, 380, 364, 318, 242, 136; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 15*r-14 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 (fifteenth 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. A051869, A104728.

Cf. sequences with formula n*(n+1)*(k*n-k+3)/6: A000217 (k=0), A000292 (k=1), A000330 (k=2), A002411 (k=3), A002412 (k=4), A002413 (k=5), A002414 (k=6), A007584 (k=7), A007585 (k=8), A007586 (k=9), A007587 (k=10), A050441 (k=11), A172073 (k=12), A177890 (k=13), A172076 (k=14), this sequence (k=15), A172078(k=16), A237617 (k=17), A172082 (k=18), A237618 (k=19), A172117(k=20), A256718 (k=21), A256716 (k=22), A256645 (k=23), A256646(k=24), A256647 (k=25), A256648 (k=26), A256649 (k=27), A256650(k=28).

List([0..40], n-> n*(n+1)*(5*n-4)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(n+1)*(5*n-4)/2: n in [0..40]];
```

I:=[0,1,18,66]; [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)*(5*n-4)/2, n=0..40); # G. C. Greubel, Aug 30 2019

Table[n(n+1)(5n-4)/2, {n, 0, 40}]
CoefficientList[Series[x (1+14x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,66},50] (* Harvey P. Dale, Jan 11 2015 *)

a(n)=n*(n+1)*(5*n-4)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+1)*(5*n-4)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1 + 14*x)/(1 - x)^4.
For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(15*i+1). More generally, the sequence with the closed form n*(n+1)*(k*n-k+3)/6 is also given by Sum_{i=0..n-1} (n-i)*(k*i+1) for n>0.
a(n) = A104728(A001844(n-1)) for n>0.
Sum_{n>=1} 1/a(n) = (2*sqrt(5*(5 + 2*sqrt(5)))*Pi + 10*sqrt(5)*arccoth(sqrt(5)) + 25*log(5) - 16)/72 = 1.086617842136293176... . - Vaclav Kotesovec, Dec 07 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020
E.g.f.: exp(x)*x*(2 + 16*x + 5*x^2)/2. - Elmo R. Oliveira, Aug 04 2025

A172047 n(n+1)(15*n^2-n-8)/12.

Original entry on oeis.org

0, 1, 25, 124, 380, 905, 1841, 3360, 5664, 8985, 13585, 19756, 27820, 38129, 51065, 67040, 86496, 109905, 137769, 170620, 209020, 253561, 304865, 363584, 430400, 506025, 591201, 686700, 793324, 911905, 1043305, 1188416, 1348160, 1523489

Offset: 0

Vincenzo Librandi, Jan 24 2010

This sequence is related to A007587 by a(n) = n*A007587(n)-sum(i=0..n-1, A007587(i)).
This is the case d=5 in the general formula n*(n*(n+1)*(2*d*n-2*d+3)/6)-sum(i=0..n-1, i*(i+1)*(2*d*i-2*d+3)/6) = n*(n+1)*(3*d*n^2-d*n+4*n-2*d+2)/12.  - Bruno Berselli, Dec 07 2010
The inverse binomial transform yields 0, 1, 23, 52, 30, 0, 0 (0 continued).  - R. J. Mathar, Dec 09 2010

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).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A007587.

[n*(n+1)*(15*n^2-n-8)/12: n in [0..50]]; // Vincenzo Librandi, Jan 01 2014

CoefficientList[Series[x (1 + 20 x + 9 x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)

G.f.: -x*(1+20*x+9*x^2)/(x-1)^5. - R. J. Mathar, Dec 09 2010

a(n)-a(-n) = A063521(n). - Bruno Berselli, Aug 26 2011

A266088 Alternating sum of 12-gonal (or dodecagonal) numbers.

Original entry on oeis.org

0, -1, 11, -22, 42, -63, 93, -124, 164, -205, 255, -306, 366, -427, 497, -568, 648, -729, 819, -910, 1010, -1111, 1221, -1332, 1452, -1573, 1703, -1834, 1974, -2115, 2265, -2416, 2576, -2737, 2907, -3078, 3258, -3439, 3629, -3820, 4020, -4221, 4431, -4642

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

More generally, the ordinary generating function for the alternating sum of k-gonal numbers is -x*(1 - (k - 3)*x)/((1 - x)*(1 + x)^3).

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A006578, A007587, A035608, A051624, A083392, A089594.

[1+(-1)^n*(5*n^2+n-2)/2: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[1 + (-1)^n (5 n^2 + n - 2)/2, {n, 0, 43}]
CoefficientList[Series[-x (1 - 9 x)/((1 - x) (1 + x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

x='x+O('x^100); concat(0, Vec(-x*(1-9*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 9*x)/((1 - x)*(1 + x)^3).
a(n) = 1 + (-1)^n*(5*n^2 + n - 2)/2.
a(n) = Sum_{k = 0..n} (-1)^k*A051624(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.

cp's OEIS Frontend

A051799 Partial sums of A007587.

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A051624 12-gonal (or dodecagonal) numbers: a(n) = n*(5*n-4).

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A093645 (10,1) Pascal triangle.

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A237616 a(n) = n*(n + 1)*(5*n - 4)/2.

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A172047 n*(n+1)*(15*n^2-n-8)/12.

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A266088 Alternating sum of 12-gonal (or dodecagonal) numbers.

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A051624 12-gonal (or dodecagonal) numbers: a(n) = n(5n-4).

A237616 a(n) = n(n + 1)(5*n - 4)/2.

A172047 n(n+1)(15*n^2-n-8)/12.