A051624 -id:A051624 - cp's OEIS frontend

A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

1, 1, 7, 7, 9, 5, 6, 0, 5, 7, 9, 2, 2, 6, 6, 3, 8, 5, 8, 7, 3, 5, 1, 7, 3, 9, 6, 8, 0, 9, 1, 8, 8, 7, 4, 1, 8, 4, 4, 5, 8, 5, 7, 2, 3, 4, 5, 6, 6, 6, 7, 9, 8, 0, 2, 8, 4, 2, 5, 2, 2, 8, 5, 7, 3, 2, 6, 6, 8, 9, 2, 5, 6, 8, 2, 8, 4, 8, 8, 7, 4, 5, 4, 0, 2, 4, 0, 7, 6, 9, 0, 2, 5, 6, 9, 5, 5, 9, 0, 3, 2, 2, 4, 4, 4

Offset: 1

Robert G. Wilson v, Jul 03 2014

From Wolfdieter Lang, Nov 09 2017: (Start)
In the Downey et al. link this is the instance k = 5 of the formula given there for S_{2*k+2}. A simpler formula is given in the Koecher reference as (5/4)*v_5(1) on p. 192. See the Kotesovec formula given below.
The partial sums are given in A294520/A294521. (End)

			1.1779560579226638587351739680918874184458572345666798028425228573...

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
Wikipedia, Polygonal number

Cf. A000038, A013661, A051624, A244639, A244644, A244645, A244646, A244647, A244648, A294520/A294521.

RealDigits[ Sum[1/(5n^2 - 4n), {n, 1 , Infinity}], 10, 111][[1]]

Equals Sum_{n>=1} 1/(5n^2 - 4n).
Equals Pi/8*sqrt(1+2/sqrt(5)) + (5*log(5) + sqrt(5)*log((3+sqrt(5))/2))/16. - Vaclav Kotesovec, Jul 04 2014
This is the value given in  the Koecher reference (see a comment above), and rewritten with the golden section phi = (1 + sqrt(5))/2 this becomes
  ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8. - Wolfdieter Lang, Nov 09 2017

A210982 Zero together with A126264 and positive terms of A051624 interleaved.

Original entry on oeis.org

0, 1, 8, 12, 26, 33, 54, 64, 92, 105, 140, 156, 198, 217, 266, 288, 344, 369, 432, 460, 530, 561, 638, 672, 756, 793, 884, 924, 1022, 1065, 1170, 1216, 1328, 1377, 1496, 1548, 1674, 1729, 1862, 1920, 2060, 2121, 2268, 2332, 2486, 2553, 2714, 2784, 2952, 3025, 3200, 3276, 3458, 3537, 3726

Offset: 0

Omar E. Pol, Aug 03 2012

Vertex number of a square spiral similar to A195162.

This is the case k=5 of the formula b(n,k) = ( 2*(k+5)*n^2+2*(k+3)*n-(k+1)+(2*(k-1)*n+k+1)*(-1)^n )/16. Sequences of the same family: A093025 (k=-1, with an initial 0), A210977 (k=0), A006578 (k=1), A210978 (k=2), A181995 (k=3, with one 0 only), A210981 (k=4). - Luce ETIENNE, Oct 30 2014

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

Members of this family are A093005, A210977, A006578, A210978, A181995, A210981, this sequence.

Cf. A001622, A051624, A126264, A195162.

[(10*n^2+8*n-3+(4*n+3)*(-1)^n )/8: n in [0..60]]; // Vincenzo Librandi, Oct 31 2014

Table[(10*n^2 + 8*n - 3 + (4*n + 3)*(-1)^n)/8, {n, 0, 50}] (* G. C. Greubel, Aug 23 2017 *)

my(x='x+O('x^50)); Vec(x*(1+7*x+2*x^2)/((1+x)^2*(1-x)^3)) \\ G. C. Greubel, Aug 23 2017

G.f.: x*(1+7*x+2*x^2) / ( (1+x)^2*(1-x)^3 ). - R. J. Mathar, Aug 07 2012
a(n) = (10*n^2 +8*n -3 +(4*n+3)*(-1)^n)/8. - Luce ETIENNE, Oct 14 2014
E.g.f.: (1/8)*((10*x^3 + 18*x -3)*exp(x) - (4*x - 3)*exp(-x)). - G. C. Greubel, Aug 23 2017
Sum_{n>=1} 1/a(n) = 5/9 + (sqrt(1-2/sqrt(5))/6 + sqrt(1+2/sqrt(5))/8)*Pi + 7*log(phi)*sqrt(5)/24 - 5*log(5)/48, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 21 2022

A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)(5k + 1) = A051624(k+1), for k >= 0.

Original entry on oeis.org

1, 13, 49, 795, 84179, 366829, 11417459, 103067441, 4235695001, 97604192047, 1661825059679, 1663957022369, 101611584435869, 101706166053389, 7226964017429851, 17176158550059533, 154681745346189277, 6654999228519884521, 6658297729691103841, 21316057915886595965, 2153790894613123442641

Offset: 0

Wolfdieter Lang, Nov 15 2017

The corresponding denominators are given in A294521.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [5,1].
The limit of the series is V(5,1) = lim_{n -> oo} V(5,1;n) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8, with the golden section phi:= (1 + sqrt(5))/2. The value is 1.17795605792266... given in A244649.

			The rationals V(5,1;n), n >= 0, begin: 1, 13/12, 49/44, 795/704, 84179/73920, 366829/320320, 11417459/9929920, 103067441/89369280, 4235695001/3664140480, 97604192047/84275231040, 1661825059679/1432678927680, ...
V(5,1;10^6) = 1.177956058 (Maple, 10 digits) to be compared with 1.177956058 obtained from V(5,1) given in A244649.

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.

G. C. Greubel, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001620, A051624, A244649, A294512, A294516/A294517, A294521.

[Numerator((&+[1/((k+1)*(5*k+1)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018

Table[Numerator[Sum[1/((k + 1)*(5*k + 1)), {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Aug 29 2018 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 1)))); \\ Michel Marcus, Nov 15 2017

a(n) = numerator(V(5,1;n)) with V(5,1;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 1)) = Sum_{k=0..n} 1/A051624(k+1) = (1/4)*Sum_{k=0..n} (1/(k + 1/5) - 1/(k+1)) = (-Psi(1/5) + Psi(n+6/5) - (gamma + Psi(n+2)))/4, with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

A294521 Denominators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)(5k + 1) = A051624(k+1), for k >= 0.

Original entry on oeis.org

1, 12, 44, 704, 73920, 320320, 9929920, 89369280, 3664140480, 84275231040, 1432678927680, 1432678927680, 87393414588480, 87393414588480, 6204932435782080, 14736714534982440, 132630430814841960, 5703108525038204280, 5703108525038204280, 18249947280122253696, 1843244675292347623296

Offset: 0

Wolfdieter Lang, Nov 15 2017

The corresponding numerators are given in A294520. Details are found there.

			See A294520 for the rationals.

Cf. A294520.

a(n) = denominator(sum(k=0, n, 1/((k + 1)*(5*k + 1)))); \\ Michel Marcus, Nov 15 2017

a(n) = denominator(V(5,1;n)) with V(5,1;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 1)) = Sum_{k=0..n} 1/A051624(k+1) = (1/4)*Sum_{k=0..n} (1/(k + 1/5) - 1/(k+1)). For the formula in terms of the digamma function see A294520.

A093646 Higher dimensional figurate numbers based on 12-gonal numbers A051624.

Original entry on oeis.org

1, 19, 145, 715, 2695, 8437, 23023, 56485, 127270, 267410, 529958, 999362, 1805570, 3142790, 5293970, 8662214, 13810511, 21511325, 32807775, 49088325, 72177105, 104442195, 148924425, 209489475, 291006300, 399555156, 542668764, 729610420

Offset: 0

Wolfdieter Lang, Apr 22 2004

Cf. A093645 ((10, 1) Pascal, column m=9).

a(n)= (10*n+9)*binomial(n+8, 8)/9. G.f.: (1+9*x)/(1-x)^10.

A195162 Generalized 12-gonal numbers: k(5k-4) for k = 0, +-1, +-2, ...

Original entry on oeis.org

0, 1, 9, 12, 28, 33, 57, 64, 96, 105, 145, 156, 204, 217, 273, 288, 352, 369, 441, 460, 540, 561, 649, 672, 768, 793, 897, 924, 1036, 1065, 1185, 1216, 1344, 1377, 1513, 1548, 1692, 1729, 1881, 1920, 2080, 2121, 2289, 2332, 2508, 2553, 2737, 2784, 2976, 3025

Offset: 0

Omar E. Pol, Sep 10 2011

Also generalized dodecagonal numbers.
Second 12-gonal numbers (A135705) and positive terms of A051624 interleaved. - Omar E. Pol, Aug 04 2012
The characteristic function of this sequence is A205988. - Jason Kimberley, Nov 15 2012
Also, integer values of m*(m+4)/5. - Bruno Berselli, Dec 05 2012
Also, numbers h such that 5*h + 4 is a square. - Bruno Berselli, Oct 10 2013
Exponents in expansion of Product_{n >= 1} (1 + x^(10*n-9))*(1 + x^(10*n-1))*(1 - x^(10*n)) = 1 + x + x^9 + x^12 + x^28 + .... - Peter Bala, Dec 10 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares, Discrete Math. , Vol. 274, No. 1-3 (2004), pp. 9-24. See E(q).
John Elias, Generalized 12-gonal & 20-gonal Cross Configurations
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Partial sums of A195161.
Column 8 of A195152.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), this sequence (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Cf. A001622, A051624, A135705.
Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

List([0..50], n-> (10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8); # G. C. Greubel, Jul 04 2019

[0] cat &cat[[5*n^2-4*n, 5*n^2+4*n]: n in [1..25]]; // Vincenzo Librandi, Sep 26 2011

nn = 25; Sort[Table[n*(5*n - 4), {n, -nn, nn}]] (* T. D. Noe, Sep 23 2011 *)

vector(50, n, n--; (10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8) \\ G. C. Greubel, Jul 04 2019

[(10*n^2 +10*n -3 +3*(-1)^n*(2*n+1))/8 for n in (0..50)] # G. C. Greubel, Jul 04 2019

From R. J. Mathar, Sep 24 2011: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = A008805(n-1) + A008805(n-3) + 8*A008805(n-2). (End)
From Bruno Berselli, Sep 26 2011: (Start)
G.f.: x*(1+8*x+x^2)/((1+x)^2*(1-x)^3).
a(n) = (10*n*(n+1) + 3*(2*n+1)*(-1)^n - 3)/8.
a(n) = a(-n-1). (End)
Sum_{n>=1} 1/a(n) = (5 + 4*sqrt(1 + 2/sqrt(5))*Pi)/16. - Vaclav Kotesovec, Oct 05 2016
E.g.f.: (3*(1 - 2*x)*exp(-x) + (-3 +20*x +10*x^2)*exp(x))/8. - G. C. Greubel, Jul 04 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5)/8 + sqrt(5)*log(phi)/4 - 5/16, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 28 2022

A139600 Square array T(n,k) = n(k-1)k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.

The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011

			The square array of nonnegatives together with polygonal numbers begins:
=========================================================
....................... A   A   .   .   A    A    A    A
....................... 0   0   .   .   0    0    1    1
....................... 0   0   .   .   1    1    3    3
....................... 0   0   .   .   6    7    9    9
....................... 0   0   .   .   9    3    6    6
....................... 0   1   .   .   5    2    0    0
....................... 4   2   .   .   7    9    6    7
=========================================================
Nonnegatives . A001477: 0,  1,  2,  3,  4,   5,   6,   7, ...
Triangulars .. A000217: 0,  1,  3,  6, 10,  15,  21,  28, ...
Squares ...... A000290: 0,  1,  4,  9, 16,  25,  36,  49, ...
Pentagonals .. A000326: 0,  1,  5, 12, 22,  35,  51,  70, ...
Hexagonals ... A000384: 0,  1,  6, 15, 28,  45,  66,  91, ...
Heptagonals .. A000566: 0,  1,  7, 18, 34,  55,  81, 112, ...
Octagonals ... A000567: 0,  1,  8, 21, 40,  65,  96, 133, ...
9-gonals ..... A001106: 0,  1,  9, 24, 46,  75, 111, 154, ...
10-gonals .... A001107: 0,  1, 10, 27, 52,  85, 126, 175, ...
11-gonals .... A051682: 0,  1, 11, 30, 58,  95, 141, 196, ...
12-gonals .... A051624: 0,  1, 12, 33, 64, 105, 156, 217, ...
...
=========================================================
The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.

Alois P. Heinz, Rows n = 0..140, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.
Index to sequences related to polygonal numbers

Cf. A001477, A057145, A139601, A139617, A139618, A139620, A180662.
A formal extension negative n is in A326728.
Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011
Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).

T:= func< n,k | k*(n*(k-1)+2)/2 >;
A139600:= func< n,k | T(n-k, k) >;
[A139600(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T:= (n, k)-> n*(k-1)*k/2+k:
seq(seq(T(d-k, k), k=0..d), d=0..14);  # Alois P. Heinz, Oct 14 2018

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def A139600Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield x
        x, y = x + y + n, y + n
for n in range(8):
    R = A139600Row(n)
    print([next(R) for  in range(11)]) # _Peter Luschny, Aug 04 2019

def T(n,k): return k*(n*(k-1)+2)/2
def A139600(n,k): return T(n-k, k)
flatten([[A139600(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = n*(k-1)*k/2+k.
T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016
From Stefano Spezia, Apr 12 2024: (Start)
G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)

Edited by Omar E. Pol, Jan 05 2009

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

A016766 a(n) = (3*n)^2.

Original entry on oeis.org

0, 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296, 1521, 1764, 2025, 2304, 2601, 2916, 3249, 3600, 3969, 4356, 4761, 5184, 5625, 6084, 6561, 7056, 7569, 8100, 8649, 9216, 9801, 10404, 11025, 11664, 12321, 12996, 13689, 14400, 15129, 15876, 16641, 17424

Offset: 0

N. J. A. Sloane

Number of edges of the complete tripartite graph of order 6n, K_n, n, 4n. - Roberto E. Martinez II, Jan 07 2002
Area of a square with side 3n. - Wesley Ivan Hurt, Sep 24 2014
Right-hand side of the binomial coefficient identity Sum_{k = 0..3*n} (-1)^(n+k+1)* binomial(3*n,k)*binomial(3*n + k,k)*(3*n - k) = a(n). - Peter Bala, Jan 12 2022

Ivan Panchenko, Table of n, a(n) for n = 0..200
John Elias, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Numbers of the form 9*n^2 + k*n, for integer n: this sequence (k = 0), A132355 (k = 2), A185039 (k = 4), A057780 (k = 6), A218864 (k = 8). - Jason Kimberley, Nov 09 2012

Cf. A000217, A000290, A008585, A033428, A051624, A086089, A195042.

[(3*n)^2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 24 2014

A016766:=n->(3*n)^2: seq(A016766(n), n=0..50); # Wesley Ivan Hurt, Sep 24 2014

(3Range[0, 49])^2 (* Alonso del Arte, Sep 24 2014 *)

A016766(n):=(3*n)^2$
makelist(A016766(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=9*n^2 \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 9*n^2 = 9*A000290(n). - Omar E. Pol, Dec 11 2008
a(n) = 3*A033428(n). - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 9*(2*n-1) for n > 0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
From Wesley Ivan Hurt, Sep 24 2014: (Start)
G.f.: 9*x*(1 + x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
a(n) = A000290(A008585(n)). (End)
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/54.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/108.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/3)/(Pi/3).
Product_{n>=1} (1 - 1/a(n)) = sinh(Pi/2)/(Pi/2) = 3*sqrt(3)/(2*Pi) (A086089). (End)
a(n) = A051624(n) + 8*A000217(n).  In general, if P(k,n) = the k-th n-gonal number, then (k*n)^2 = P(k^2 + 3,n) + (k^2 - 1)*A000217(n). - Charlie Marion, Mar 09 2022
From Elmo R. Oliveira, Nov 30 2024: (Start)
E.g.f.: 9*x*(1 + x)*exp(x).
a(n) = n*A008591(n) = A195042(2*n). (End)

More terms from Zerinvary Lajos, May 30 2006

A131242 Partial sums of A059995: a(n) = sum_{k=0..n} floor(k/10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198

Offset: 0

Hieronymus Fischer, Jun 21 2007

Complementary with A130488 regarding triangular numbers, in that A130488(n)+10*a(n)=n(n+1)/2=A000217(n).

			As square array :
    0,   0,   0,   0,   0,   0,   0,   0,   0,    0
    1,   2,   3,   4,   5,   6,   7,   8,   9,   10
   12,  14,  16,  18,  20,  22,  24,  26,  28,   30
   33,  36,  39,  42,  45,  48,  51,  54,  57,   60
   64,  68,  72,  76,  80,  84,  88,  92,  96,  100
  105, 110, 115, 120, 125, 130, 135, 140, 145,  150
  156, 162, 168, 174, 180, 186, 192, 198, 204,  210
... - _Philippe Deléham_, Mar 27 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Cf. A008728, A059995, A010879, A002266, A130488, A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470.

Table[(1/2)*Floor[n/10]*(2*n - 8 - 10*Floor[n/10]), {n,0,50}] (* G. C. Greubel, Dec 13 2016 *)
Accumulate[Table[FromDigits[Most[IntegerDigits[n]]],{n,0,110}]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,1,2},120] (* Harvey P. Dale, Apr 06 2017 *)

for(n=0,50, print1((1/2)*floor(n/10)*(2n-8-10*floor(n/10)), ", ")) \\ G. C. Greubel, Dec 13 2016

a(n)=my(k=n\10); k*(n-5*k-4) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (1/2)*floor(n/10)*(2n-8-10*floor(n/10)).
a(n) = A059995(n)*(2n-8-10*A059995(n))/2.
a(n) = (1/2)*A059995(n)*(n-8+A010879(n)).
a(n) = (n-A010879(n))*(n+A010879(n)-8)/20.
G.f.: x^10/((1-x^10)(1-x)^2).
From Philippe Deléham, Mar 27 2013: (Start)
a(10n)   = A051624(n).
a(10n+1) = A135706(n).
a(10n+2) = A147874(n+1).
a(10n+3) = 2*A005476(n).
a(10n+4) = A033429(n).
a(10n+5) = A202803(n).
a(10n+6) = A168668(n).
a(10n+7) = 2*A147875(n).
a(10n+8) = A135705(n).
a(10n+9) = A124080(n). (End)
a(n) = A008728(n-10) for n>= 10. - Georg Fischer, Nov 03 2018

cp's OEIS Frontend

A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A210982 Zero together with A126264 and positive terms of A051624 interleaved.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)*(5*k + 1) = A051624(k+1), for k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A294521 Denominators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)*(5*k + 1) = A051624(k+1), for k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A093646 Higher dimensional figurate numbers based on 12-gonal numbers A051624.

Views

Author

Keywords

Crossrefs

Formula

A195162 Generalized 12-gonal numbers: k*(5*k-4) for k = 0, +-1, +-2, ...

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)(5k + 1) = A051624(k+1), for k >= 0.

A294521 Denominators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)(5k + 1) = A051624(k+1), for k >= 0.

A195162 Generalized 12-gonal numbers: k(5k-4) for k = 0, +-1, +-2, ...

A139600 Square array T(n,k) = n(k-1)k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.