A059845 -id:A059845 - cp's OEIS frontend

A151542 Generalized pentagonal numbers: a(n) = 12n + 3n*(n-1)/2.

0, 12, 27, 45, 66, 90, 117, 147, 180, 216, 255, 297, 342, 390, 441, 495, 552, 612, 675, 741, 810, 882, 957, 1035, 1116, 1200, 1287, 1377, 1470, 1566, 1665, 1767, 1872, 1980, 2091, 2205, 2322, 2442, 2565, 2691, 2820, 2952, 3087, 3225, 3366, 3510, 3657, 3807, 3960

Offset: 0

N. J. A. Sloane, May 15 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Leo Tavares, Illustration: Star Triangles
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

The generalized pentagonal numbers b*n + 3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Cf. A003154, A060544.

s=0;lst={};Do[AppendTo[lst,s+=n],{n,12,6!,3}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 05 2010 *)
LinearRecurrence[{3,-3,1}, {0,12,27}, 50] (* or *) With[{nn = 50}, CoefficientList[Series[(3/2)*(8*x + x^2)*Exp[x], {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 26 2017 *)

x='x+O('x^50); concat([0], Vec(serlaplace((3/2)*(8*x + x^2)*exp(x)))) \\ G. C. Greubel, May 26 2017

a(n)=(3*n^2+21*n)/2 \\ Charles R Greathouse IV, Jun 16 2017

a(n) = a(n-1) + 3*n + 9 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(4 - 3*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
From G. C. Greubel, May 26 2017: (Start)
a(n) = 3*n*(n+7)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*(8*x + x^2)*exp(x). (End)
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 121/490.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/21 - 319/4410. (End)
a(n) = A003154(n+1) - A060544(n). - Leo Tavares, Mar 26 2022

A140673 a(n) = 3n(n + 5)/2.

Original entry on oeis.org

0, 9, 21, 36, 54, 75, 99, 126, 156, 189, 225, 264, 306, 351, 399, 450, 504, 561, 621, 684, 750, 819, 891, 966, 1044, 1125, 1209, 1296, 1386, 1479, 1575, 1674, 1776, 1881, 1989, 2100, 2214, 2331, 2451, 2574, 2700, 2829, 2961, 3096

Offset: 0

Omar E. Pol, May 22 2008

a(n) equals the number of vertices of the A256666(n)-th graph (see Illustration of initial terms in A256666 Links). - Ivan N. Ianakiev, Apr 20 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A055998.

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Table[Sum[i + n - 3, {i, 6, n}], {n, 5, 52}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[3 n (n + 5)/2, {n, 0, 50}] (* Bruno Berselli, Sep 05 2018 *)
LinearRecurrence[{3,-3,1},{0,9,21},50] (* Harvey P. Dale, Jul 20 2023 *)

concat(0, Vec(3*x*(3 - 2*x)/(1 - x)^3 + O(x^100))) \\ Michel Marcus, Apr 20 2015

a(n) = 3*n*(n+5)/2; \\ Altug Alkan, Sep 05 2018

a(n) = A055998(n)*3 = (3*n^2 + 15*n)/2 = n*(3*n + 15)/2.
a(n) = 3*n + a(n-1) + 6 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(3 - 2*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 + 18*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 137/450.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/15 - 47/450. (End)

A140674 a(n) = n(3n + 17)/2.

Original entry on oeis.org

0, 10, 23, 39, 58, 80, 105, 133, 164, 198, 235, 275, 318, 364, 413, 465, 520, 578, 639, 703, 770, 840, 913, 989, 1068, 1150, 1235, 1323, 1414, 1508, 1605, 1705, 1808, 1914, 2023, 2135, 2250, 2368, 2489, 2613, 2740, 2870, 3003, 3139

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

[n*(3*n + 17)/2: n in [0..70]]; // Wesley Ivan Hurt, Apr 21 2021

Table[(3*n^2 + 17*n)/2, {n, 0, 50}] (* G. C. Greubel, Jul 17 2017 *)

a(n)=n*(3*n+17)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (3*n^2 + 17*n)/2.
a(n) = 7*n + 3*A000217(n). - Reinhard Zumkeller, May 28 2008
a(n) = 3*n + a(n-1) + 7 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
G.f.: x*(10 - 7*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (1/2)*(3*x^2 + 20*x)*exp(x). - G. C. Greubel, Jul 17 2017

A112414 a(n) = 3*n + 7.

Original entry on oeis.org

7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178, 181, 184

Offset: 0

Zerinvary Lajos, Dec 09 2005

At least the first 2 million terms from a(1) on coincide with the corresponding terms of A086822(n+2). - R. J. Mathar, Aug 15 2008

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016933, A059845.

3Range[0,70]+7 (* or *) LinearRecurrence[{2,-1},{7,10},70] (* Harvey P. Dale, May 12 2019 *)

a(n)=3*n+7 \\ Charles R Greathouse IV, Jul 10 2016

From Elmo R. Oliveira, Oct 30 2024: (Start)
G.f.: (7 - 4*x)/(1 - x)^2.
E.g.f.: (7 + 3*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = A016933(n+2)/2 = A059845(n+1) - A059845(n). (End)

Better definition from T. D. Noe, Nov 30 2006

A185874 Second accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 11, 10, 10, 21, 26, 20, 15, 34, 48, 50, 35, 21, 50, 76, 90, 85, 56, 28, 69, 110, 140, 150, 133, 84, 36, 91, 150, 200, 230, 231, 196, 120, 45, 116, 196, 270, 325, 350, 336, 276, 165, 55, 144, 248, 350, 435, 490, 504, 468, 375, 220, 66, 175, 306, 440, 560, 651, 700, 696, 630, 495, 286, 78, 209, 370, 540, 700, 833, 924, 960, 930, 825, 638, 364, 91, 246, 440, 650, 855, 1036, 1176, 1260, 1275, 1210, 1056, 806, 455, 105, 286, 516, 770, 1025, 1260, 1456, 1596, 1665, 1650, 1540, 1326, 1001, 560

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain: A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
.   1,   3,   6,   10,   15,   21,   28,   36,   45,   55, ...
.   4,  11,  21,   34,   50,   69,   91,  116,  144,  175, ...
.  10,  26,  48,   76,  110,  150,  196,  248,  306,  370, ...
.  20,  50,  90,  140,  200,  270,  350,  440,  540,  650, ...
.  35,  85, 150,  230,  325,  435,  560,  700,  855, 1025, ...
.  56, 133, 231,  350,  490,  651,  833, 1036, 1260, 1505, ...
.  84, 196, 336,  504,  700,  924, 1176, 1456, 1764, 2100, ...
. 120, 276, 468,  696,  960, 1260, 1596, 1968, 2376, 2820, ...
. 165, 375, 630,  930, 1275, 1665, 2100, 2580, 3105, 3675, ...
. 220, 495, 825, 1210, 1650, 2145, 2695, 3300, 3960, 4675, ...
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A144112, A051340, A141419, A185875, A185876.
Row 1 to 5: A000217, A115056, 2*A140096, 10*A000096, 5*A059845.
Column 1 to 3: A000292, A051925, A267370 and 3*A005581.
Main diagonal: A117066.

f[n_, k_] := (1/12)*k*n*(1 + n)*(1 + 3*k + 2*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten

T(n,k) = k*n*(n+1)*(2*n+3*k+1)/12 for k>=1, n>=1.

Edited by Bruno Berselli, Jan 14 2016

A095871 Triangle read by rows: T(n,k)=(n+1)(3(n+1)-1)/2-k(3k-1)/2.

Original entry on oeis.org

1, 5, 4, 12, 11, 7, 22, 21, 17, 10, 35, 34, 30, 23, 13, 51, 50, 46, 39, 29, 16, 70, 69, 65, 58, 48, 35, 19, 92, 91, 87, 80, 70, 57, 41, 22, 117, 116, 112, 105, 95, 82, 66, 47, 25, 145, 144, 140, 133, 123, 110, 94, 75, 53, 28, 176, 175, 171, 164, 154, 141, 125, 106, 84, 59

Offset: 1

Gary W. Adamson, Jun 10 2004, Jul 28 2008

Octagonal pyramidal number triangle, read by rows.
The triangle is generated from the product B*A of the infinite lower triangular matrices A =
  1 0 0 0...
  1 1 0 0...
  1 1 1 0...
  1 1 1 1...
and B =
  1 0 0 0...
  1 4 0 0...
  1 4 7 0...
  1 4 7 10...
T(n,0)=A000326(n+1)
T(n,2)=A059845(n+2)
T(n,n)=3*n+1

			Column 3 = A059845: 7, 17, 30, 46, 65...; while rightmost terms of rows are 1, 4, 7, 10...
First few rows of the triangle =
  1;
  5, 4;
  12, 11, 7;
  22, 21, 17, 10;
  35, 34, 30, 23, 13;
  51, 50, 46, 39, 29, 16;
  70, 69, 65, 58, 48, 35, 19;
  ...

Cf. A095872, A000326, A059845, A002414 (row sums)

T(n, k) = local(i); if(k>n,0,(n+1)*(3*(n+1)-1)/2-k*(3*k-1)/2)
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print()) \\ Lambert Klasen

Triangle read by rows, T(n,k) = sum {j=k..n} 3*j - 2 = A000012 * ((3*j - 2) * 0^(n-k)) * A000012; 1<=k<=n. E.g. T(5,3) = 30 = (7 + 10 + 13).

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 21 2005

A095872 Square of the lower triangular matrix T[i,j] = 3j-2 for 1<=j<=i, read by rows.

Original entry on oeis.org

1, 5, 16, 12, 44, 49, 22, 84, 119, 100, 35, 136, 210, 230, 169, 51, 200, 322, 390, 377, 256, 70, 176, 455, 580, 624, 560, 361, 70, 276, 455, 580, 624, 560, 361, 92, 364, 609, 800, 910, 912, 779, 484, 117, 464, 784, 1050, 1235, 1312, 1254, 1034, 625, 145, 576, 980, 1330, 1599

Offset: 1

Gary W. Adamson, Jun 10 2004

Arranged by flush left columns (k=1,2,3...), (k=1) column = A000326, the pentagonal numbers (1, 5, 12, 22, 35...). The Octagonal pyramidal number triangle of A095871 is generated from A095872 by dividing the k-th row by the n-th term in the series 1, 4, 7, 10...(k starting with 1). Dividing the 3rd column of A095872 (49, 119, 210, 322, 455...) by 7 generates A059845: 7, 17, 30, 46, 65... Rightmost terms of each row of A095872 are A016778 (1, 16, 49, 100, 169...); i.e. squares of 1, 4, 7, 10... Row sums of A095872 are 1, 21, 105, 325, 780, 1596, 2926... Row sums of A095871 are the octagonal pyramidal numbers, A002414: 1, 9, 30, 70, 135, 231, 364...

[Editor's note: OEIS' "TABL" format (fmt=2) rather displays the transposed matrix as upper triangular matrix.]

			Let M = the infinite lower triangular matrix in the format exemplified by a 3rd order matrix: [1 0 0 / 1 4 0 / 1 4 7]: i.e. for the n-th order matrix, each row has n terms in the series 1, 4, 7, 10... with the rest of the spaces filled in with zeros. Square the matrix and delete the zeros; then read by rows.
[1 0 0 / 1 4 0 / 1 4 7]^2 = [1 0 0 / 5 16 0 / 12 44 49]; then delete the zeros and read by rows: 1, 5, 16, 12, 44, 49...

Cf. A095871, A000326, A059845, A002414, A016778.

A095802(n)={ my( r=sqrtint(2*n)+1, T=matrix(r,r,i,j,if(j>=i,3*j-2))^2); concat(vector(#T,i,vecextract(T[,i],2^i-1)))[n] } \\ M. F. Hasler, Apr 18 2009

a(k(k+1)/2) = (3k-2)^2 (diagonal elements: squares of the initial series), a(k(k-1)/2+1) = A000326(k) (1st column: pentagonal numbers). - M. F. Hasler, Apr 18 2009

Edited and extended by M. F. Hasler, Apr 18 2009

A370238 a(n) = n(3n + 23)/2.

Original entry on oeis.org

0, 13, 29, 48, 70, 95, 123, 154, 188, 225, 265, 308, 354, 403, 455, 510, 568, 629, 693, 760, 830, 903, 979, 1058, 1140, 1225, 1313, 1404, 1498, 1595, 1695, 1798, 1904, 2013, 2125, 2240, 2358, 2479, 2603, 2730, 2860, 2993, 3129, 3268, 3410, 3555, 3703, 3854, 4008

Offset: 0

Torlach Rush, Feb 12 2024

a(a(1)) = A000566(a(1)). This is also true for each of the sequences provided in the formulae below; e.g., A151542(A151542(1)) = A000566(A151542(1)).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A000566, A008594, A005449, A045943, A059845, A115067, A140090, A140091, A140672, A140673, A140674, A140675, A151542, A277976.

Table[n(3n+23)/2,{n,0,48}] (* James C. McMahon, Feb 20 2024 *)

Python
```
def a(n): return n*(3*n+23)//2
```

a(n) = n*(3*n + 23)/2 = A277976(n)/2.
G.f.: x*(13-10*x)/(1-x)^3.
a(n) = A151542(n) + n.
a(n) = A140675(n) + 2*n.
a(n) = A140674(n) + 3*n.
a(n) = A140673(n) + 4*n.
a(n) = A140672(n) + 5*n.
a(n) = A059845(n) + 6*n.
a(n) = A140091(n) + 7*n.
a(n) = A140090(n) + 8*n.
a(n) = A115067(n) + 9*n.
a(n) = A045943(n) + 10*n.
a(n) = A005449(n) + 11*n.
a(n) = A000326(n) + A008594(n).
Sum_{n>=1} 1/a(n) = 823467/2769844 + sqrt(3)*Pi/69 -3*log(3)/23 = 0.2328608... - R. J. Mathar, Apr 23 2024
E.g.f.: exp(x)*x*(26 + 3*x)/2. - Stefano Spezia, Apr 26 2024

cp's OEIS Frontend

A151542 Generalized pentagonal numbers: a(n) = 12*n + 3*n*(n-1)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A140673 a(n) = 3*n*(n + 5)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A140674 a(n) = n*(3*n + 17)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A112414 a(n) = 3*n + 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A185874 Second accumulation array of A051340, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A095871 Triangle read by rows: T(n,k)=(n+1)*(3*(n+1)-1)/2-k*(3*k-1)/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A095872 Square of the lower triangular matrix T[i,j] = 3j-2 for 1<=j<=i, read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A151542 Generalized pentagonal numbers: a(n) = 12n + 3n*(n-1)/2.

A140673 a(n) = 3n(n + 5)/2.

A140674 a(n) = n(3n + 17)/2.

A095871 Triangle read by rows: T(n,k)=(n+1)(3(n+1)-1)/2-k(3k-1)/2.

A370238 a(n) = n(3n + 23)/2.