A140672 -id:A140672 - 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

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

A343125 Triangle T(k, n) = (n+3)*(k-n) - 4, k >= 2, 1 <= n <= k-1, read by rows.

Original entry on oeis.org

0, 4, 1, 8, 6, 2, 12, 11, 8, 3, 16, 16, 14, 10, 4, 20, 21, 20, 17, 12, 5, 24, 26, 26, 24, 20, 14, 6, 28, 31, 32, 31, 28, 23, 16, 7, 32, 36, 38, 38, 36, 32, 26, 18, 8, 36, 41, 44, 45, 44, 41, 36, 29, 20, 9, 40, 46, 50, 52, 52, 50, 46, 40, 32, 22, 10

Offset: 2

Russell Jay Hendel, Apr 06 2021

T(k, n) is even if k is odd.
T(k, n) = T(k, n+1) for n = k/2 - 2 if k >= 6 is even.
T(k, n) = T(k, n+2) for n = (k-1)/2 - 2 if k >= 7 is odd.
For fixed n, T(k, n) is linear in k.
The T(k, j) contribute coefficients to a closed formula for the sum of the first n+1 squares of the k-generalized Fibonacci numbers, F(k, j) = A092921(k, j). See A343138 for sums of squares of F(k, j). See the Formula section for closed formula. Although other sequences occur in coefficients in the closed formula for sums of squares, they are linear in nature. All coefficient sequences are mentioned in the arXiv link. The closed formula generalizes results of Schumacher (see References) for the cases k=3 and k=4 with a uniform proof method (see arXiv link).

			Triangle T(k, n) begins:
   k \ n|  1  2  3  4  5  6  7  8  9  10 11
  ------+----------------------------------
   2    |  0
   3    |  4  1
   4    |  8  6  2
   5    | 12 11  8  3
   6    | 16 16 14 10  4
   7    | 20 21 20 17 12  5
   8    | 24 26 26 24 20 14  6
   9    | 28 31 32 31 28 23 16  7
  10    | 32 36 38 38 36 32 26 18  8
  11    | 36 41 44 45 44 41 36 29 20  9
  12    | 40 46 50 52 52 50 46 40 32 22 10
.
The following are the closed formulas for k = 3, 4 for A(k, n) = Sum_{m=0..n} F(k, m)^2, with F(k, n) = A092921(k, n), the k-generalized Fibonacci numbers, and A(k, n) = A343138(k, n), the sum of squares of F(k, n). These formulas are derived from the closed formula in the formula section. Of course further simplifications are possible. For k = 2, T(2, 1) = 0 so illustrations start with k = 3.
k | Formula
--+--------------------------------------------------------
3 | Sum_{m=0..n} F(3,m)^2 = (1/4)*(2*F(3,n)*F(3,n+2) + 4*F(3,n+1)*F(3,n+2) - (k - 2)*F(3,n)^2 - T(3,1)*F(3,n+1)^2 - T(3,2)*F(3,n+2)^2 + 1).
4 | Sum_{m=0..n} F(3,m)^2 = (1/6)*(-2*F(4,n)*F(4,n+1) + 2*F(4,n)*F(4,n+3) + 4*F(4,n+1)*F(4,n+3) + 6*F(4,n+2)*F(4,n+3) - (k-2)*F(4,n)^2 - T(4,1)*F(4,n+1)^2 - T(4, 2)*F(4,n+2)^2 - T(4,3)*F(4,n+3)^2 + 2).

Raphael Schumacher, How to Sum the Squares of the Tetranacci Numbers and the Fibonacci m-step Numbers, Fibonacci Quarterly, 57, (2019), 168-175.
Raphael Schumacher, Explicit Formulas for Sums Involving the Squares of the First n Tribonacci Numbers, Fibonacci Quarterly, 58 (2020), 194-202.

Russell Jay Hendel, Sums of Squares: Methods for Proving Identity Families, arXiv:2103.16756 [math.NT], 2021.
Proof Wiki, Sum of Sequence of Squares of Fibonacci Numbers

Cf. A028557, A085697, A092921, A140672, A343138.

T := (k, n) -> (n + 3)*(k - n) - 4:
seq(print(seq(T(k, n), n=1..k-1)), k = 2..12); # Peter Luschny, Apr 02 2021

Table[(n + 3) (k - n) - 4, {k, 2, 12}, {n, k - 1}] // Flatten (* Michael De Vlieger, Apr 06 2021 *)

T(k,n)=(n + 3)*(k - n) - 4
for(k = 2,12,for(n = 1,k - 1, print1(T(k,n),", ")))

flatten([[(n+3)*(k-n) -4 for n in (1..k-1)] for k in (2..15)]) # G. C. Greubel, Nov 22 2021

Let F(k, n) = A092921(k, n), the k-generalized Fibonacci numbers. Let A(k, n) = A343138(k, n) = Sum_{m=0..n} F(k, m)^2, the sum of the first m+1 k-generalized Fibonacci numbers. Then, for k >= 2, a closed formula for A(k, n) is:
A(k, n) = (1/(2*k-2)) * (Sum_{j=0..k-2, m=j+1..k-1} 2*(j+1)*(m-k+1) * F(k, n+j) * F(k, n+m)) - (k-2)*F(k, n)^2 - Sum_{j=1..k}(T(k, j) * F(k, n+j)^2) + (k-2)).
From G. C. Greubel, Nov 22 2021: (Start)
T(2*n-2, n) = A028557(n-2), n >= 2.
T(4*n-6, n) = 2*A140672(n-2), n >= 2. (End)

cp's OEIS Frontend

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

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

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

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

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A343125 Triangle T(k, n) = (n+3)*(k-n) - 4, k >= 2, 1 <= n <= k-1, read by rows.

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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.

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