A060544 -id:A060544 - cp's OEIS frontend

A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

1, 4, 11, 26, 55, 105, 184, 301, 466, 690, 985, 1364, 1841, 2431, 3150, 4015, 5044, 6256, 7671, 9310, 11195, 13349, 15796, 18561, 21670, 25150, 29029, 33336, 38101, 43355, 49130, 55459, 62376, 69916, 78115, 87010, 96639, 107041, 118256, 130325

Offset: 1

Dan Graybill (clopen(AT)comcast.net), Aug 12 2008

Consider the array of triangular, square and centered polygonal numbers (irregular variant of A086272 and A086273):
 3   6  10  15  21  28  36  45  55  A000217
 4   9  16  25  36  49  64  81 100  A000290
 6  16  31  51  76 106 141 181 226  A005891
 7  19  37  61  91 127 169 217 271  A003215
 8  22  43  71 106 148 197 253 316  A069099
 9  25  49  81 121 169 225 289 361  A016754
10  28  55  91 136 190 253 325 406  A060544
11  31  61 101 151 211 281 361 451  A062786
12  34  67 111 166 232 309 397 496  A069125
13  37  73 121 181 253 337 433 541  A003154
14  40  79 131 196 274 365 469 586  A069126
15  43  85 141 211 295 393 505 631  A069127
etc. The sequence contains the antidiagonal sums of this array. - R. J. Mathar, Jun 05 2011
For comparison, the antidiagonal sums of A086270 are essentially A006522 starting at the 4th term. - R. J. Mathar, Sep 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217.

a(n) = (n-1)*(n^3+11*n^2-38*n+120)/24, n>1. - R. J. Mathar, Sep 12 2008

G.f.: x*(1-x+x^2+x^3-x^5)/(1-x)^5. - Alexander R. Povolotsky, Jun 06 2011

A144410 a(n) = 4(3n+1)(3n+2).

Original entry on oeis.org

8, 80, 224, 440, 728, 1088, 1520, 2024, 2600, 3248, 3968, 4760, 5624, 6560, 7568, 8648, 9800, 11024, 12320, 13688, 15128, 16640, 18224, 19880, 21608, 23408, 25280, 27224, 29240, 31328, 33488, 35720, 38024, 40400, 42848, 45368, 47960, 50624, 53360, 56168, 59048, 62000, 65024, 68120, 71288, 74528, 77840, 81224, 84680, 88208, 91808, 95480

Offset: 0

Paul Curtz, Sep 30 2008

The sequence lists all numbers k such that k+1 is a square and k+4 is divisible by 12. - Bruno Berselli, Sep 28 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A060544, A136016, A244977.

[4*(3*n+1)*(3*n+2): n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

A144410:= n-> 4*(3*n+1)*(3*n+2); seq(A144410(n), n=0..60); # G. C. Greubel, Mar 27 2021

Table[4 (3 n + 1) (3 n + 2), {n, 0, 51}] (* or *)
CoefficientList[Series[8 (1 + 7 x + x^2)/(1 - x)^3, {x, 0, 51}], x] (* Michael De Vlieger, Sep 29 2017 *)

a(n)=4*(3*n+1)*(3*n+2) \\ Charles R Greathouse IV, Jun 17 2017

[4*(3*n+1)*(3*n+2) for n in (0..60)] # G. C. Greubel, Mar 27 2021

G.f.: 8*(1 + 7*x + x^2)/(1 - x)^3. - Michael De Vlieger, Sep 29 2017
a(n) = 8*A060544(n+1).
a(n) = A136016(2*n+1).
a(n) = a(m) + 36*(n - m)*(n + m + 1). For m = n-1, a(n) = a(n-1) + 72*n. - Bruno Berselli, Sep 29 2017
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3. - Klaus Purath, Jul 05 2020
E.g.f.: 4*(2 +18*x +9*x^2)*exp(x). - G. C. Greubel, Mar 27 2021
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi/(12*sqrt(3)) (A244977).
Sum_{n>=0} (-1)^n/a(n) = log(2)/6. (End)

A178034 a(n) = binomial(n*Omega(n),Omega(n)) / n.

Original entry on oeis.org

1, 1, 1, 7, 1, 11, 1, 253, 17, 19, 1, 595, 1, 27, 29, 39711, 1, 1378, 1, 1711, 41, 43, 1, 138415, 49, 51, 3160, 3403, 1, 3916, 1, 25637001, 65, 67, 69, 477191, 1, 75, 77, 657359, 1, 7750, 1, 8515, 8911, 91, 1, 132563501, 97, 11026, 101, 11935, 1, 1633355

Offset: 1

Michel Lagneau, May 17 2010

nonn

Omega(.) = A001222(.) is the number of prime divisors of n (counted with multiplicity).
binomial(nk,k)= n*binomial(nk-1,k-1) ensures that all entries are integers.
Subcases for this sequence:
If n is prime, Omega(n) = 1, and a(n) = binomial (n,1) / n = 1.
If n and n+1 are products of two primes (A070552), then Omega(n) = Omega(n+1) = 2, and binomial(n*Omega(n), Omega(n)) / n = binomial(2*n, 2) / n = 2*n-1 and binomial(2*(n+1), 2) / (n+1) = 2*n+1, and we obtain two consecutive numbers of the form (x, x+2), for example (17,19), (27,29), (41,43),... at n =9, 14...
Chaining this property: If n, n+1, and n+2 are semiprimes (A056809) , we find three consecutive numbers of the form (x, x+2,x+4), for example (65, 67, 69), (169, 171, 173), at n=33, 85.
At places where Omega(n)=3, we find the subsequence A060544, for example a(8) = A060544(8).
At places where Omega(n)=4, we find the subsequence A015219.

			a(8) = binomial(8*Omega(8),Omega(8))/8 = binomial(8*3,3)/8 = 2024/8 = 253.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A038456

A178034 := proc(n)
        local o ;
        o := numtheory[bigomega](n) ;
        binomial(n*o,o)/n ;
end proc: # R. J. Mathar, Jul 08 2012

bon[n_]:=Module[{o=PrimeOmega[n]},Binomial[n*o,o]/n]; Array[bon,60] (* Harvey P. Dale, Jul 22 2014 *)

a(n)=my(b=bigomega(n));binomial(n*b,b)/n \\ Charles R Greathouse IV, Oct 25 2012

A180570 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the graph \|/\/\/...\/_\|/ having n nodes on the horizontal path. The entries in row n are the coefficients of the Wiener polynomial of the graph.

Original entry on oeis.org

7, 12, 9, 10, 18, 18, 9, 13, 24, 27, 18, 9, 16, 30, 36, 27, 18, 9, 19, 36, 45, 36, 27, 18, 9, 22, 42, 54, 45, 36, 27, 18, 9, 25, 48, 63, 54, 45, 36, 27, 18, 9, 28, 54, 72, 63, 54, 45, 36, 27, 18, 9, 31, 60, 81, 72, 63, 54, 45, 36, 27, 18, 9, 34, 66, 90, 81, 72, 63, 54, 45, 36, 27

Offset: 2

Emeric Deutsch, Sep 16 2010

Row n has n+1 entries.
Sum of entries in row n = (2 + 9n + 9n^2)/2 =A060544(n+1).
Sum_{k>=0} k*T(n,k) = A180571(n) (the Wiener indices of the graphs).

			T(2,3)=9 because in the graph \|/_\|/ there are 9 unordered pairs of vertices at distance 3.
Triangle starts:
   7, 12,  9;
  10, 18, 18,  9;
  13, 24, 27, 18,  9;
  16, 30, 36, 27, 18,  9;

I. Gutman, SL Lee, CH Chu. YLLuo, Indian J. Chem., 33A, 603.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte fur Chemie, 131, 421-427 (see Eq. between (10) and (11); replace n with n+2).

Cf. A060544, A180571.

for n from 2 to 11 do P[n] := sort(expand(simplify(t*(9*t^(n+2)-3*n*t^3-8*t^2-2*t+1+3*n)/(1-t)^2))) end do: for n from 2 to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form

The generating polynomial of row n is t*(9t^(n+2) - 3nt^3 - 8t^2 - 2t + 1 + 3n)/(1-t)^2.

The bivariate g.f. is G = tz^2*(7 + 12t + 9t^2 - 4z - 13tz + 4tz^2 + 6t^2*z^2 - 12t^2*z)/((1-z)^2*(1-tz)).

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 7, 1, 1, 5, 10, 13, 11, 1, 1, 6, 13, 19, 21, 16, 1, 1, 7, 16, 25, 31, 31, 22, 1, 1, 8, 19, 31, 41, 46, 43, 29, 1, 1, 9, 22, 37, 51, 61, 64, 57, 37, 1, 1, 10, 25, 43, 61, 76, 85, 85, 73, 46, 1, 1, 11, 28, 49, 71, 91, 106, 113, 109, 91

Offset: 0

Kival Ngaokrajang, Jul 07 2014

T(n,k) is the total number of boxes, when we start with 1 center box (n = 0) then expand 1 box on k-arms for each n iteration. See illustration in links.
It seems that column C(k) = centered k-gonal numbers, and row R(n) = A000217(n)*k + 1.
The triangle under the main diagonal is A121722.
Column N (CN) is the Narayana transform (A001263) of (1, N, 0, 0, 0, ...). Example: C2 (1, 3, 7, 13, ...) is the Narayana transform of (1, 2, 0, 0, 0, ...). - Gary W. Adamson, Oct 01 2015

			Table begins:
       C0  C1  C2  C3  C4  C5
  n/k  0   1   2   3   4   5   ...
R0 0   1   1   1   1   1   1   ...
R1 1   1   2   3   4   5   6   ...
R2 2   1   4   7   10  13  16  ...
R3 3   1   7   13  19  25  31  ...
R4 4   1   11  21  31  41  51  ...
R5 5   1   16  31  46  61  76  ...
R6 6   1   22  43  64  85  106 ...
R7 7   1   29  57  85  113 141 ...
R8 8   1   37  73  109 145 181 ...
R9 9   1   46  91  136 181 226 ...
  ...  ... ... ... ... ... ... ...
C1 = A000124, C2 = A002061, C3 = A005448, C4 = A001844, C5 = A005891, C6 = A003215, C7 = A069099, C8 = A016754, C9 = A060544, C10 = A062786, C11 = A069125, C12  =  A003154.
R1 = A000027, R2 = A016777, R3 = A016921, R4 = A017281, R5 = 15*k + 1, R6 = A215146, R7 = A161714.

Kival Ngaokrajang, Illustration for n = 0..3, k = 1..4

Cf. A000217, A121722, A000124, A002061, A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A000027, A016777, A016921, A017281, A215146, A161714.

T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

A131828 Square of lower triangular matrix in A131821, read by rows.

Original entry on oeis.org

1, 6, 4, 14, 5, 9, 25, 7, 7, 16, 39, 9, 9, 9, 25, 56, 11, 11, 11, 11, 36, 76, 13, 13, 13, 13, 13, 49, 99, 15, 15, 15, 15, 15, 15, 64, 125, 17, 17, 17, 17, 17, 17, 17, 81, 154, 19, 19, 19, 19, 19, 19, 19, 19, 100

Offset: 1

Gary W. Adamson, Jul 20 2007

Left column = A095794: (1, 6, 14, 25, 39, 56, ...).

Row sums = A060544: (1, 10, 28, 55, 91, 190, ...).

			First few rows of the triangle:
   1;
   6,  4;
  14,  5,  9;
  25,  7,  7, 16;
  39,  9,  9,  9, 25;
  56, 11, 11, 11, 11, 36;
  76, 13, 13, 13, 13, 13, 49;
  ...

Cf. A131821, A095794, A060544.

A131821^2.

A361949 Triangle read by rows. T(n, k) = binomial(3n - 1, 3k - 1).

Original entry on oeis.org

1, 10, 1, 28, 56, 1, 55, 462, 165, 1, 91, 2002, 3003, 364, 1, 136, 6188, 24310, 12376, 680, 1, 190, 15504, 125970, 167960, 38760, 1140, 1, 253, 33649, 490314, 1352078, 817190, 100947, 1771, 1, 325, 65780, 1562275, 7726160, 9657700, 3124550, 230230, 2600, 1

Offset: 1

Peter Luschny, Mar 31 2023

			Table T(n, k) starts:
  [1]   1;
  [2]  10,     1;
  [3]  28,    56,       1;
  [4]  55,   462,     165,       1;
  [5]  91,  2002,    3003,     364,       1;
  [6] 136,  6188,   24310,   12376,     680,       1;
  [7] 190, 15504,  125970,  167960,   38760,    1140,      1;
  [8] 253, 33649,  490314, 1352078,  817190,  100947,   1771,    1;
  [9] 325, 65780, 1562275, 7726160, 9657700, 3124550, 230230, 2600, 1.

Cf. A082365 (row sums), A228888 (subdiagonal), A060544 (column 1), A066802 (central column).

T := (n, k) -> binomial(3*n - 1, 3*k - 1):
seq(print(seq(T(n, k), k = 1..n)), n = 1..8);

cp's OEIS Frontend

A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

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A144410 a(n) = 4*(3*n+1)*(3*n+2).

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A178034 a(n) = binomial(n*Omega(n),Omega(n)) / n.

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A180570 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the graph \|/\/\/...\/_\|/ having n nodes on the horizontal path. The entries in row n are the coefficients of the Wiener polynomial of the graph.

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A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

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A131828 Square of lower triangular matrix in A131821, read by rows.

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A361949 Triangle read by rows. T(n, k) = binomial(3*n - 1, 3*k - 1).

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A144410 a(n) = 4(3n+1)(3n+2).

A361949 Triangle read by rows. T(n, k) = binomial(3n - 1, 3k - 1).