A051865 -id:A051865 - cp's OEIS frontend

A152997 Twice 13-gonal numbers: a(n) = n(11n - 9).

0, 2, 26, 72, 140, 230, 342, 476, 632, 810, 1010, 1232, 1476, 1742, 2030, 2340, 2672, 3026, 3402, 3800, 4220, 4662, 5126, 5612, 6120, 6650, 7202, 7776, 8372, 8990, 9630, 10292, 10976, 11682, 12410, 13160, 13932, 14726, 15542, 16380, 17240, 18122, 19026, 19952, 20900, 21870, 22862, 23876

Offset: 0

Omar E. Pol, Dec 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051865 (13-gonal numbers).

Cf. numbers of the form n*(n*k - k + 4)/2 listed in A226488 (this sequence is the case k=22). - Bruno Berselli, Jun 10 2013

List([0..50], n-> n*(11*n-9)); # G. C. Greubel, Sep 01 2019

Magma
```
[n*(11*n-9): n in [0..50]];
```

seq(n*(11*n-9), n=0..50); # G. C. Greubel, Sep 01 2019

Table[n*(11*n-9), {n,0,50}] (* G. C. Greubel, Sep 01 2019 *)

a(n)=n*(11*n-9) \\ Charles R Greathouse IV, Jun 17 2017

[n*(11*n-9) for n in (0..50)] # G. C. Greubel, Sep 01 2019

a(n) = 11*n^2 - 9*n = A051865(n)*2.
a(n) = a(n-1) + 22*n - 20 (with a(0)=0). - Vincenzo Librandi, Nov 27 2010
From G. C. Greubel, Sep 01 2019: (Start)
G.f.: 2*x*(1 + 10*x)/(1-x)^3.
E.g.f.: x*(2 + 11*x)*exp(x). (End)

Terms a(39) onward added by G. C. Greubel, Sep 01 2019

A227321 a(n) is the least r>=3 such that the difference between the nearest r-gonal number >= n and n is an r-gonal number.

Original entry on oeis.org

3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 5, 3, 8, 3, 3, 4, 5, 3, 11, 3, 3, 3, 5, 4, 3, 10, 3, 3, 11, 3, 17, 4, 3, 5, 3, 3, 7, 14, 3, 4, 15, 3, 23, 3, 3, 5, 11, 4, 3, 5, 5, 3, 19, 3, 3, 3, 8, 5, 21, 3, 32, 14, 3, 4, 3, 3, 15, 3, 5, 5, 25, 3, 38, 7, 3, 6, 3, 3, 13, 4, 3

Offset: 0

Vladimir Shevelev, Jul 30 2013

nonn

The n-th r-gonal numbers is n((n-1)r-2(n-2))/2, such that 3-gonal numbers are triangular numbers, 4-gonal numbers are squares, etc.

Peter J. C. Moses, Table of n, a(n) for n = 0..1999

Cf. A000217 (r=3), A000290 (r=4), A000326 (r=5), A000384 (r=6), A000566 (r=7), A000567 (r=8), A001106-7 (r=9,10), A051682 (r=11), A051624 (r=12), A051865-A051876 (r=13-24).

rGonalQ[r_,0]:=True; rGonalQ[r_,n_]:=IntegerQ[(Sqrt[((8r-16)n+(r-4)^2)]+r-4)/(2r-4)]; nthrGonal[r_,n_]:=(n (r-2)(n-1))/2+n; nextrGonal[r_,n_]:=nthrGonal[r,Ceiling[(Sqrt[((8r-16)n+(r-4)^2)]+r-4)/(2r-4)]]; (* next r-gonal number greater than or equal to n *) Table[NestWhile[#+1&,3,!rGonalQ[#,nextrGonal[#,n]-n]&],{n,0,99}] (* Peter J. C. Moses, Aug 03 2013 *)

If n is prime, then n == 1 or 2 mod (a(n)-2). If n >= 13 is the greater of a pair of twin primes (A006512), then a(n) = (n+3)/2. - Vladimir Shevelev, Aug 07 2013

More terms from Peter J. C. Moses, Jul 30 2013

A236257 a(n) = 2n^2 - 7n + 9.

Original entry on oeis.org

9, 4, 3, 6, 13, 24, 39, 58, 81, 108, 139, 174, 213, 256, 303, 354, 409, 468, 531, 598, 669, 744, 823, 906, 993, 1084, 1179, 1278, 1381, 1488, 1599, 1714, 1833, 1956, 2083, 2214, 2349, 2488, 2631, 2778, 2929, 3084, 3243, 3406, 3573, 3744, 3919, 4098, 4281, 4468

Offset: 0

Vladimir Shevelev, Jan 21 2014

If zero polygonal numbers are ignored, then for n >= 3, the a(n)-th n-gonal number is a sum of the (a(n)-1)-th n-gonal number and the (2*n-3)-th n-gonal number.

			a(7)=58. This means that the 58th heptagonal number 8323 (cf. A000566) is a sum of two heptagonal numbers. We have 8323 = 8037 + 286 with indices in A000566 58,57,11.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
W. Burrows, C. Tuffley, Maximising common fixtures in a round robin tournament with two divisions, arXiv preprint arXiv:1502.06664 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051624, A051682, A051865-A051876, A152948.

Table[2 n^2 - 7 n + 9, {n, 0, 48}] (* Michael De Vlieger, Apr 19 2015 *)
LinearRecurrence[{3,-3,1},{9,4,3},50] (* Harvey P. Dale, Nov 24 2017 *)

Vec(-(18*x^2-23*x+9)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jan 21 2014

From Colin Barker, Jan 21 2014: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(18*x^2 - 23*x + 9)/(x-1)^3. (End)
E.g.f.: exp(x)*(9 - 5*x + 2*x^2). - Elmo R. Oliveira, Nov 13 2024

A256716 a(n) = n(n+1)(22*n-19)/6.

Original entry on oeis.org

0, 1, 25, 94, 230, 455, 791, 1260, 1884, 2685, 3685, 4906, 6370, 8099, 10115, 12440, 15096, 18105, 21489, 25270, 29470, 34111, 39215, 44804, 50900, 57525, 64701, 72450, 80794, 89755, 99355, 109616, 120560, 132209, 144585, 157710, 171606, 186295, 201799, 218140

Offset: 0

Bruno Berselli, Apr 09 2015

This sequence is related to the tridecagonal numbers (A051865) by a(n) = n*A051865(n) - Sum_{i=0..n-1} A051865(i).

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (22nd row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A051876.
Cf. similar sequences listed in A237616.
Cf. A051865.

Magma
```
[n*(n+1)*(22*n-19)/6: n in [0..40]];
```

Table[n (n + 1) (22 n - 19)/6, {n, 0, 40}]

vector(40, n, n--; n*(n+1)*(22*n-19)/6)

Sage
```
[n*(n+1)*(22*n-19)/6 for n in (0..40)]
```

G.f.: x*(1 + 21*x)/(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with n>3, a(0)=0, a(1)=1, a(2)=25, a(3)=94.
a(n) = Sum_{i=0..n-1} (n-i)*(22*i+1) for n>0.
E.g.f.: exp(x)*x*(6 + 69*x + 22*x^2)/6. - Elmo R. Oliveira, Aug 04 2025

A081928 Sum of the n smallest numbers having the sum of their digits equal to n.

Original entry on oeis.org

1, 13, 36, 70, 115, 171, 238, 316, 405, 604, 868, 1197, 1591, 2158, 2844, 3829, 5140, 6939, 9415, 12100, 14994, 18493, 26062, 34650, 49414, 69535, 96534, 129412, 164299, 201195, 240154, 281122, 414036, 584635, 852634, 1212633, 1629532

Offset: 1

Amarnath Murthy, Apr 01 2003

Sum of n-th row of A081926.

			The two smallest numbers with digit sum 2 are 2 and 11, whose sum is 13.
For seven, 7+16+25+34+43+52+61=238.

Cf. A081926, A081927, A051865.

a[n_] := Module[{}, co = 0; in = 1; su = 0; While[co < n, If[Plus @@ IntegerDigits[in] == n, co++; su = su + in]; in++ ]; su]; Table[a[n], {n, 1, 30}] (Steinerberger)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A236333 The (n-2)-th (n>=3) triple of terms gives coefficients of double trinomial P_n(x) = ((n-2)^2x^2 + nx + 2)/2 (see comment).

Original entry on oeis.org

1, 3, 2, 4, 4, 2, 9, 5, 2, 16, 6, 2, 25, 7, 2, 36, 8, 2, 49, 9, 2, 64, 10, 2, 81, 11, 2, 100, 12, 2, 121, 13, 2, 144, 14, 2, 169, 15, 2, 196, 16, 2, 225, 17, 2, 256, 18, 2, 289, 19, 2, 324, 20, 2, 361, 21, 2, 400, 22, 2, 441, 23, 2, 484, 24, 2, 529, 25, 2, 576, 26, 2, 625, 27, 2, 676, 28, 2, 729, 29, 2

Offset: 3

Vladimir Shevelev, Jan 22 2014

Let {G_n(k)}_(k>=0) be sequence of n-gonal numbers. Then G_n(P_n(k)) = G_n(P_n(k)-1) + G_n((n-2)*k+1).

			Let n=5, k=4. Then G_5(k)=k*(3*k-1)/2 (Cf. A000326) and the double trinomial 2*P_5(x)= 9*x^2+5*x+2, P_5(4)=(9*4^2+5*4+2)/2=83,
Thus, we have G_5(83)=G_5(82)+G_5(13), or 83*124 = 41*245 + 13*19 = 10292.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865-A051876.

I:=[1,3,2,4,4,2,9,5,2]; [n le 9 select I[n] else 3*Self(n-3)-3*Self(n-6)+Self(n-9): n in [1..90]]; // Vincenzo Librandi, Feb 02 2014

a[n_]:=Which[Mod[n,3]==0,n^2/9,Mod[n,3]==1,(n+5)/3,True,2]; Map[a,Range[3,103]]
CoefficientList[Series[(-1-3 x-2 x^2-x^3+5 x^4+4 x^5-2 x^7-2 x^8)/((-1+x)^3 (1+x+x^2)^3),{x,0,100}],x]

Vec(-x^3*(2*x^8+2*x^7-4*x^5-5*x^4+x^3+2*x^2+3*x+1)/((x-1)^3*(x^2+x+1)^3) + O(x^100)) \\ Colin Barker, Jan 23 2014

If n==0 (mod 3), then a(n) = n^2/9;
if n==1 (mod 3), then a(n) = (n+5)/3;
if n==2 (mod 3), then a(n) = 2.
G.f.: -x^3*(2*x^8+2*x^7-4*x^5-5*x^4+x^3+2*x^2+3*x+1) / ((x-1)^3*(x^2+x+1)^3). - Colin Barker, Jan 23 2014

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^jx^(kj*(j-1)/2 + j^2).

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0

Offset: 0

Ilya Gutkovskiy, Aug 13 2017

			Square array begins:
   1,   1,   1,   1,   1,   1, ...
  -2,  -1,  -1,  -1,  -1,  -1, ...
   0,  -1,   0,   0,   0,   0, ...
   0,   0,  -1,   0,   0,   0, ...
   2,   0,   0,  -1,   0,   0, ...
   0,   1,   0,   0,  -1,   0, ...

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.

Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.

Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

A365522 Decimal expansion of (Pisqrt(3) + 9log(3))/24.

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Sep 08 2023

This sequence is also the decimal expansion of Sum_{k>=1} 1/(f(k) +g(k)), where f(k) and g(k) are respectively the k-th triangular and the 13-gonal numbers (A000217 and A051865).

			0.63870452877981836559747674605121660577831724019512...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Ian Shamos, A catalog of the real numbers (2011), p. 544.
Wikipedia, Polygonal number.

Cf. A000217, A000796, A002194, A002391, A051865.

Cf. A244639, A244641, A244645, A244646, A244647, A244648, A244649.

SetDefaultRealField(RealField(139)); R:= RealField(); (Pi(R)*Sqrt(3)+9*Log(3))/24; // G. C. Greubel, Mar 24 2024

RealDigits[(Pi*Sqrt[3] + 9*Log[3])/24, 10 , 100][[1]] (* Amiram Eldar, Sep 08 2023 *)

PARI
```
(Pi*sqrt(3)+9*log(3))/24
```

numerical_approx((pi*sqrt(3)+9*log(3))/24, digits=139) # G. C. Greubel, Mar 24 2024

Equals Sum_{k>=1} 1/(6*k^2 - 4*k) = A244645/2 [Shamos].

Equals - Integral_{x=0..1} log(1-x^6)/x^5 dx [Shamos].

A373616 Positive integers that cannot be written as a sum of a practical number and a 13-gonal number.

Original entry on oeis.org

10, 11, 22, 23, 26, 27, 34, 35, 39, 46, 47, 50, 51, 58, 59, 62, 63, 70, 75, 83, 87, 95, 99, 107, 111, 115, 122, 137, 138, 146, 149, 152, 159, 165, 167, 184, 185, 188, 194, 206, 218, 222, 245, 248, 263, 282, 284, 287, 314, 329, 345, 354, 357, 359, 362, 383, 386

Offset: 1

Duc Van Khanh Tran, Jun 11 2024

Somu and Tran (2024) conjectured that there are finitely many such integers. It was also conjectured that 1612172 is the largest such integer. This conjecture was checked up to 10^8.

			11 is in the sequence as 13-gonal numbers below 11 are 0, 1 and both 11-0 and 11-1 are not practical. - _David A. Corneth_, Jun 11 2024

Duc Van Khanh Tran, Table of n, a(n) for n = 1..609
Sai Teja Somu and Duc Van Khanh Tran, On sums of practical numbers and polygonal numbers, Journal of Integer Sequences, 27(5), 2024.

Cf. A005153, A051865.

cp's OEIS Frontend

A152997 Twice 13-gonal numbers: a(n) = n*(11*n - 9).

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A227321 a(n) is the least r>=3 such that the difference between the nearest r-gonal number >= n and n is an r-gonal number.

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A236257 a(n) = 2*n^2 - 7*n + 9.

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A256716 a(n) = n*(n+1)*(22*n-19)/6.

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A081928 Sum of the n smallest numbers having the sum of their digits equal to n.

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A236333 The (n-2)-th (n>=3) triple of terms gives coefficients of double trinomial P_n(x) = ((n-2)^2*x^2 + n*x + 2)/2 (see comment).

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A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2 + j^2).

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A152997 Twice 13-gonal numbers: a(n) = n(11n - 9).

A236257 a(n) = 2n^2 - 7n + 9.

A256716 a(n) = n(n+1)(22*n-19)/6.

A236333 The (n-2)-th (n>=3) triple of terms gives coefficients of double trinomial P_n(x) = ((n-2)^2x^2 + nx + 2)/2 (see comment).

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^jx^(kj*(j-1)/2 + j^2).

A365522 Decimal expansion of (Pisqrt(3) + 9log(3))/24.