A057145 -id:A057145 - cp's OEIS frontend

A090467 Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form 1 + km(m-1)/2 - (m-1)^2 where k > 2 and m > 2.

1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 17, 19, 20, 23, 26, 29, 31, 32, 37, 38, 41, 43, 44, 47, 50, 53, 56, 59, 61, 62, 67, 68, 71, 73, 74, 77, 79, 80, 83, 86, 89, 97, 98, 101, 103, 104, 107, 109, 110, 113, 116, 119, 122, 127, 128, 131, 134, 137, 139, 140, 143, 146, 149, 151, 152

Offset: 1

Robert G. Wilson v, Dec 01 2003

The m-th k-gonal number is 1 + k*m*(m-1)/2 - (m-1)^2 = A057145(k,m).

Numbers that are strictly trivially polygonal: numbers m that are only 2-gonal and m-gonal. - Daniel Mondot, Jun 13 2024

			3 is a triangular number, but is not a k-gonal number for any other k, so 3 is a term.
6 is both a triangular number and a hexagonal number, so 6 is not a term.

Albert H. Beiler, Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains, Dover, NY, 1964, pp. 185-199.

Michel Marcus, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Figurate Number.
Index to sequences related to polygonal numbers

Complement is A090466.

Cf. A057145, A176774, A176775.

Complement[ Table[i, {i, 300}], Take[ Union[ Flatten[ Table[1 + k*n(n - 1)/2 - (n - 1)^2, {n, 3, 40}, {k, 3, 300}]]], 300]]

isok(n) = (n < 3) || (vecsum(vector(n-2, k, k+=2; ispolygonal(n, k))) == 1); \\ Michel Marcus, May 01 2016

An integer n >= 3 is in this sequence iff A176774(n) = n (or, equivalently, A176775(n) = 2). - Max Alekseyev, Apr 24 2018

Verified by Don Reble, Mar 12 2006

A334466 Square array read by antidiagonals upwards: T(n,k) is the total number of parts in all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

Original entry on oeis.org

1, 3, 1, 4, 1, 1, 7, 3, 1, 1, 6, 1, 1, 1, 1, 12, 3, 3, 1, 1, 1, 8, 4, 1, 1, 1, 1, 1, 15, 3, 3, 3, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 18, 6, 3, 3, 3, 1, 1, 1, 1, 1, 12, 5, 4, 1, 1, 1, 1, 1, 1, 1, 1, 28, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 14, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 3, 6, 3, 3, 3, 3, 1

Offset: 1

Omar E. Pol, May 01 2020

The one-part partition n = n is included in the count.
The column k is related to (k+2)-gonal numbers, assuming that 2-gonals are the nonnegative numbers, 3-gonals are the triangular numbers, 4-gonals are the squares, 5-gonals are the pentagonal numbers, and so on.
Note that the number of parts for T(n,0) = A000203(n), equaling the sum of the divisors of n.
For fixed k>0, Sum_{j=1..n} T(j,k) ~ 2^(3/2) * n^(3/2) / (3*sqrt(k)). - Vaclav Kotesovec, Oct 23 2024

			Square array starts:
   n\k|   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, 1, ...
   2  |   3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   3  |   4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   4  |   7, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   5  |   6, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   6  |  12, 4, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, ...
   7  |   8, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, ...
   8  |  15, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, ...
   9  |  13, 6, 4, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, ...
  10  |  18, 5. 3. 1. 3. 1, 3, 1, 3, 1, 1, 1, 1, ...
  11  |  12, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, ...
  12  |  28, 4, 6, 4, 3, 1, 3, 1, 3, 1, 3, 1, 1, ...
  ...
For n = 9 we have that:
For k = 0 the partitions of 9 into consecutive parts that differ by 0 (or simply: the partitions of 9 into equal parts) are [9], [3,3,3], [1,1,1,1,1,1,1,1,1]. In total there are 13 parts, so T(9,0) = 13.
For k = 1 the partitions of 9 into consecutive parts that differ by 1 (or simply: the partitions of 9 into consecutive parts) are [9], [5,4], [4,3,2]. In total there are six parts, so T(9,1) = 6.
For k = 2 the partitions of 9 into consecutive parts that differ by 2 are [9], [5, 3, 1]. In total there are four parts, so T(9,2) = 4.

Columns k: A000203 (k=0), A204217 (k=1), A066839 (k=2), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6), A377300 (k=7), A377301 (k=8).
Triangles whose row sums give the column k: A127093 (k=0), A285914 (k=1), A330466 (k=2) (conjectured), A330888 (k=3), A334462 (k=4), A334540 (k=5), A339947 (k=6).
Sequences of number of partitions related to column k: A000005 (k=0), A001227 (k=1), A038548 (k=2), A117277 (k=3), A334461 (k=4), A334541 (k=5), A334948 (k=6).
Tables of partitions related to column k: A010766 (k=0), A286001 (k=1), A332266 (k=2), A334945 (k=3), A334618 (k=4).
Polygonal numbers related to column k: A001477 (k=0), A000217 (k=1), A000290 (k=2), A000326 (k=3), A000384 (k=4), A000566 (k=5), A000567 (k=6).
Cf. A051735, A237048, A303300, A330887, A334460, A334465, A334946.
Cf. A038040, A245579, A060872, A334467, A327262, A334733, A334953.
Cf. A057145, A323345.

nmax = 14;
col[k_] := col[k] = CoefficientList[Sum[n x^(n(k n - k + 2)/2)/(1 - x^n), {n, 1, nmax}] + O[x]^(nmax+1), x];
T[n_, k_] := col[k][[n+1]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)

The g.f. for column k is Sum_{n>=1} n*x^(n*(k*n-k+2)/2)/(1-x^n). (For proof, see A330889. - N. J. A. Sloane, Nov 21 2020)

A350405 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

37, 142, 285, 536, 911, 1268, 1909, 2713, 3876, 5179, 6891, 8901, 11190, 14384, 18087, 21697, 27055, 32166, 39111, 46560, 53892, 64412, 73949, 86778, 98202, 113635, 130088, 148051, 167505, 190968, 214955, 240143, 269775, 297615, 331201, 367429, 409179, 451340, 497830

Offset: 3

Ilya Gutkovskiy, Dec 29 2021

nonn

			For n = 3: 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.

David A. Corneth, Table of n, a(n) for n = 3..102
Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A006484, A025443, A057145, A307598, A350207, A350241, A350288.

Do[i=1;While[b=PolygonalNumber[n,Range@i++];!IntegerQ[t=Min[First/@Select[Tally[Select[Total/@Subsets[b,{n}],#<=Max@b&]],Last@#==n&]]]];Print@t,{n,3,10}] (* Giorgos Kalogeropoulos, Dec 30 2021 *)

a(n) >= A006484(n). - David A. Corneth, Dec 30 2021

a(10)-a(31) from Michael S. Branicky, Dec 29 2021

More terms from David A. Corneth, Dec 30 2021

A139606 a(n) = 15*n + 6.

Original entry on oeis.org

6, 21, 36, 51, 66, 81, 96, 111, 126, 141, 156, 171, 186, 201, 216, 231, 246, 261, 276, 291, 306, 321, 336, 351, 366, 381, 396, 411, 426, 441, 456, 471, 486, 501, 516, 531, 546, 561, 576, 591, 606, 621, 636, 651, 666, 681, 696, 711, 726, 741, 756, 771, 786

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 6th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

6th transversal numbers (or 6-transversal numbers): (A000217(6)-6)*n + 6.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A067076, A144562, A153238.

Cf. A000217, A008597, A016873, A057145, A139600.

[15*n+6 : n in [0..60]]; // Vincenzo Librandi, Oct 06 2011

Range[6, 1000, 15] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
15*Range[0,60]+6 (* or *) LinearRecurrence[{2,-1},{6,21},60] (* Harvey P. Dale, Aug 09 2019 *)

a(n)=15*n+6 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,6).
G.f.: 3*(2+3*x)/(x-1)^2 . - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 12 2024: (Start)
E.g.f.: 3*exp(x)*(2 + 5*x).
a(n) = 3*A016873(n) = A008597(n) + 6.
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A254407 a(n) = n(n+1)(11*n +10)/6.

Original entry on oeis.org

0, 7, 32, 86, 180, 325, 532, 812, 1176, 1635, 2200, 2882, 3692, 4641, 5740, 7000, 8432, 10047, 11856, 13870, 16100, 18557, 21252, 24196, 27400, 30875, 34632, 38682, 43036, 47705, 52700, 58032, 63712, 69751, 76160, 82950, 90132, 97717, 105716, 114140, 123000

Offset: 0

Bruno Berselli, Jan 30 2015

Similar sequences of the type m*P(s,m) - Sum_{i=1..m} P(s-1,i), where P(s,m) is the m-th s-gonal number:
s=3: A027480(n) = (n+1)*A000217(n+1) - Sum_{i=1..n+1} i;
s=4: A162148(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i);
s=5: A245301(n) = (n+1)*A000326(n+1) - Sum_{i=1..n+1} A000290(i);
s=6: A085788(n) = (n+1)*A000384(n+1) - Sum_{i=1..n+1} A000326(i);
s=7:       a(n) = (n+1)*A000566(n+1) - Sum_{i=1..n+1} A000384(i).

			532 is the 7th term because A000566(7)=112 and Sum_{i=1..7} A000384(i)=252, therefore 7*112-252 = 532.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Polygonal numbers: Table of values.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A011875, A027480, A057145, A085788, A132112, A162148, A245301, A254963.

Magma
```
[n*(n+1)*(11*n+10)/6: n in [0..40]];
```

A254407:= n-> n*(n+1)*(11*n+10)/6; seq(A254407(n), n=0..50); # G. C. Greubel, Mar 31 2021

Table[n (n + 1) (11 n + 10)/6, {n, 0, 40}]
Column[CoefficientList[Series[x (7 + 4 x) / (1 - x)^4, {x, 0, 60}], x]] (* Vincenzo Librandi, Jan 31 2015 *)

makelist(n*(n+1)*(11*n+10)/6, n, 0, 40);

vector(40, n, n--; n*(n+1)*(11*n+10)/6)

Sage
```
[n*(n+1)*(11*n+10)/6 for n in (0..40)]
```

G.f.: x*(7 + 4*x)/(1 - x)^4.
a(-n) = -A132112(n-1).
a(n) = Sum_{k=0..n} A011875(11*k+2).
Equivalently, partial sums of A254963.
E.g.f.: x*(42 + 54*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

A162607 a(n) = n(n^2 - 4n + 5)/2.

Original entry on oeis.org

0, 1, 1, 3, 10, 25, 51, 91, 148, 225, 325, 451, 606, 793, 1015, 1275, 1576, 1921, 2313, 2755, 3250, 3801, 4411, 5083, 5820, 6625, 7501, 8451, 9478, 10585, 11775, 13051, 14416, 15873, 17425, 19075, 20826, 22681, 24643, 26715, 28900, 31201, 33621

Offset: 0

R. J. Mathar and Omar E. Pol, Jul 21 2009

Positive values are the row sums of triangle A159798.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 25 2012
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Cf. A159798.

[n*(n^2 - 4*n + 5)/2: n in [0..50]]; // Vincenzo Librandi, Dec 19 2012

f[n_]:=(n^3-n^2)/2+1; Table[f[n],{n,-1,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
CoefficientList[Series[x*(1 - 3*x + 5*x^2)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 19 2012 *)

for(n=0,50, print1(n*(n^2-4*n+5)/2, ", ")) \\ G. C. Greubel, Apr 21 2018

G.f.: x*(1 - 3*x + 5*x^2)/(1 - x)^4. - Vincenzo Librandi, Dec 19 2012
a(n) = A057145(n-1,n) = A072277(n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: x*(x^2 - x + 2)*exp(x)/2. - G. C. Greubel, Apr 21 2018

A320943 Numbers that have exactly 26 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

1559439365121, 2468046593376, 7760419091425

Offset: 1

Hugh Erling, Oct 24 2018

			a(1): 1559439365121 has representations P(n,k) = P(3, 519813121708)=P(6, 103962624343)=P(9, 43317760144)=P(11, 28353443004)=P(18, 10192414153)=P(27, 4442847196)=P(33, 2953483648)=P(57, 977092336)=P(66, 727011361)=P(69, 664722664)=P(81, 481308448)=P(86, 426659199)=P(129, 188885584)=P(131, 183140268)=P(171, 107288572)=P(209, 71744544)=P(237, 55761976)=P(414, 18240979)=P(473, 13969968)=P(513, 11874388)=P(711, 6178324)=P(729, 5876784)=P(1881, 881968)=P(3537, 249376)=P(16899, 10924)=P(720981, 8).
a(2): 2468046593376 has representations P(n,k) = P(3, 822682197793)=P(6, 164536439560)=P(12, 37394645356)=P(18, 16131023488)=P(24, 8942197804)=P(26, 7593989520)=P(39, 3330697159)=P(42, 2866488496)=P(56, 1602627660)=P(72, 965589436)=P(84, 707988124)=P(96, 541238290)=P(116, 370021980)=P(126, 313402744)=P(392, 32204796)=P(416, 28591830)=P(576, 14903665)=P(647, 11809911)=P(783, 8061483)=P(936, 5640220)=P(1827, 1479601)=P(2912, 582306)=P(4302, 266776)=P(5823, 145603)=P(7056, 99160)=P(145551, 235).
a(3): 7760419091425 has representations P(n,k) = P(5, 776041909144)=P(7, 369543766260)=P(10, 172453757589)=P(13, 99492552456)=P(19, 45382567788)=P(25, 25868063640)=P(35, 13042721164)=P(37, 11652280920)=P(49, 6598995828)=P(55, 5225871444)=P(65, 3730970719)=P(82, 2336771785)=P(143, 764347396)=P(145, 743335164)=P(154, 658723293)=P(205, 371134344)=P(290, 185190769)=P(325, 147396376)=P(475, 68935548)=P(1225, 10351368)=P(1378, 8179601)=P(1729, 5194893)=P(2755, 2045644)=P(7585, 269814)=P(1969825, 6)=P(3939649, 3).

Hugh Erling, Python program

Cf. A057145, A063778, A129654, A139601, A177029, A195527, A195528.

A subsequence of A090466.

PARI
```
\\ See Hugh Erling, Oct 29 2018 link.
```

A323345 Square array read by ascending antidiagonals: T(n, k) is the number of partitions of n where parts, if sorted in ascending order, form an arithmetic progression (AP) with common difference of k; n >= 1, k >= 0.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 4, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 4, 2, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Luc Rousseau, Jan 11 2019

T(n, k) is the number of positive integers in the sequence defined, for all i >= 1, by x_1 = n and x_i = (i-1)*(x_(i-1)-k)/i; or defined equivalently by x_i=n/i-(k/2)*(i-1). An x_i positive and integer characterizes the AP-partition with smallest part x_i and number of parts i.

T(n, k) is the number of i, positive integers, such that n - P(k+2,i) is both nonnegative and divisible by i, where P(r,i) denotes the i-th r-gonal number (see A057145).

			There are 4 partitions of 150 such that the parts form an arithmetic progression with common difference of 9:
150 = 150
150 = 41 + 50 + 59
150 = 24 + 33 + 42 + 51
150 = 12 + 21 + 30 + 39 + 48
Then, T(150,9) = 4.
Array begins:
     k 0 1 2 3 4 5 6 7 8 9
   n +--------------------
   1 | 1 1 1 1 1 1 1 1 1 1
   2 | 2 1 1 1 1 1 1 1 1 1
   3 | 2 2 1 1 1 1 1 1 1 1
   4 | 3 1 2 1 1 1 1 1 1 1
   5 | 2 2 1 2 1 1 1 1 1 1
   6 | 4 2 2 1 2 1 1 1 1 1
   7 | 2 2 1 2 1 2 1 1 1 1
   8 | 4 1 2 1 2 1 2 1 1 1
   9 | 3 3 2 2 1 2 1 2 1 1
  10 | 4 2 2 1 2 1 2 1 2 1

Cf. A000005, A001227, A038548, A117277, A334461, A334541, A334948.

Cf. A057145.

T[n_, k_] :=
Module[{c = 0, i = 1, x = n},
  While[x >= 1, If[IntegerQ[x], c++]; i++; x = (i-1)*(x-k)/i]; c]
A004736[n_] := Binomial[Floor[3/2 + Sqrt[2*n]], 2] - n + 1
A002260[n_] := n - Binomial[Floor[1/2 + Sqrt[2*n]], 2]
a[n_] := T[A004736[n], A002260[n] - 1]
Table[a[n], {n, 1, 91}]
(* Second program: *)
nmax = 14;
col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
T[n_, k_] := col[k][[n]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)

T(n,k)=c=0;i=1;x=n;while(x>=1,if(frac(x)==0,c++);i++;x=n/i-(k/2)*(i-1));c
for(s=1,13,for(k=0,s-1,n=s-k;print1(T(n,k),", ")))

T(n, 0) = A000005(n), the number of divisors of n.
T(n, 1) = A001227(n), the number of odd divisors of n.
T(n, 2) = A038548(n), the number of divisors of n that are at most sqrt(n).
T(n, 3) = A117277(n).
The g.f. for column d is Sum_{k>=1} x^(k*(d*k-d+2)/2)/(1-x^k) [information taken from A117277]. - Joerg Arndt, May 05 2020

A373169 Square array read by ascending antidiagonals: T(n,k) = noz(T(n,k-1) + (k-1)*(n-2) + 1), with T(n,1) = 1, n >= 2, k >= 1, where noz(n) = A004719(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 9, 1, 5, 1, 6, 12, 16, 6, 6, 1, 7, 15, 22, 25, 12, 7, 1, 8, 18, 28, 35, 36, 19, 8, 1, 9, 21, 34, 45, 51, 49, 27, 9, 1, 1, 24, 4, 55, 66, 7, 64, 36, 1, 1, 11, 18, 46, 29, 81, 91, 29, 81, 46, 2, 1, 12, 3, 43, 75, 6, 112, 12, 54, 1, 57, 3

Offset: 2

Paolo Xausa, May 27 2024

Row n is the zeroless analog of the positive n-gonal numbers.

			The array begins:
  n\k|  1  2   3   4   5    6    7    8    9   10  ...
  ----------------------------------------------------
   2 |  1, 2,  3,  4,  5,   6,   7,   8,   9,   1, ... = A177274
   3 |  1, 3,  6,  1,  6,  12,  19,  27,  36,  46, ... = A243658 (from n = 1)
   4 |  1, 4,  9, 16, 25,  36,  49,  64,  81,   1, ... = A370812
   5 |  1, 5, 12, 22, 35,  51,   7,  29,  54,  82, ... = A373171
   6 |  1, 6, 15, 28, 45,  66,  91,  12,  45,  82, ... = A373172
   7 |  1, 7, 18, 34, 55,  81, 112, 148, 189, 235, ...
   8 |  1, 8, 21,  4, 29,   6,  43,  86, 135,  19, ...
   9 |  1, 9, 24, 46, 75, 111, 154,  24,  81, 145, ...
  10 |  1, 1, 18, 43, 76, 117, 166, 223, 288, 361, ...
  ...      |                                     \______ A373170 (main diagonal)
        A004719 (from n = 2)

Paolo Xausa, Table of n, a(n) for n = 2..11326 (first 150 antidiagonals, flattened).
Wikipedia, Polygonal number.

Cf. rows 2..6: A177274, A243658, A370812, A373171, A373172.
Cf. A373170 (main diagonal).
Cf. A004719, A057145.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A373169[n_, k_] := A373169[n, k] = If[k == 1, 1, noz[A373169[n, k-1] + (k-1)*(n-2) + 1]];
Table[A373169[n - k + 1, k], {n, 2, 15}, {k, n - 1}]

noz(n) = fromdigits(select(sign, digits(n)));
T(n,k) = if (k==1, 1, noz(T(n,k-1) + (k-1)*(n-2) + 1));
matrix(7,7,n,k,T(n+1,k)) \\ Michel Marcus, May 30 2024

A139618 a(n) = 153*n + 18.

Original entry on oeis.org

18, 171, 324, 477, 630, 783, 936, 1089, 1242, 1395, 1548, 1701, 1854, 2007, 2160, 2313, 2466, 2619, 2772, 2925, 3078, 3231, 3384, 3537, 3690, 3843, 3996, 4149, 4302, 4455, 4608, 4761, 4914, 5067, 5220, 5373, 5526, 5679, 5832, 5985

Offset: 0

Omar E. Pol, May 21 2008

Numbers of the 18th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 18th column in the square array A057145.

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

Cf. A008599, A057145, A139600.

[9*(17*n + 2): n in [0..50]]; // Vincenzo Librandi, Jul 23 2011

153*Range[0,40]+18 (* or *) LinearRecurrence[{2,-1},{18,171},40] (* Harvey P. Dale, May 25 2016 *)

a(n)=153*n+18 \\ Charles R Greathouse IV, Oct 05 2011

From Elmo R. Oliveira, Apr 04 2024: (Start)
G.f.: 9*(2+15*x)/(x-1)^2.
E.g.f.: 9*exp(x)*(2 + 17*x).
a(n) = 9*(A008599(n) + 2).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

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A090467 Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form 1 + k*m*(m-1)/2 - (m-1)^2 where k > 2 and m > 2.

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A334466 Square array read by antidiagonals upwards: T(n,k) is the total number of parts in all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

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A350405 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such number exists.

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A139606 a(n) = 15*n + 6.

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A254407 a(n) = n*(n+1)*(11*n +10)/6.

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A162607 a(n) = n*(n^2 - 4*n + 5)/2.

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A320943 Numbers that have exactly 26 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

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A090467 Numbers which are not regular figurative or polygonal numbers of order greater than 2. That is, numbers not of the form 1 + km(m-1)/2 - (m-1)^2 where k > 2 and m > 2.

A254407 a(n) = n(n+1)(11*n +10)/6.

A162607 a(n) = n(n^2 - 4n + 5)/2.

A320943 Numbers that have exactly 26 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.