A053698 -id:A053698 - cp's OEIS frontend

A056106 Second spoke of a hexagonal spiral.

1, 3, 11, 25, 45, 71, 103, 141, 185, 235, 291, 353, 421, 495, 575, 661, 753, 851, 955, 1065, 1181, 1303, 1431, 1565, 1705, 1851, 2003, 2161, 2325, 2495, 2671, 2853, 3041, 3235, 3435, 3641, 3853, 4071, 4295, 4525, 4761, 5003, 5251, 5505, 5765, 6031, 6303

Offset: 0

Henry Bottomley, Jun 09 2000

First differences of A027444. - J. M. Bergot, Jun 04 2012
Numbers of the form ((h^2+h+1)^2+(-h^2+h+1)^2+(h^2+h-1)^2)/(h^2-h+1) for h=n-1. - Bruno Berselli, Mar 13 2013
For n > 0: 2*a(n) = A058331(n) + A001105(n) + A001844(n-1) = A251599(3*n-2) + A251599(3*n-1) + A251599(3*n). - Reinhard Zumkeller, Dec 13 2014
For all n >= 6, a(n+1) expressed in base n is "353". - Mathew Englander, Jan 06 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Henry Bottomley, Spokes of a hexagonal spiral (illustration of initial terms).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A053698, A027444, and A188947.
Cf. A113524 (semiprime terms), A002061.
Cf. A001105, A001844, A058331, A251599.
Other spokes: A003215, A056105, A056107, A056108, A056109.
Other spirals: A054552.

a056106 n = n * (3 * n - 1) + 1  -- Reinhard Zumkeller, Dec 13 2014

I:=[1,3]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2)+6: n in [1..50]]; // Vincenzo Librandi, Nov 14 2011

Table[3*n^2 - n + 1, {n,0,50}] (* G. C. Greubel, Jul 19 2017 *)

PARI
```
a(n) = 3*n^2-n+1;
```

a(n) = 3*n^2 - n + 1.
a(n) = a(n-1) + 6*n - 4 = 2*a(n-1) - a(n-2) + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (1+2*x+3*x^2)*exp(x). - Paul Barry, Mar 13 2003
a(n) = A096777(3*n) for n>0. - Reinhard Zumkeller, Dec 29 2007
G.f.: (1+5*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
a(n) = n*A002061(n+1) - (n-1)*A002061(n). - Bruno Berselli, Jan 15 2013
a(-n) = A056108(n). - Bruno Berselli, Mar 13 2013

A055129 Repunits in different bases: table by antidiagonals of numbers written in base k as a string of n 1's.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 13, 15, 5, 1, 6, 21, 40, 31, 6, 1, 7, 31, 85, 121, 63, 7, 1, 8, 43, 156, 341, 364, 127, 8, 1, 9, 57, 259, 781, 1365, 1093, 255, 9, 1, 10, 73, 400, 1555, 3906, 5461, 3280, 511, 10, 1, 11, 91, 585, 2801, 9331, 19531, 21845, 9841, 1023, 11

Offset: 1

Henry Bottomley, Jun 14 2000

			T(3,5)=31 because 111 base 5 represents 25+5+1=31.
      1       1       1       1       1       1       1
      2       3       4       5       6       7       8
      3       7      13      21      31      43      57
      4      15      40      85     156     259     400
      5      31     121     341     781    1555    2801
      6      63     364    1365    3906    9331   19608
      7     127    1093    5461   19531   55987  137257
Starting with the second column, the q-th column list the numbers that are written as 11...1 in base q. - _John Keith_, Apr 12 2021

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (1 <= n <= 150).

Rows include A000012, A000027, A002061, A053698, A053699, A053700. Columns (see recurrence) include A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002275, A016123, A016125. Diagonals include A023037, A031973. Numbers in the table (apart from the first column and first two rows) are ordered in A053696.

A055129 := proc(n,k)
    add(k^j,j=0..n-1) ;
end proc: # R. J. Mathar, Dec 09 2015

Table[FromDigits[ConstantArray[1, #], k] &[n - k + 1], {n, 11}, {k, n, 1, -1}] // Flatten (* or *)
Table[If[k == 1, n, (k^# - 1)/(k - 1) &[n - k + 1]], {n, 11}, {k, n, 1, -1}] // Flatten (* Michael De Vlieger, Dec 11 2016 *)

T(n, k) = (k^n-1)/(k-1) [with T(n, 1) = n] = T(n-1, k)+k^(n-1) = (k+1)*T(n-1, k)-k*T(n-2, k) [with T(0, k) = 0 and T(1, k) = 1].
From Werner Schulte, Aug 29 2021 and Sep 18 2021: (Start)
T(n,k) = 1 + k * T(n-1,k) for k > 0 and n > 1.
Sum_{m=2..n} T(m-1,k)/Product_{i=2..m} T(i,k) = (1 - 1/Product_{i=2..n} T(i,k))/k for k > 0 and n > 1.
Sum_{n > 1} T(n-1,k)/Product_{i=2..n} T(i,k) = 1/k for k > 0.
Sum_{i=1..n} k^(i-1) / (T(i,k) * T(i+1,k)) = T(n,k) / T(n+1,k) for k > 0 and n > 0. (End)

A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

Original entry on oeis.org

3, 4, 4, 5, 6, 5, 6, 8, 10, 6, 7, 10, 17, 14, 7, 8, 12, 26, 26, 22, 8, 9, 14, 37, 42, 53, 30, 9, 10, 16, 50, 62, 106, 80, 46, 10, 11, 18, 65, 86, 187, 170, 161, 62, 11, 12, 20, 82, 114, 302, 312, 426, 242, 94, 12, 13, 22, 101, 146, 457, 518, 937, 682, 485, 126, 13

Offset: 1

Jason Kimberley, Oct 27 2011

k >= 2; g >= 3.
The base k-1 reading of the base 10 string of A094626(g).
Exoo and Jajcay Theorem 1: M(k,g) <= A054760(k,g) with equality if and only if: k = 2 and g >= 3; g = 3 and k >= 2; g = 4 and k >= 2; g = 5 and k = 2, 3, 7 or possibly 57; or g = 6, 8, or 12, and there exists a symmetric generalized n-gon of order k - 1.

			This is the table formed from the antidiagonals for k+g = 5..20:
3   4   5   6    7    8    9     10    11    12    13    14    15   16  17 18
4   6  10  14   22   30    46    62    94   126   190   254   382  510 766
5   8  17  26   53   80   161   242   485   728  1457  2186  4373 6560
6  10  26  42  106  170   426   682  1706  2730  6826 10922 27306
7  12  37  62  187  312   937  1562  4687  7812 23437 39062
8  14  50  86  302  518  1814  3110 10886 18662 65318
9  16  65 114  457  800  3201  5602 22409 39216
10 18  82 146  658 1170  5266  9362 42130
11 20 101 182  911 1640  8201 14762
12 22 122 222 1222 2222 12222
13 24 145 266 1597 2928
14 26 170 314 2042
15 28 197 366
16 30 226
17 32
18

E. Bannai and T. Ito, On finite Moore graphs, J. Fac. Sci. Tokyo, Sect. 1A, 20 (1973) 191-208.
R. M. Damerell, On Moore graphs, Proc. Cambridge Phil. Soc. 74 (1973) 227-236.

Jason Kimberley, Table of n, a(n) for n = 1..20100 (k+g = 5..204)
Jason Kimberley, Table of n, k+g, k, g, M(k,g)=a(n) for k+g = 5..204 (n = 1..20100)
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency

Moore lower bound on the order of a (k,g) cage: this sequence (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7), 2*A053698 (g=8), 2*A053699 (g=10), 2*A053700 (g=12), 2*A053716 (g=14), 2*A053716 (g=16), 2*A102909 (g=18), 2*A103623 (g=20), 2*A060885 (g=22), 2*A105067 (g=24), 2*A060887 (g=26), 2*A104376 (g=28), 2*A104682 (g=30), 2*A105312 (g=32).

Cf. A054760 (the actual order of a (k,g)-cage).

ExtendedStringToInt:=func;
M:=func;
k_:=2;g_:=3;
anti:=func;
[anti(kg):kg in[5..15]];

Table[Function[g, FromDigits[#, k - 1] &@ IntegerDigits@ SeriesCoefficient[x (1 + x)/((1 - x) (1 - 10 x^2)), {x, 0, g}]][n - k + 3], {n, 2, 12}, {k, n, 2, -1}] // Flatten (* Michael De Vlieger, May 15 2017 *)

M(k,2i) = 2 sum_{j=0}^{i-1}(k-1)^j =  string "2"^i read in base k-1.
M(k,2i+1) = (k-1)^i +  2 sum_{j=0}^{i-1}(k-1)^j = string "1"*"2"^i read in base k-1.
Recurrence:
M(k,3) = k + 1,
M(k,2i) = M(k,2i-1) + (k-1)^(i-1),
M(k,2i+1) = M(k,2i) + (k-1)^i.

A060885 a(n) = Sum_{j=0..10} n^j.

Original entry on oeis.org

1, 11, 2047, 88573, 1398101, 12207031, 72559411, 329554457, 1227133513, 3922632451, 11111111111, 28531167061, 67546215517, 149346699503, 311505013051, 617839704241, 1172812402961, 2141993519227, 3780494710543, 6471681049901, 10778947368421, 17513875027111, 27824681019587

Offset: 0

N. J. A. Sloane, May 05 2001

a(n) = Phi_11(n), where Phi_k is the k-th cyclotomic polynomial.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are:

A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), this sequence (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..10]]): n in [0..20]]; // G. C. Greubel, Apr 15 2019

A060885 := proc(n)
        numtheory[cyclotomic](11,n) ;
end proc:
seq(A060885(n),n=0..20) ; # R. J. Mathar, Feb 07 2014

Join[{1},Table[Total[n^Range[0,10]],{n,20}]] (* Harvey P. Dale, Jun 19 2011 *)

a(n) = n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 \\ Harry J. Smith, Jul 14 2009

a(n) = polcyclo(11, n); \\ Michel Marcus, Apr 06 2016

[sum(n^j for j in (0..10)) for n in (0..20)] # G. C. Greubel, Apr 15 2019

a(n) = n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.

G.f.: (1+x^2*(1981+x*(66496+x*(534898+x*(1364848+x*(1233970+ x*(389104+x*(36829+x*(672+x)))))))))/(1-x)^11. - Harvey P. Dale, Jun 19 2011

A059722 a(n) = n(2n^2 - 2*n + 1).

Original entry on oeis.org

0, 1, 10, 39, 100, 205, 366, 595, 904, 1305, 1810, 2431, 3180, 4069, 5110, 6315, 7696, 9265, 11034, 13015, 15220, 17661, 20350, 23299, 26520, 30025, 33826, 37935, 42364, 47125, 52230, 57691, 63520, 69729, 76330, 83335, 90756, 98605, 106894, 115635, 124840

Offset: 0

Henry Bottomley, Feb 07 2001

Mean of the first four nonnegative powers of 2n+1, i.e., ((2n+1)^0 + (2n+1)^1 + (2n+1)^2 + (2n+1)^3)/4. E.g., a(2) = (1 + 3 + 9 + 27)/4 = 10.
Equatorial structured meta-diamond numbers, the n-th number from an equatorial structured n-gonal diamond number sequence. There are no 1- or 2-gonal diamonds, so 1 and (n+2) are used as the first and second terms since all the sequences begin as such. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = row sums of triangle A143803. - Gary W. Adamson, Sep 01 2008
Form an array from the antidiagonals containing the terms in A002061 to give antidiagonals 1; 3,3; 7,4,7; 13,8,8,13; 21,14,9,14,21; and so on. The difference between the sum of the terms in n+1 X n+1 matrices and those in n X n matrices is a(n) for n>0. - J. M. Bergot, Jul 08 2013
Sum of the numbers from (n-1)^2 to n^2. - Wesley Ivan Hurt, Sep 08 2014

Harry J. Smith, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of sequences with the formula m*(m+1)^2 - (k+2)*m^2 (table includes this sequence for k = 2-m, m >= 0).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A005900, A081436, A100178, A100179, A059722: "equatorial" structured diamonds; A000447: "polar" structured meta-diamond; A006484 for other structured meta numbers; and A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Cf. A000217, A000290, A001844, A002061, A002414, A053698, A059721, A059723, A136392, A143803.

[n*(2*n^2 - 2*n + 1) : n in [0..50]]; // Wesley Ivan Hurt, Sep 08 2014

A059722:=n->n*(2*n^2 - 2*n + 1): seq(A059722(n), n=0..50); # Wesley Ivan Hurt, Sep 08 2014

Table[n (2 n^2 - 2 n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Sep 08 2014 *)

a(n) = { n*(2*n^2 - 2*n + 1) } \\ Harry J. Smith, Jun 28 2009

a(n) = A053698(2*n-1)/4.
a(n) = Sum_{j=1..n} ((n+j-1)^2-j^2+1). - Zerinvary Lajos, Sep 13 2006
From R. J. Mathar, Sep 02 2008: (Start)
G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^4.
a(n) = A002414(n-1) + A002414(n).
a(n+1) - a(n) = A136392(n+1). (End)
a(n) = (A000290(n) + A000290(n+1)) * (A000217(n+1) - A000217(n)). - J. M. Bergot, Nov 02 2012
a(n) = n * A001844(n-1). - Doug Bell, Aug 18 2015
a(n) = A000217(n^2) - A000217(n^2-2*n). - Bruno Berselli, Jun 26 2018
E.g.f.: exp(x)*x*(1 + 4*x + 2*x^2). - Stefano Spezia, Jun 20 2021

Edited with new definition by N. J. A. Sloane, Aug 29 2008

A060887 a(n) = Sum_{j=0..12} n^j.

Original entry on oeis.org

1, 13, 8191, 797161, 22369621, 305175781, 2612138803, 16148168401, 78536544841, 317733228541, 1111111111111, 3452271214393, 9726655034461, 25239592216021, 61054982558011, 139013933454241, 300239975158033, 619036127056621, 1224880286215951, 2336276859014281, 4311578947368421

Offset: 0

N. J. A. Sloane, May 05 2001

a(n) = Phi_13(n) where Let Phi_k is the k-th cyclotomic polynomial.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A053698, A053699, A053700, A053716, A053717, A102909, A103623, A060885, A102909, A104376, A104682, A105067.

[(&+[n^j: j in [0..12]]): n in [0..20]]; // G. C. Greubel, Apr 14 2019

A060887 := proc(n)
        numtheory[cyclotomic](13,n) ;
end proc:
seq(A060887(n),n=0..20) ; # R. J. Mathar, Feb 11 2014

Table[1 + Sum[n^j, {j, 1, 12}], {n, 0, 20}] (* G. C. Greubel, Apr 14 2019 *)

a(n) = { n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 } \\ Harry J. Smith, Jul 14 2009

A060887(n)=polcyclo(13,n) \\ M. F. Hasler, Dec 31 2012

[sum(n^j for j in (0..12)) for n in (0..20)] # G. C. Greubel, Apr 14 2019

a(n) = n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.
G.f.: (x^12 + 2718*x^11 + 363156*x^10 + 8452952*x^9 + 59276439*x^8 + 155812164*x^7 + 167537592*x^6 + 74214648*x^5 + 12642423*x^4 + 691406*x^3 + 8100*x^2 + 1)/(1-x)^13. - Colin Barker, Oct 29 2012
a(n) = (n^13-1)/(n-1) with a(1) = 13 = lim_{x->1} a(x). - M. F. Hasler, Dec 31 2012

Name changed by G. C. Greubel, Apr 14 2019

A062158 a(n) = n^3 - n^2 + n - 1 = (n-1) * (n^2 + 1).

Original entry on oeis.org

-1, 0, 5, 20, 51, 104, 185, 300, 455, 656, 909, 1220, 1595, 2040, 2561, 3164, 3855, 4640, 5525, 6516, 7619, 8840, 10185, 11660, 13271, 15024, 16925, 18980, 21195, 23576, 26129, 28860, 31775, 34880, 38181, 41684, 45395, 49320, 53465, 57836, 62439, 67280, 72365, 77700, 83291, 89144, 95265, 101660

Offset: 0

Henry Bottomley, Jun 08 2001

Number of walks of length 4 between any two distinct vertices of the complete graph K_{n+1} (n >= 1). Example: a(2) = 5 because in the complete graph ABC we have the following walks of length 4 between A and B: ABACB, ABCAB, ACACB, ACBAB and ACBCB. - Emeric Deutsch, Apr 01 2004
1/a(n) for n >= 2, is in base n given by 0.repeat(0,0,1,1), due to (1/n^3 + 1/n^4)*(1/(1-1/n^4)) = 1/((n-1)*(n^2+1)). - Wolfdieter Lang, Jun 20 2014
For n>3, a(n) is 1220 in base n-1. - Bruno Berselli, Jan 26 2016
For odd n, a(n) * (n+1) / 2 + 1 also represents the first integer in a sum of n^4 consecutive integers that equals n^8. - Patrick J. McNab, Dec 26 2016

			a(4) = 4^3 - 4^2 + 4 - 1 = 64 - 16 + 4 - 1 = 51.

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

Cf. A002061, A023443, A053698, A060884, A060888, A062159, A062160, A268086.

[n^3 - n^2 + n - 1 : n in [0..50]]; // Wesley Ivan Hurt, Dec 26 2016

[seq(n^3-n^2+n-1,n=0..49)]; # Zerinvary Lajos, Jun 29 2006
a:=n->sum(1+sum(n, k=1..n), k=2..n):seq(a(n), n=0...43); # Zerinvary Lajos, Aug 24 2008

Table[n^3 - n^2 + n - 1, {n, 0, 49}] (* Alonso del Arte, Apr 30 2014 *)

a(n) = { n*(n*(n - 1) + 1) - 1 } \\ Harry J. Smith, Aug 02 2009

a(n) = round(n^4/(n+1)) for n >= 2.
a(n) = A062160(n, 4), for n > 2.
G.f.: (4*x-1)*(1+x^2)/(1-x)^4 (for the signed sequence). - Emeric Deutsch, Apr 01 2004
a(n) = floor(n^5/(n^2+n)) for n > 0. - Gary Detlefs, May 27 2010
a(n) = -A053698(-n). - Bruno Berselli, Jan 26 2016
Sum_{n>=2} 1/a(n) = A268086. - Amiram Eldar, Nov 18 2020
E.g.f.: exp(x)*(x^3 + 2*x^2 + x - 1). - Stefano Spezia, Apr 22 2023

More terms from Emeric Deutsch, Apr 01 2004

A098547 a(n) = n^3 + n^2 + 1.

Original entry on oeis.org

1, 3, 13, 37, 81, 151, 253, 393, 577, 811, 1101, 1453, 1873, 2367, 2941, 3601, 4353, 5203, 6157, 7221, 8401, 9703, 11133, 12697, 14401, 16251, 18253, 20413, 22737, 25231, 27901, 30753, 33793, 37027, 40461, 44101, 47953, 52023, 56317, 60841, 65601, 70603, 75853

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Oct 26 2004

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A001093, A011379, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568.

Cf. A000217, A081423.

[(n^3+n^2+1): n in [1..60]]; // Vincenzo Librandi, Apr 06 2011

with(combinat): seq(fibonacci(3,n)+n^3, n=0..40); # Zerinvary Lajos, May 25 2008

Table[n^3+n^2+1,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)

Vec(-(x^3-7*x^2+x-1)/(x-1)^4 + O(x^100)) \\ Colin Barker, Aug 29 2014

From Colin Barker, Aug 29 2014: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (1 - x + 7*x^2 - x^3)/(1-x)^4. (End)
a(n) = A081423(n) + A000217(n-1). - Bruce J. Nicholson, Jan 06 2019
E.g.f.: exp(x)*(1 + 2*x + 4*x^2 + x^3). - Elmo R. Oliveira, Apr 20 2025

A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 3, 1, 1, 5, 7, 7, 4, 1, 1, 6, 9, 13, 9, 4, 1, 1, 7, 11, 21, 16, 11, 4, 1, 1, 8, 13, 31, 25, 22, 13, 4, 1, 1, 9, 15, 43, 36, 37, 28, 15, 5, 1, 1, 10, 17, 57, 49, 56, 49, 40, 17, 5, 1, 1, 11, 19, 73, 64, 79, 76, 85, 43, 19, 5, 1, 1, 12, 21, 91, 81, 106, 109, 156, 89, 49, 21, 5, 1

Offset: 1

Omar E. Pol, Mar 05 2015

			The corner of the square array with the first 15 terms of the first 12 rows looks like this:
-------------------------------------------------------------------------
A000012: 1, 1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,   1,   1,   1
A070941: 1, 2, 3,  3,  4,  4,  4,   4,   5,   5,   5,   5,   5,   5,   5
A005408: 1, 3, 5,  7,  9, 11, 13,  15,  17,  19,  21,  23,  25,  27,  29
A151788: 1, 4, 7, 13, 16, 22, 28,  40,  43,  49,  55,  67,  73,  85,  97
A147562: 1, 5, 9, 21, 25, 37, 49,  85,  89, 101, 113, 149, 161, 197, 233
A151790: 1, 6,11, 31, 36, 56, 76, 156, 161, 181, 201, 281, 301, 381, 461
A151781: 1, 7,13, 43, 49, 79,109, 259, 265, 295, 325, 475, 505, 655, 805
A151792: 1, 8,15, 57, 64,106,148, 400, 407, 449, 491, 743, 785,1037,1289
A151793: 1, 9,17, 73, 81,137,193, 585, 593, 649, 705,1097,1153,1545,1937
A255764: 1,10,19, 91,100,172,244, 820, 829, 901, 973,1549,1621,2197,2773
A255765: 1,11,21,111,121,211,301,1111,1121,1211,1301,2111,2201,3011,3821
A255766: 1,12,23,133,144,254,364,1464,1475,1585,1695,2795,2905,4005,5105
...

Index entries for sequences related to cellular automata

Rows 1-12: A000012, A070941, A005408, A151788, A147562, A151790, A151781, A151792, A151793, A255764, A255765, A255766.
Columns 1-10: A000012, A000027, A005408, A002061, A000290, A084849, A056107, A053698, A100705, A100104.
Cf. A000120, A255740.

A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451

Offset: 0

Paul Barry, Mar 21 2003

			The array starts
  1  1  3 10 ...
  1  2  6 16 ...
  1  3  9 22 ...
  1  4 12 28 ...
The triangle starts
  1;
  1,  1;
  1,  2,  3;
  1,  3,  6, 10;
  1,  4,  9, 16, 25;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Polygonal Number

Rows include A060354, A064808, A006000, A006003, A002411.
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
Antidiagonals are composed of n-gonal numbers.

Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019

[[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018

Table[PolygonalNumber[n,i],{n,0,10},{i,n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)

tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

[[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019

Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.

T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015

cp's OEIS Frontend

A056106 Second spoke of a hexagonal spiral.

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A055129 Repunits in different bases: table by antidiagonals of numbers written in base k as a string of n 1's.

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A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

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A060885 a(n) = Sum_{j=0..10} n^j.

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A059722 a(n) = n*(2*n^2 - 2*n + 1).

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A060887 a(n) = Sum_{j=0..12} n^j.

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A059722 a(n) = n(2n^2 - 2*n + 1).