A017281 -id:A017281 - cp's OEIS frontend

A017305 a(n) = 10*n + 3.

3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, 113, 123, 133, 143, 153, 163, 173, 183, 193, 203, 213, 223, 233, 243, 253, 263, 273, 283, 293, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 403, 413, 423, 433, 443, 453, 463, 473, 483, 493, 503, 513, 523, 533

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(38).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008592, A016885, A017198, A017281, A017293, A156677.

[10*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011

Range[3, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n) = A017198(n) - A156677(n+2). - Reinhard Zumkeller, Jul 13 2010
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 28 2011
G.f.: (3+7*x)/(x-1)^2. - R. J. Mathar, Apr 11 2016
E.g.f.: exp(x)*(3 + 10*x). - Stefano Spezia, Aug 22 2023
a(n) = A016885(2*n). - Elmo R. Oliveira, Apr 10 2025

A017293 a(n) = 10*n + 2.

Original entry on oeis.org

2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 122, 132, 142, 152, 162, 172, 182, 192, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 302, 312, 322, 332, 342, 352, 362, 372, 382, 392, 402, 412, 422, 432, 442, 452, 462, 472, 482, 492, 502, 512, 522, 532

Offset: 0

N. J. A. Sloane, Dec 11 1996

Number of 5 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1A008574; m=3: A016933; m=4: A022144; m=6: A017569. - Sergey Kitaev, Nov 13 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A139222, A139245, A017329, A139249, A139264, A139279 and A139280.

Cf. A008574, A008592, A016861, A016873, A016933, A017569, A022144.

[10*n+2: n in [0..50]]; // Vincenzo Librandi, May 04 2011

A017293:=n->10*n+2; seq(A017293(n), n=0..100); # Wesley Ivan Hurt, May 03 2014

Range[2, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
CoefficientList[Series[(2 + 8 x) / (1 - x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Jul 23 2016 *)
10 Range[0,60]+2 (* or *) LinearRecurrence[{2,-1},{2,12},60] (* Harvey P. Dale, Jul 04 2019 *)

a(n) = 2*A016861(n) = A008592(n) + 2. - Wesley Ivan Hurt, May 03 2014
G.f.: 2*(1 + 4*x)/(1-x)^2. - Vincenzo Librandi, Jul 23 2016
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 2*exp(x)*(1 + 5*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = A016873(2*n). (End)

A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 7, 4, 1, 1, 11, 13, 10, 5, 1, 1, 16, 21, 19, 13, 6, 1, 1, 22, 31, 31, 25, 16, 7, 1, 1, 29, 43, 46, 41, 31, 19, 8, 1, 1, 37, 57, 64, 61, 51, 37, 22, 9, 1, 1, 46, 73, 85, 85, 76, 61, 43, 25, 10, 1, 1, 56, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 1, 67

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004

Row n gives the centered figurate numbers of the n-gon.

Antidiagonal sums are in A101338.

			The upper left corner of the infinite array T is
|0| 1   1   1   1   1   1   1   1   1   1 ... A000012
|1| 1   2   4   7  11  16  22  29  37  46 ... A000124
|2| 1   3   7  13  21  31  43  57  73  91 ... A002061
|3| 1   4  10  19  31  46  64  85 109 136 ... A005448
|4| 1   5  13  25  41  61  85 113 145 181 ... A001844
|5| 1   6  16  31  51  76 106 141 181 226 ... A005891
|6| 1   7  19  37  61  91 127 169 217 271 ... A003215
|7| 1   8  22  43  71 106 148 197 253 316 ... A069099
|8| 1   9  25  49  81 121 169 225 289 361 ... A016754
|9| 1  10  28  55  91 136 190 253 325 406 ... A060544

Cf. A000217, A001263, A101338.

A101321 := proc(n,k)
        n*k*(k+1)/2+1 ;
end proc: # R. J. Mathar, Jul 28 2016

T[n_, m_] := 1 + n m (m + 1)/2;
Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 23 2020 *)

T(n,m) = 1 + n*m*(m+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

T(n,2) = A016777(n). T(n,3) = A016921(n). T(n,4) = A017281(n).
T(10,m) = A062786(m+1).
T(11,m) = A069125(m+1).
T(12,m) = A003154(m+1).
T(13,m) = A069126(m+1).
T(14,m) = A069127(m+1).
T(15,m) = A069128(m+1).
T(16,m) = A069129(m+1).
T(17,m) = A069130(m+1).
T(18,m) = A069131(m+1).
T(19,m) = A069132(m+1).
T(20,m) = A069133(m+1).
T(n+1,m) = T(n,m) + m*(m+1)/2. - Gary W. Adamson and Michel Marcus, Oct 13 2015

Edited by R. J. Mathar, Oct 21 2009

A161705 a(n) = 18*n + 1.

Original entry on oeis.org

1, 19, 37, 55, 73, 91, 109, 127, 145, 163, 181, 199, 217, 235, 253, 271, 289, 307, 325, 343, 361, 379, 397, 415, 433, 451, 469, 487, 505, 523, 541, 559, 577, 595, 613, 631, 649, 667, 685, 703, 721, 739, 757, 775, 793, 811, 829, 847, 865, 883, 901, 919, 937, 955

Offset: 0

Reinhard Zumkeller, Jun 17 2009

Digital root of a(n) is 1. - Alexander R. Povolotsky, Jun 13 2012

These numbers can be written as the sum of four integer cubes as a(n) = (2*n + 14)^3 + (3*n + 30)^3 + (- 2*n - 23)^3 + (- 3*n - 26)^3. - Arkadiusz Wesolowski, Aug 15 2013

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A016813, A016921, A017281, A017533, A128470, A158057, A161700, A161709, A161714, A287326 (fourth column).

Cf. A016777, A017173.

List([0..60], n-> 18*n+1); # G. C. Greubel, Sep 18 2019

[18*n +1: n in [0..60]]; // G. C. Greubel, Sep 18 2019

seq(18*n+1, n=0..60); # G. C. Greubel, Sep 18 2019

Range[1, 1000, 18] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
LinearRecurrence[{2,-1},{1,19}, 60] (* G. C. Greubel, Feb 17 2017 *)

vector(60, n, 18*n-17) \\ G. C. Greubel, Feb 17 2017

[18*n+1 for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 18*n + 1, n >= 0.
a(n) = a(n-1) + 18 (with a(0)=1). - Vincenzo Librandi, Dec 27 2010
From G. C. Greubel, Feb 17 2017: (Start)
G.f.: (1 + 17*x)/(1-x)^2.
E.g.f.: (1 + 18*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2). (End)
a(n) = A017173(2*n) = A016777(6*n). - Elmo R. Oliveira, Apr 12 2025

A106621 a(n) = numerator of n/(n+20).

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 3, 7, 2, 9, 1, 11, 3, 13, 7, 3, 4, 17, 9, 19, 1, 21, 11, 23, 6, 5, 13, 27, 7, 29, 3, 31, 8, 33, 17, 7, 9, 37, 19, 39, 2, 41, 21, 43, 11, 9, 23, 47, 12, 49, 5, 51, 13, 53, 27, 11, 14, 57, 29, 59, 3, 61, 31, 63, 16, 13, 33, 67, 17, 69, 7, 71, 18, 73, 37, 15, 19, 77, 39, 79

Offset: 0

N. J. A. Sloane, May 15 2005

Contains as subsequences A026741, A017281, A017305, A005408, A017353, and A017377. - Luce ETIENNE, Nov 04 2018

Multiplicative and also a strong divisibility sequence: gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Feb 24 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A000010, A010879.

Cf. Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106620 (k = 13 thru 19).

List([0..80],n->NumeratorRat(n/(n+20))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+20)): n in [0..100]]; // Vincenzo Librandi, Mar 06 2018

seq(numer(n/(n+20)),n=0..80); # Muniru A Asiru, Feb 19 2019

f[n_]:=Numerator[n/(n+20)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

a(n) = numerator(n/(n+20)); \\ Michel Marcus, Mar 07 2018

[lcm(20,n)/20 for n in range(0, 80)] # Zerinvary Lajos, Jun 12 2009

a(n) = lcm(20, n)/20. - Zerinvary Lajos, Jun 12 2009
a(n) = n/gcd(n, 20). - Andrew Howroyd, Jul 25 2018
From Luce ETIENNE, Nov 04 2018: (Start)
a(n) = 9*a(n-20) - 36*a(n-40) + 84*a(n-60) - 126*a(n-80) + 126*a(n-100) - 84*a(n-120) + 36*a(n-140) - 9*a(n-160) + a(n-180).
a(n) = (5*(119*m^9 - 4923*m^8 + 86250*m^7 - 832230*m^6 + 4807887*m^5 - 16882299*m^4 + 34770400*m^3 - 37855620m^2 + 16581744*m + 54432)*floor(n/10) + 72*m*(3*m^8 - 120*m^7 + 2030*m^6 - 18900*m^5 + 105329*m^4 - 356580*m^3 + 706220*m^2 - 733200*m + 300258) + ((19*m^9 - 855*m^8 + 15810*m^7 - 154350*m^6 + 849387*m^5 - 2597175*m^4 + 4037840*m^3 - 2600100*m^2 + 540144*m - 90720)*floor(n/10) - 72*m*(m^7 - 35*m^6 + 490*m^5 - 3500*m^4 + 13489*m^3 - 27335*m^2 + 26340*m - 9450))*(-1)^floor(n/10))/362880 where m = (n mod 10). (End)
From Peter Bala, Feb 24 2019: (Start)
a(n) = n/gcd(n,20) is a quasi-polynomial in n since gcd(n,20) is a purely periodic sequence of period 20.
O.g.f.: F(x) - F(x^2) - F(x^4) - 4*F(x^5) + 4*F(x^10) + 4*F(x^20), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 20} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (1/2)*log(1/(1 - x^2)) + (2/4)*log(1/(1 - x^4)) + (4/5)*log(1/(1 - x^5)) + (4/10)*log(1/(1 - x^10)) + (8/20)*log(1/(1 - x^20)), where phi(n) denotes the Euler totient function A000010. (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 2^max(0, e-2), a(5^e) = 5^max(0,e-1), and a(p^e) = p^e otherwise.
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/4^s - 4/5^s + 4/10^s + 4/20^s).
Sum_{k=1..n} a(k) ~ (231/800) * n^2. (End)

Keyword:mult added by Andrew Howroyd, Jul 25 2018

A139279 a(n) = 40*n - 32.

Original entry on oeis.org

8, 48, 88, 128, 168, 208, 248, 288, 328, 368, 408, 448, 488, 528, 568, 608, 648, 688, 728, 768, 808, 848, 888, 928, 968, 1008, 1048, 1088, 1128, 1168, 1208, 1248, 1288, 1328, 1368, 1408, 1448, 1488, 1528, 1568, 1608, 1648, 1688, 1728, 1768, 1808, 1848

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 8 with unit digit equal to 8.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139249, A139264 and A139280.

[40*n-32: n in [1..50]]; // Vincenzo Librandi, Jun 19 2011

NestList[#+40&,8,50] (* Harvey P. Dale, Oct 02 2013 *)

a(n)=40*n-32 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 40.
From G. C. Greubel, Jul 18 2017: (Start)
G.f.: (72*x - 32)/(1-x)^2.
E.g.f.: (40*x - 32)*exp(x). (End)

More terms from Reinhard Zumkeller, Jun 22 2008

New definition from Paolo P. Lava, Sep 06 2010

A139280 a(n) = 90*n - 81.

Original entry on oeis.org

9, 99, 189, 279, 369, 459, 549, 639, 729, 819, 909, 999, 1089, 1179, 1269, 1359, 1449, 1539, 1629, 1719, 1809, 1899, 1989, 2079, 2169, 2259, 2349, 2439, 2529, 2619, 2709, 2799, 2889, 2979, 3069, 3159, 3249, 3339, 3429, 3519, 3609, 3699, 3789, 3879, 3969

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 9 with final digit 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139249, A139264 and A139279.

Cf. A017281, A139222.

[90*n-81: n in [1..50]]; // Vincenzo Librandi, Jun 19 2011

Table[90*n-81, {n, 0, 50}] (* G. C. Greubel, Jul 18 2017 *)

for(n=1, 1e2, a=90*n-81; print1(a, ", ")) \\ Felix Fröhlich, Jul 07 2014

a(n) = a(n-1) + 90.
From G. C. Greubel, Jul 18 2017: (Start)
G.f.: 9*(19*x-9)/(x-1)^2.
E.g.f.: 81 + 9*(10*x - 9)*exp(x). (End) [G.f. corrected by Georg Fischer, May 12 2019]; [E.g.f. corrected by Elmo R. Oliveira, Apr 04 2025]
From Elmo R. Oliveira, Apr 04 2025: (Start)
a(n) = 9*A017281(n-1) = 3*A139222(n).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

A128470 a(n) = 30*n + 1.

Original entry on oeis.org

1, 31, 61, 91, 121, 151, 181, 211, 241, 271, 301, 331, 361, 391, 421, 451, 481, 511, 541, 571, 601, 631, 661, 691, 721, 751, 781, 811, 841, 871, 901, 931, 961, 991, 1021, 1051, 1081, 1111, 1141, 1171, 1201, 1231, 1261, 1291, 1321, 1351, 1381, 1411, 1441, 1471

Offset: 0

Cino Hilliard, May 06 2007

Possible upper bounds of twin primes pairs ending in 1: For a 30k + r "wheel", k > 0, r = 1, 13, 19 are the only possible values that can form an upper bound of a twin prime pair. The 30k+r wheel gives the sequence 1, 7, 11, 13, 17, 19, 23, 29 31, 37, 41, 43, 47, 49, 53, 59, ... which is frequently used in prime number sieves to skip multiples of 2, 3, 5. The fact that subtracting 2 from 30k+7, 11, 17, 23 will give us a multiple of 3 or 5 precludes these numbers from being an upper bound of a twin prime pair. This leaves us with r = 1, 13, 19 for k > 0 as the only possible cases to form an upper bound of a twin prime pair. 1, 13, 19 concludes the 6 numbers of the 8 number wheel that can form part of a twin prime pair.

			61 = 30 * 2 + 1, the upper part of the twin prime pair 59, 61.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Albert van der Horst, Counting Twin Primes
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A161700, A005408, A016813, A016921, A017281, A017533, A158057, A161705, A161709, A161714.

[30*n+1: n in [0..50]]; // Vincenzo Librandi, Jun 16 2011

Range[1, 3001, 30] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
CoefficientList[Series[(1 + 29 x) / (1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 30 2014 *)
LinearRecurrence[{2, -1}, {1, 31}, 100] (* G. C. Greubel, Apr 04 2016 *)

a(n)=30*n+1 \\ Charles R Greathouse IV, Oct 07 2015

(0 to 49).map(30 *  + 1) // _Alonso del Arte, Jun 02 2019

a(n) = 2*a(n-1) - a(n-2) for n > 1. - Vincenzo Librandi, Dec 30 2014
G.f.: (1 + 29*x)/(1 - x)^2. - Vincenzo Librandi, Dec 30 2014
E.g.f.: (1 + 30*x)*exp(x). - G. C. Greubel, Apr 04 2016

A139245 a(n) = 20*n - 16.

Original entry on oeis.org

4, 24, 44, 64, 84, 104, 124, 144, 164, 184, 204, 224, 244, 264, 284, 304, 324, 344, 364, 384, 404, 424, 444, 464, 484, 504, 524, 544, 564, 584, 604, 624, 644, 664, 684, 704, 724, 744, 764, 784, 804, 824, 844, 864, 884, 904, 924, 944, 964, 984, 1004, 1024, 1044

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 4 with the unit digit equal to 4.

Positive integers that are the product of two integers ending with 2 (see A017293). - Stefano Spezia, Jul 25 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A017329, A139249, A139264, A139279 and A139280.

Cf. A017293.

[20*n-16: n in [1..60]]; // Vincenzo Librandi, Jun 19 2011

A139245:=n->20*n-16; seq(A139245(n), n=1..100); # Wesley Ivan Hurt, Jan 17 2014

Range[4, 1500, 20] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

a(n)=20*n-16 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 20.
a(n) = 4*A016861(n-1). - Wesley Ivan Hurt, Jan 17 2014
From Stefano Spezia, Jul 25 2021: (Start)
O.g.f.: 4*x*(1 + 4*x)/(1 - x)^2.
E.g.f.: 16 + 4*exp(x)*(5*x - 4).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

A017401 a(n) = 11n + 1.

Original entry on oeis.org

1, 12, 23, 34, 45, 56, 67, 78, 89, 100, 111, 122, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 243, 254, 265, 276, 287, 298, 309, 320, 331, 342, 353, 364, 375, 386, 397, 408, 419, 430, 441, 452, 463, 474, 485, 496, 507, 518, 529, 540, 551, 562, 573, 584, 595, 606, 617, 628, 639, 650, 661

Offset: 0

N. J. A. Sloane

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=11, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1)=-charpoly(A,x^(n-1)). - Milan Janjic, Feb 21 2010

Sequence lists all nonnegative solutions to x^k == 1 (mod 11), where k is a member of A045572. - Bruno Berselli, Jan 18 2016

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

Cf. A008593, A017281, A017413.

[11*n+1: n in [0..60]]; // Vincenzo Librandi, Jul 30 2011

Range[1, 1000, 11] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

a(n)=11*n+1 \\ Charles R Greathouse IV, Jan 18 2016

G.f.: (1+10*x)/(1-x)^2.

E.g.f.: exp(x)*(1 + 11*x). - Stefano Spezia, Oct 08 2022

cp's OEIS Frontend

A017305 a(n) = 10*n + 3.

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A017293 a(n) = 10*n + 2.

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A101321 Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.

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A161705 a(n) = 18*n + 1.

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A106621 a(n) = numerator of n/(n+20).

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A139279 a(n) = 40*n - 32.

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A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.