A051624 -id:A051624 - cp's OEIS frontend

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66

Offset: 0

Omar E. Pol, Apr 27 2008

A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number. - Omar E. Pol, Dec 21 2008

			The square array of polygonal numbers begins:
========================================================
Triangulars .. A000217: 0, 1,  3,  6, 10,  15,  21,  28,
Squares ...... A000290: 0, 1,  4,  9, 16,  25,  36,  49,
Pentagonals .. A000326: 0, 1,  5, 12, 22,  35,  51,  70,
Hexagonals ... A000384: 0, 1,  6, 15, 28,  45,  66,  91,
Heptagonals .. A000566: 0, 1,  7, 18, 34,  55,  81, 112,
Octagonals ... A000567: 0, 1,  8, 21, 40,  65,  96, 133,
9-gonals ..... A001106: 0, 1,  9, 24, 46,  75, 111, 154,
10-gonals .... A001107: 0, 1, 10, 27, 52,  85, 126, 175,
11-gonals .... A051682: 0, 1, 11, 30, 58,  95, 141, 196,
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217,
And so on ..............................................
========================================================

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.

Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.
Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.
Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.
Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.
Cf. A139619, A139620.

T:= func< n,k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n,k | T(n-k, k) >;
[A139601(n,k): k in  [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)

def T(n,k): return k*((n+1)*(k-1)+2)/2
def A139601(n,k): return T(n-k, k)
flatten([[A139601(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

Original entry on oeis.org

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151

Offset: 0

Bruno Berselli, Dec 08 2012

Partial sums of A172073.

Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A135021, A172073.
Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):
k= 2 (A000027 ): A000292;
k= 3 (A000217 ): A000332 (after the third term);
k= 4 (A000290 ): A002415 (after the first term);
k= 5 (A000326 ): A001296;
k= 6 (A000384*): A002417;
k= 7 (A000566 ): A002418;
k= 8 (A000567*): A002419;
k= 9 (A001106*): A051740;
k=10 (A001107*): A051797;
k=11 (A051682*): A051798;
k=12 (A051624*): A051799;
k=13 (A051865*): A055268.
Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

A051866:=func; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];

I:=[0,1,16,70,200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x*(1+11*x)/(1-x)^5.
a(n) = n*(n+1)*(n+2)*(3*n-2)/6.
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)

A147874 a(n) = (5n-7)(n-1).

Original entry on oeis.org

0, 3, 16, 39, 72, 115, 168, 231, 304, 387, 480, 583, 696, 819, 952, 1095, 1248, 1411, 1584, 1767, 1960, 2163, 2376, 2599, 2832, 3075, 3328, 3591, 3864, 4147, 4440, 4743, 5056, 5379, 5712, 6055, 6408, 6771, 7144, 7527, 7920, 8323, 8736, 9159, 9592, 10035

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Zero followed by partial sums of A017305.
Appears to be related to various other sequences: a(n) = A036666(2*n-2) for n>1; a(n) = A115006(2*n-3) for n>1; a(n) = A118015(5*n-6) for n>1; a(n) = A008738(5*n-7) for n>1.
Even dodecagonal numbers divided by 4. - Omar E. Pol, Aug 19 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Leo Tavares, Illustration: Triangular Badges
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017305 (10n+3), A036666, A115006, A118015 (floor(n^2/5)), A008738 (floor((n^2+1)/5)), A294830.
Cf. A051624, A193872. - Omar E. Pol, Aug 19 2011
Cf. A014105, A002378.

List([1..50], n-> (5*n-7)*(n-1)); # G. C. Greubel, Jul 30 2019

[ 0 ] cat [ &+[ 10*k+3: k in [0..n-1] ]: n in [1..50] ]; // Klaus Brockhaus, Nov 17 2008

Magma
```
[ 5*n^2-2*n: n in [0..50] ];
```

s=0;lst={s};Do[s+=n++ +3;AppendTo[lst,s],{n,0,6!,10}];lst
Table[5n^2-12n+7,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{0,3,16},50] (* or *) PolygonalNumber[12,Range[0,100,2]]/4 (* Harvey P. Dale, Aug 08 2021 *)

{m=50; a=7; for(n=0, m, print1(a=a+10*(n-1)+3, ","))} \\ Klaus Brockhaus, Nov 17 2008

[(5*n-7)*(n-1) for n in (1..50)] # G. C. Greubel, Jul 30 2019

a(n) = Sum_{k=0..n-2} 10*k+3 = Sum_{k=0..n-2} A017305(k).
G.f.: x*(3 + 7*x)/(1-x)^3.
a(n) = 10*(n-2) + 3 + a(n-1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A193872(n-1)/4. - Omar E. Pol, Aug 19 2011
a(n+1) = A131242(10n+2). - Philippe Deléham, Mar 27 2013
E.g.f.: -7 + (7 - 7*x + 5*x^2)*exp(x). - G. C. Greubel, Jul 30 2019
Sum_{n>=2} 1/a(n) = A294830. - Amiram Eldar, Nov 15 2020
a(n) = A014105(n-1) + 3*A002378(n-2). - Leo Tavares, Mar 31 2025

Edited by R. J. Mathar and Klaus Brockhaus, Nov 17 2008, Nov 20 2008

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

Original entry on oeis.org

0, 3, 36, 99, 192, 315, 468, 651, 864, 1107, 1380, 1683, 2016, 2379, 2772, 3195, 3648, 4131, 4644, 5187, 5760, 6363, 6996, 7659, 8352, 9075, 9828, 10611, 11424, 12267, 13140, 14043, 14976, 15939, 16932, 17955, 19008, 20091, 21204

Offset: 0

Omar E. Pol, Dec 26 2008

This sequence is related to A172117 by 3*A172117(n) = n*a(n) - Sum_{i=0..n-1} a(i) and this is the case d=10 in the identity n*(3*n*(d*n - d + 2)/2) - Sum_{k=0..n-1} 3*k*(d*k - d + 2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/2. - Bruno Berselli, Aug 26 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051624, A152965.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153875, A172117.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=30: see Comments lines of A226492.

Table[3n(5n-4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,3,36},40] (* Harvey P. Dale, Jun 18 2014 *)
3*PolygonalNumber[12,Range[0,60]] (* Harvey P. Dale, May 13 2022 *)

a(n)=3*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 15*n^2 - 12*n = A051624(n)*3.
a(n) = 30*n + a(n-1) - 27 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 9*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=36. - Harvey P. Dale, Jun 18 2014
E.g.f.: 3*x*(1 + 5*x)*exp(x). - G. C. Greubel, Aug 21 2016
a(n) = (4*n-2)^2 - (n-2)^2. In general, if P(k,n) is the k-th n-gonal number, then (2*k+1)*P(8*k+4,n) = ((3*k+1)*n-2*k)^2 - (k*n-2*k)^2. - Charlie Marion, Jul 29 2021

A255184 25-gonal numbers: a(n) = n(23n-21)/2.

Original entry on oeis.org

0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, 8722, 9367, 10035, 10726, 11440, 12177, 12937, 13720, 14526, 15355, 16207, 17082, 17980

Offset: 0

Luciano Ancora, Apr 03 2015

If b(n,k) = n*((k-2)*n-(k-4))/2 is n-th k-gonal number, then b(n,k) = A000217(n) + (k-3)* A000217(n-1) (see Deza in References section, page 21, where the formula is attributed to Bachet de Méziriac).
Also, b(n,k) = b(n,k-1) + A000217(n-1) (see Deza and Picutti in References section, page 20 and 137 respectively, where the formula is attributed to Nicomachus). Some examples:
for k=4, A000290(n) = A000217(n) + A000217(n-1);
for k=5, A000326(n) = A000290(n) + A000217(n-1);
for k=6, A000384(n) = A000326(n) + A000217(n-1), etc.
This is the case k=25.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (23rd row of the table).
E. Picutti, Sul numero e la sua storia, Feltrinelli Economica (1977), pages 131-147.

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

Cf. k-gonal numbers: A000217 (k=3), A000290 (k=4), A000326 (k=5), A000384 (k=6), A000566 (k=7), A000567 (k=8), A001106 (k=9), A001107 (k=10), A051682 (k=11), A051624 (k=12), A051865 (k=13), A051866 (k=14), A051867 (k=15), A051868 (k=16), A051869 (k=17), A051870 (k=18), A051871 (k=19), A051872 (k=20), A051873 (k=21), A051874 (k=22), A051875 (k=23), A051876 (k=24), this sequence (k=25), A255185 (k=26), A255186 (k=27), A161935 (k=28), A255187 (k=29), A254474 (k=30).

k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // Bruno Berselli, Apr 10 2015

Mathematica
```
Table[n (23 n - 21)/2, {n, 40}]
```

a(n)=n*(23*n-21)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(-1 - 22*x)/(-1 + x)^3.
a(n) = A000217(n) + 22*A000217(n-1) = A051876(n) + A000217(n-1), see comments.
Product_{n>=2} (1 - 1/a(n)) = 23/25. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 23*x^2/2). - Nikolaos Pantelidis, Feb 05 2023

A051876 24-gonal numbers: a(n) = n(11n-10).

Original entry on oeis.org

0, 1, 24, 69, 136, 225, 336, 469, 624, 801, 1000, 1221, 1464, 1729, 2016, 2325, 2656, 3009, 3384, 3781, 4200, 4641, 5104, 5589, 6096, 6625, 7176, 7749, 8344, 8961, 9600, 10261, 10944, 11649, 12376, 13125, 13896, 14689, 15504, 16341

Offset: 0

N. J. A. Sloane, Dec 15 1999

Sequence found by reading the line from 0, in the direction 0, 24,... and the parallel line from 1, in the direction 1, 69,..., in the square spiral whose vertices are the generalized 24-gonal numbers. - Omar E. Pol, Jul 18 2012

This is also a star dodecagonal number: a(n) = A051624(n) + 12*A000217(n-1). - Luciano Ancora, Mar 30 2015

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

A051876:=n->n*(11*n - 10); seq(A051876(n), n=0..50); # Wesley Ivan Hurt, Feb 25 2014

Table[n (11 n - 10), {n, 0, 50}] (* Wesley Ivan Hurt, Feb 25 2014 *)

a(n)=n*(11*n-10) \\ Charles R Greathouse IV, Jul 19 2011

a(n) = 22*n+a(n-1)-21, with n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
G.f.: x*(1+21*x)/(1-x)^3. - Colin Barker, Jan 10 2012
a(22*a(n)+232*n+1) = a(22*a(n)+232*n) + a(22*n+1). - Vladimir Shevelev, Jan 24 2014
Product_{n>=2} (1 - 1/a(n)) = 11/12. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 11*x^2). - Nikolaos Pantelidis, Feb 05 2023

A135703 a(n) = n(7n-2).

Original entry on oeis.org

0, 5, 24, 57, 104, 165, 240, 329, 432, 549, 680, 825, 984, 1157, 1344, 1545, 1760, 1989, 2232, 2489, 2760, 3045, 3344, 3657, 3984, 4325, 4680, 5049, 5432, 5829, 6240, 6665, 7104, 7557, 8024, 8505, 9000, 9509, 10032, 10569, 11120, 11685, 12264, 12857, 13464

Offset: 0

N. J. A. Sloane, Mar 04 2008

G. C. Greubel, Table of n, a(n) for n = 0..2500
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. index to numbers of the form n*(d*n+10-d)/2 in A014106.

Cf. A185019.

List([0..50], n-> n*(7*n-2)); # G. C. Greubel, Jul 04 2019

[n*(7*n-2): n in [0..50]]; // G. C. Greubel, Jul 04 2019

Array[ #*(7*# - 2) &, 50, 0] (* Zerinvary Lajos, Jul 10 2009 *)

a(n)=n*(7*n-2) \\ Charles R Greathouse IV, Oct 07 2015

[n*(7*n-2) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = 5*n + 14*binomial(n,2).
From R. J. Mathar, Apr 21 2008: (Start)
O.g.f. x*(5+9*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = a(n-1) + 14*n - 9 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
a(n) = 4*A000217(n) + A051624(n). - Bruno Berselli, Feb 11 2011
E.g.f.: x*(5 + 7*x)*exp(x). - G. C. Greubel, Oct 29 2016

A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n(n+1)(10*n-7)/6.

Original entry on oeis.org

0, 1, 13, 46, 110, 215, 371, 588, 876, 1245, 1705, 2266, 2938, 3731, 4655, 5720, 6936, 8313, 9861, 11590, 13510, 15631, 17963, 20516, 23300, 26325, 29601, 33138, 36946, 41035, 45415, 50096, 55088, 60401, 66045, 72030, 78366, 85063, 92131, 99580, 107420, 115661

Offset: 0

N. J. A. Sloane, R. K. Guy

Binomial transform of [1, 12, 21, 10, 0, 0, 0, ...] = (1, 13, 46, 110, ...). - Gary W. Adamson, Nov 28 2007

This sequence is related to A000566 by a(n) = n*A000566(n) - Sum_{i=0..n-1} A000566(i) and this is the case d=5 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/6. - Bruno Berselli, Oct 18 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A093645 ((10, 1) Pascal, column m=3). Partial sums of A051624.
Cf. A000566.
See similar sequences listed in A237616.

List([0..45], n-> n*(n+1)*(10*n-7)/6); # G. C. Greubel, Aug 30 2019

[ n eq 1 select 0 else Self(n-1)+(n-1)*(5*n-9): n in [1..45] ]; // Klaus Brockhaus, Nov 20 2008

A007587:=n->n*(n+1)*(10*n-7)/6: seq(A007587(n), n=0..50); # Wesley Ivan Hurt, Oct 23 2014

CoefficientList[Series[x(1+9x)/(1-x)^4, {x,0,45}], x] (* Vincenzo Librandi, Jun 20 2013 *)
Table[n(n+1)(10n-7)/6,{n,0,50}] (* Harvey P. Dale, Nov 13 2013 *)

a(n)=if(n,([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;1;13;46])[1,1],0) \\ Charles R Greathouse IV, Oct 07 2015

vector(45, n, n*(n-1)*(10*n-17)/6) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(10*n-7)/6 for n in (0..45)] # G. C. Greubel, Aug 30 2019

a(n) = (10*n-7)*binomial(n+1, 2)/3.
G.f.: x*(1+9*x)/(1-x)^4.
a(n) = Sum_{k=0..n} k*(5*k-4). - Klaus Brockhaus, Nov 20 2008
a(n) = Sum_{i=0..n-1} (n-i)*(10*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 23 2014
E.g.f.: exp(x)*x*(6 + 33*x + 10*x^2)/6. - Elmo R. Oliveira, Aug 04 2025

A190816 a(n) = 5n^2 - 4n + 1.

Original entry on oeis.org

1, 2, 13, 34, 65, 106, 157, 218, 289, 370, 461, 562, 673, 794, 925, 1066, 1217, 1378, 1549, 1730, 1921, 2122, 2333, 2554, 2785, 3026, 3277, 3538, 3809, 4090, 4381, 4682, 4993, 5314, 5645, 5986, 6337, 6698, 7069, 7450, 7841, 8242, 8653, 9074

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 20 2011

For n >= 2, hypotenuses of primitive Pythagorean triangles with m = 2*n-1, where the sides of the triangle are a = m^2 - n^2, b = 2*n*m, c = m^2 + n^2; this sequence is the c values, short sides (a) are A045944(n-1), and long sides (b) are A002939(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Short sides (a) A045944(n-1), long sides (b) A002939(n).
Cf. A017281 (first differences), A051624 (a(n)-1), A202141.
Sequences of the form m*n^2 - 4*n + 1: -A131098 (m=0), A028872 (m=1), A056220 (m=2), A045944 (m=3), A016754 (m=4), this sequence (m=5), A126587 (m=6), A339623 (m=7), A080856 (m=8).

[5*n^2 - 4*n + 1: n in [0..50]]; // Vincenzo Librandi, Jun 19 2011

Table[5*n^2 - 4*n + 1, {n, 0, 100}]
LinearRecurrence[{3,-3,1},{1,2,13},100] (* or *) CoefficientList[ Series[ (-10 x^2+x-1)/(x-1)^3,{x,0,100}],x] (* Harvey P. Dale, May 24 2011 *)

a(n)=5*n^2-4*n+1 \\ Charles R Greathouse IV, Oct 16 2015

[5*n^2-4*n+1 for n in range(41)] # G. C. Greubel, Dec 03 2023

From Harvey P. Dale, May 24 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=13.
G.f.: (1 - x + 10*x^2)/(1-x)^3. (End)
E.g.f.: (1 + x + 5*x^2)*exp(x). - G. C. Greubel, Dec 03 2023

Edited by Franklin T. Adams-Watters, May 20 2011

A152965 Twice 12-gonal numbers: a(n) = 2n(5*n-4).

Original entry on oeis.org

0, 2, 24, 66, 128, 210, 312, 434, 576, 738, 920, 1122, 1344, 1586, 1848, 2130, 2432, 2754, 3096, 3458, 3840, 4242, 4664, 5106, 5568, 6050, 6552, 7074, 7616, 8178, 8760, 9362, 9984, 10626, 11288, 11970, 12672, 13394, 14136, 14898, 15680, 16482, 17304, 18146, 19008

Offset: 0

Omar E. Pol, Dec 21 2008

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

Cf. A051624 (12-gonal numbers), A051874.

Cf. numbers of the form n*(n*k - k + 4)/2 listed in A226488 (this sequence is the case k=20). - Bruno Berselli, Jun 10 2013

[10*n^2-8*n: n in [0..50]]; // Klaus Brockhaus, Nov 27 2010

Table[10*n^2-8*n,{n,0,50}] (* Vincenzo Librandi, Jul 10 2012 *)
LinearRecurrence[{3,-3,1},{0,2,24},60] (* Harvey P. Dale, Apr 18 2016 *)

a(n)=2*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*A051624(n).
From Vincenzo Librandi, Jul 10 2012: (Start)
G.f.: 2*x*(1+9*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 2*exp(x)*x*(1 + 5*x).
a(n) = n + A051874(n). (End)

cp's OEIS Frontend

A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.

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A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

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A147874 a(n) = (5*n-7)*(n-1).

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A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3*n*(5*n-4).

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A255184 25-gonal numbers: a(n) = n*(23*n-21)/2.

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A051876 24-gonal numbers: a(n) = n*(11*n-10).

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A135703 a(n) = n*(7*n-2).

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A139601 Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)(k - 1)k/2 + k.

A147874 a(n) = (5n-7)(n-1).

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

A255184 25-gonal numbers: a(n) = n(23n-21)/2.

A051876 24-gonal numbers: a(n) = n(11n-10).

A135703 a(n) = n(7n-2).

A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n(n+1)(10*n-7)/6.

A190816 a(n) = 5n^2 - 4n + 1.

A152965 Twice 12-gonal numbers: a(n) = 2n(5*n-4).