A051682 -id:A051682 - cp's OEIS frontend

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

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

A276819 a(n) = (9*n^2 - n)/2 + 1.

Original entry on oeis.org

1, 5, 18, 40, 71, 111, 160, 218, 285, 361, 446, 540, 643, 755, 876, 1006, 1145, 1293, 1450, 1616, 1791, 1975, 2168, 2370, 2581, 2801, 3030, 3268, 3515, 3771, 4036, 4310, 4593, 4885, 5186, 5496, 5815, 6143, 6480, 6826, 7181, 7545, 7918, 8300, 8691, 9091, 9500, 9918, 10345, 10781, 11226, 11680, 12143, 12615

Offset: 0

Yuriy Sibirmovsky, Sep 18 2016

Diagonal of triangular spiral in A051682. The other 5 diagonals are given by A140064, A117625, A081267, A064225, A006137. See the link as well.
First differences are given by A017209.
72*a(n) - 71 is a perfect square. - Klaus Purath, Jan 14 2022

Colin Barker, Table of n, a(n) for n = 0..1000
Yuriy Sibirmovsky, Six diagonals of the triangular spiral.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002378, A003215, A006137, A017209, A051682, A064225, A081267, A117625, A140064.

Mathematica
```
Table[(9*n^2-n)/2+1, {n,0,100}]
```

Vec((1+2*x+6*x^2)/(1-x)^3 + O(x^60)) \\ Colin Barker, Sep 18 2016

a(n) = (9*n^2 - n)/2 + 1; \\ Altug Alkan, Sep 18 2016

a(n) = (9*n^2 - n)/2 + 1.
a(n) = a(n-1) + 9*n - 5 with a(0) = 1.
From Colin Barker, Sep 18 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: (1 + 2*x + 6*x^2)/(1 - x)^3. (End)
From Klaus Purath, Jan 14 2022: (Start)
a(n) = A006137(n) - n.
A003215(a(n)) - A003215(a(n)-3) = A002378(9*n-1). (End)
E.g.f.: exp(x)*(2 + 8*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A077081 Fixed point when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

Original entry on oeis.org

1, 2, 2, 6, 6, 6, 6, 864, 864, 10, 10, 864, 864, 864, 864, 864, 864, 864, 864, 20, 20, 22, 22, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 48, 864, 46, 46, 48, 864, 864, 48, 864, 864, 48, 48, 48, 48, 58, 58

Offset: 1

Labos Elemer, Oct 28 2002

nonn

A065387 when iterated seems to converge [tested for initial values below 1024]. On the other hand iterating A051682 often ends in cycle.
Iteration of phi(A065387())=phi(sigma()+phi()) seems to converge. Tested below n=1024. Critical values however arise. For example: n=534,556,557,580,624,702,710, etc. These initial values generate very large terms and i was unable to decide if they converge.
For n=1..1024 no more but 27 distinct fixed points arised:{1,2,6,10,..,3552,570240}

			n=225: results in iteration sequence of 44 terms: {225,522,444,...,471744,653312,570240}, a[25]=570240.

Cf. A000010, A000203, A065387, A077080, A077082, A077083, A077084, A077085.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Table[FixedPoint[f, w], {w, 1, 256}]

a(n) = FixedPoint[A077080, n].

A131432 Triangle read by rows: (A000012 * A131431) + (A131431 * A000012) - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 4, 7, 7, 10, 13, 10, 13, 16, 19, 13, 16, 19, 22, 25, 16, 19, 22, 25, 28, 31, 19, 22, 25, 28, 31, 34, 37, 22, 25, 28, 31, 34, 37, 40, 43, 25, 28, 31, 34, 37, 40, 43, 46, 49, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67

Offset: 0

Gary W. Adamson, Jul 10 2007

Left border = A016777, (1, 4, 7, 10, 13, ...).
Right border = A016921: (1, 7, 13, 19, 25, ...).
Row sums = A051682, 11-gonal numbers: (1, 11, 30, 58, 95, ...).

			First few rows of the triangle:
   1;
   4,  7;
   7, 10, 13;
  10, 13, 16, 19;
  13, 16, 19, 22, 25;
  16, 19, 22, 25, 28, 31;
  19, 22, 25, 28, 31, 34, 37;
  ...

Cf. A016777, A051682, A131431, A016921.

a(28) ff. corrected and more terms from Georg Fischer, Jun 05 2023

A214420 Numbers not representable as the sum of three 11-gonal numbers.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 40, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 62, 63, 64, 65, 66, 67, 68, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 91, 92

Offset: 1

T. D. Noe, Jul 17 2012

It is conjectured that 12453 positive numbers are not the sum of three 11-gonal numbers.

R. K. Guy, Unsolved Problems in Number Theory, D3.

T. D. Noe, Table of n, a(n) for n = 1..12453
R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Cf. A051682 (11-gonal numbers).

Cf. A118278, A118279.

nn = 900; hen = Table[n*(9*n-7)/2, {n, 0, nn}]; t = Table[0, {hen[[-1]]}]; Do[n = hen[[i]] + hen[[j]] + hen[[k]]; If[n <= hen[[-1]], t[[n]] = 1], {i, nn}, {j, i, nn}, {k, j, nn}]; Flatten[Position[t, 0]]

A235537 Expansion of (6 + 13x - 8x^2 - 8x^3 + 6x^4)/((1 + x)^2*(1 - x)^3).

Original entry on oeis.org

6, 19, 23, 41, 49, 72, 84, 112, 128, 161, 181, 219, 243, 286, 314, 362, 394, 447, 483, 541, 581, 644, 688, 756, 804, 877, 929, 1007, 1063, 1146, 1206, 1294, 1358, 1451, 1519, 1617, 1689, 1792, 1868, 1976, 2056, 2169, 2253, 2371, 2459, 2582, 2674, 2802, 2898

Offset: 0

Bruno Berselli, Jan 23 2014

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Triangular spiral which contains the terms of A235537 (see A051682).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A235332.

[(6*n*(3*n+17)-(2*n+43)*(-1)^n+11)/16+8: n in [0..50]];

Table[(6 n (3 n + 17) - (2 n + 43) (-1)^n + 11)/16 + 8, {n, 0, 50}]
LinearRecurrence[{1,2,-2,-1,1},{6,19,23,41,49},80] (* Harvey P. Dale, Aug 22 2015 *)

G.f.: (6 + 13*x - 8*x^2 - 8*x^3 + 6*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (6*n*(3*n + 17) - (2*n + 43)*(-1)^n + 11)/16 + 8. The terms a(2k) are in A235332; the closed form of the terms a(2k+1) is n*(9*n+35)/2+19.

A253254 Largest prime factor of the n-th 11-gonal number.

Original entry on oeis.org

11, 5, 29, 19, 47, 7, 13, 37, 83, 23, 101, 13, 17, 5, 137, 73, 31, 41, 173, 13, 191, 23, 19, 109, 227, 59, 7, 127, 263, 31, 281, 29, 23, 11, 317, 163, 67, 43, 353, 181, 53, 43, 389, 199, 37, 47, 17, 31, 443, 113, 461, 53, 479, 61, 71, 23, 103, 131, 41, 271, 31, 7, 569, 17, 587, 149, 17, 307, 89, 79

Offset: 2

Gionata Neri, May 31 2015

nonn

a(A024907(n)) = A061238(n).

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A006530, A051682, A069902 (similar, with triangular numbers).

gpf:= n -> max(numtheory:-factorset(n));
seq(gpf(n*(9*n-7)/2),n=2..100); # Robert Israel, Jun 24 2015

a(n) = my(f = factor(n*(9*n-7)/2)); f[#f~,1]; \\ Michel Marcus, May 31 2015

a(n) = A006530(A051682(n)).

A183223 Complement of the 11-gonal numbers.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102

Offset: 1

Clark Kimberling, Jan 01 2011

nonn

Cf. A051682 (11-gonal numbers).

 Table[n+Floor[1/2+(2n/9+2/9)^(1/2)], {n,100}]

from math import isqrt
def A183223(n): return n+(isqrt((n<<3|4)//9)+1>>1) # Chai Wah Wu, Oct 06 2024

a(n) = n + floor(1/2+(2n/9+2/9)^(1/2)).

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

A266087 Alternating sum of 11-gonal (or hendecagonal) numbers.

Original entry on oeis.org

0, -1, 10, -20, 38, -57, 84, -112, 148, -185, 230, -276, 330, -385, 448, -512, 584, -657, 738, -820, 910, -1001, 1100, -1200, 1308, -1417, 1534, -1652, 1778, -1905, 2040, -2176, 2320, -2465, 2618, -2772, 2934, -3097, 3268, -3440, 3620, -3801, 3990, -4180

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A006578, A007586, A035608, A051682, A083392, A089594.

[(18*(-1)^n*n^2 + 4*(-1)^n*n - 7*(-1)^n + 7)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[((18 n^2 + 4 n - 7) (-1)^n + 7)/8, {n, 0, 43}]
CoefficientList[Series[(x - 8 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
Accumulate[Times@@@Partition[Riffle[PolygonalNumber[11,Range[0,50]],{1,-1},{2,-1,2}],2]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{-2,0,2,1},{0,-1,10,-20},50] (* Harvey P. Dale, Aug 27 2019 *)

x='x+O('x^100); concat(0, Vec(-x*(1-8*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 8*x)/((1 - x)*(1 + x)^3).
a(n) = ((18*n^2 + 4*n - 7)*(-1)^n + 7)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A051682(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.
E.g.f.: (1/4)*(9*x^2 - 11*x)*cosh(x) - (1/4)*(9*x^2 - 11*x - 7)*sinh(x). - G. C. Greubel, Jan 27 2016

cp's OEIS Frontend

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

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A276819 a(n) = (9*n^2 - n)/2 + 1.

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A077081 Fixed point when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

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A131432 Triangle read by rows: (A000012 * A131431) + (A131431 * A000012) - A000012 as infinite lower triangular matrices.

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A214420 Numbers not representable as the sum of three 11-gonal numbers.

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A235537 Expansion of (6 + 13*x - 8*x^2 - 8*x^3 + 6*x^4)/((1 + x)^2*(1 - x)^3).

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A253254 Largest prime factor of the n-th 11-gonal number.

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A183223 Complement of the 11-gonal numbers.

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

A235537 Expansion of (6 + 13x - 8x^2 - 8x^3 + 6x^4)/((1 + x)^2*(1 - x)^3).

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).