A017461 -id:A017461 - cp's OEIS frontend

A017471 a(n) = (11*n + 6)^11.

362797056, 34271896307633, 8293509467471872, 317475837322472439, 4882812500000000000, 43513917611435838661, 269561249468963094528, 1287831418538085836267, 5062982072492057196544, 17103393581163134765625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Powers of the form (11*n+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), A017464 (m=4), A017465 (m=5), A017466 (m=6), A017467 (m=7), A017468 (m=8), A017469 (m=9), A017470 (m=10), this sequence (m=11), A017472 (m=12).

List([0..20], n-> (11*n+6)^11); # G. C. Greubel, Sep 19 2019

[(11*n+6)^11: n in [0..10]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+6)^11, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[20] -5)^11 (* G. C. Greubel, Sep 19 2019 *)

vector(20, n, (11*n-5)^11) \\ G. C. Greubel, Sep 19 2019

[(11*n+6)^11 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (362797056 +34267542742961*x +7882270656385972*x^2 + 220215589053761433*x^3 +1612934439380337744*x^4 +4065965093212217778*x^5 +3893323100536505064*x^6 +1409984186533172778*x^7 +173024396961630192* x^8 +5347957556678781*x^9 +17591600106916*x^10 +48828125 x^11)/(1-x)^12.
E.g.f.: (362797056 +34271533510577*x +4112483018826831*x^2 + 48783020707697111*x^3 +152605546678854500*x^4 +184932081242538212*x^5 + 104853627173466171*x^6 +30701237124182097*x^7 +4849119426541500*x^8 + 411237867048855*x^9 +17404011907271*x^10 +285311670611*x^11)*exp(x). (End)

A017472 a(n) = (11*n + 6)^12.

Original entry on oeis.org

2176782336, 582622237229761, 232218265089212416, 12381557655576425121, 244140625000000000000, 2654348974297586158321, 19408409961765342806016, 106890007738661124410161, 475920314814253376475136, 1795856326022129150390625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Powers of the form (11*n+6)^m: A017461 (m=1), A017462 (m=2), A017463 (m=3), A017464 (m=4), A017465 (m=5), A017466 (m=6), A017467 (m=7), A017468 (m=8), A017469 (m=9), A017470 (m=10), A017471 (m=11), this sequence (m=12).

List([0..20], n-> (11*n+6)^12); # G. C. Greubel, Sep 19 2019

[(11*n+6)^12: n in [0..10]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+6)^12, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[0,20]+6)^12 (* Harvey P. Dale, Sep 22 2012 *)

vector(20, n, (11*n-5)^12) \\ G. C. Greubel, Sep 19 2019

[(11*n+6)^12 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (2176782336 +582593939059393*x +224644345794247731*x^2 + 9408164121360836975*x^3 +101126771751016700469*x^4 + 380284494715132979466*x^5 +569002784856695826846*x^6 + 350628073514443644414*x^7 +85353518454518704170*x^8 + 7136462627993219301*x^9 +146435479642729343*x^10 +281471802882531*x^11 +244140625*x^12)/(1-x)^13.
E.g.f.: (2176782336 +582620060447425*x +115526511395767615*x^2 + 1947775120807282470*x^3 +8166890561727393221*x^4 + 12959517969262230432 *x^5 +9583714050157484644*x^6 +3701592580215241932*x^7 + 793530834460904067*x^8 +96520289732086275*x^9 +6542481918780841*x^10 + 227678713147578*x^11 +3138428376721*x^12)*exp(x). (End)

A076394 a(n) = p(11n+6)/11 where p(n) = number of partitions of n (A000041).

Original entry on oeis.org

1, 27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805845, 1117485621, 3366123200, 9767105406, 27398618368, 74534264393, 197147918679, 508189847045, 1279140518117, 3149375120229, 7596463993261

Offset: 0

Jeff Burch, Nov 07 2002

nonn

That p(11n+6) == 0 (mod 11) is one of the congruences stated by Ramanujan. - Omar E. Pol, Jan 18 2013

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

Cf. A000041, A071734, A071746, A213256, A182668.

seq(combinat:-numbpart(11*n+6)/11, n=0..30); # Robert Israel, Jan 07 2015

PartitionsP[(11*Range[0,30]+6)]/11 (* Harvey P. Dale, May 28 2015 *)

a(n) = numbpart(11*n+6)/11; \\ Michel Marcus, Jan 07 2015

a(n) = A000041(A017461(n))/11 = A213256(n)/11. - Omar E. Pol, Jan 18 2013

A213256 p(11n+6) where p(k) = number of partitions of k = A000041(k).

Original entry on oeis.org

11, 297, 3718, 31185, 204226, 1121505, 5392783, 23338469, 92669720, 342325709, 1188908248, 3913864295, 12292341831, 37027355200, 107438159466, 301384802048, 819876908323, 2168627105469, 5590088317495, 14070545699287, 34643126322519, 83561103925871, 197726516681672, 459545750448675, 1050197489931117

Offset: 0

N. J. A. Sloane, Jun 07 2012

nonn

It is known that a(n) is divisible by 11 (see A076394).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
K. Ono, On the Circular Summation of the Eleventh Powers of Ramanujan's Theta Function, Journal of Number Theory, Volume 76, Issue 1, May 1999, Pages 62-65.
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56-59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

Cf. A000041, A076394.

PartitionsP[11Range[0,30]+6] (* Paolo Xausa, Nov 08 2023 *)

a(n) = numbpart(11*n+6); \\ Michel Marcus, Jan 07 2015

a(n) = A000041(A017461(n)). - Omar E. Pol, Jan 18 2013

A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

Original entry on oeis.org

5, 8, 13, 11, 18, 25, 14, 23, 32, 41, 17, 28, 39, 50, 61, 20, 33, 46, 59, 72, 85, 23, 38, 53, 68, 83, 98, 113, 26, 43, 60, 77, 94, 111, 128, 145, 29, 48, 67, 86, 105, 124, 143, 162, 181, 32, 53, 74, 95, 116, 137, 158, 179, 200, 221, 35, 58, 81, 104, 127, 150, 173, 196, 219, 242, 265

Offset: 1

Vincenzo Librandi, Jan 13 2009

First column: A016789, second column: A016885, third column: A017029, fourth column: A017221, fifth column: A017461. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   5;
   8, 13;
  11, 18, 25;
  14, 23, 32, 41;
  17, 28, 39, 50,  61;
  20, 33, 46, 59,  72,  85;
  23, 38, 53, 68,  83,  98, 113;
  26, 43, 60, 77,  94, 111, 128, 145;
  29, 48, 67, 86, 105, 124, 143, 162, 181;
  32, 53, 74, 95, 116, 137, 158, 179, 200, 221; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A001105, A001844, A003307, A007742, A104275, A142463, A160378.

Columns k: A016789 (k=1), A016885 (k=2), A017029 (k=3), A017221 (k=4), A017461 (k=5).

[2*n*k + n + k + 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

T[n_,k_]:= 2 n*k + n + k + 1; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*n*k+n+k+1 for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 14 2023

Sum_{n=1..m} T(m, n) = m*(2*m+3)*(m+1)/2 = A160378(n+1) (row sums). - R. J. Mathar, Jan 15 2009, Jan 05 2011
From G. C. Greubel, Oct 14 2023: (Start)
T(n, n) = A001844(n).
T(n, n-1) = A001105(n), n >= 2.
T(n, n-2) = A142463(n-1), n >= 3.
T(n, n-3) = (-1)*A147973(n+2), n >= 4.
Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^n*A007742(floor((n+1)/2)).
G.f.: x*y*(5 - 2*x - 2*x*y - 2*x^2*y + x^2*y^2)/((1-x)^2*(1-x*y)^3). (End)

A141694 a(n) = 22*n + 12.

Original entry on oeis.org

12, 34, 56, 78, 100, 122, 144, 166, 188, 210, 232, 254, 276, 298, 320, 342, 364, 386, 408, 430, 452, 474, 496, 518, 540, 562, 584, 606, 628, 650, 672, 694, 716, 738, 760, 782, 804, 826, 848, 870, 892, 914, 936, 958, 980, 1002, 1024, 1046, 1068, 1090, 1112

Offset: 0

Paul Curtz, Sep 10 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008604, A010861 (first differences), A017461.

[22*n + 12: n in [0..50]]; // Vincenzo Librandi, Aug 08 2011

Range[12, 1500, 22] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
LinearRecurrence[{2,-1},{12,34},60] (* Harvey P. Dale, Sep 06 2024 *)

for(n=0,50, print1(2*(11*n + 6), ", ")) \\ G. C. Greubel, Jun 03 2018

From G. C. Greubel, Jun 03 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: 2*(6 + 5*x)/(1 - x)^2.
E.g.f.: 2*(6 + 11*x)*exp(x). (End)
a(n) = 2*A017461(n). - Elmo R. Oliveira, Apr 11 2025

Edited by R. J. Mathar, Oct 24 2008

Offset changed from 1 to 0 by Vincenzo Librandi, Aug 08 2011

A182719 Numbers of the form 5k + 4, 7k + 5, or 11k + 6.

Original entry on oeis.org

4, 5, 6, 9, 12, 14, 17, 19, 24, 26, 28, 29, 33, 34, 39, 40, 44, 47, 49, 50, 54, 59, 61, 64, 68, 69, 72, 74, 75, 79, 82, 83, 84, 89, 94, 96, 99, 103, 104, 105, 109, 110, 114, 116, 117, 119, 124, 127, 129, 131, 134, 138, 139, 144, 145, 149, 152, 154, 159, 160, 164

Offset: 1

Omar E. Pol, Feb 08 2011

Numbers such that the Ramanujan congruences apply, making p(a(n)) divisible by at least one of 5, 7, or 11, where p is A000041.
Union of A016897, A017041 and A017461.
First differences are periodic with period length 145.

Wikipedia, Ramanujan's congruences
Index entries for linear recurrences with constant coefficients, order 146.

Cf. A000041, A016897, A017041, A017461.

IsA182719:=func< n | exists{ k: k in [0..n div 5] | n in [5*k+4, 7*k+5, 11*k+6] } >; [ n: n in [1..160] | IsA182719(n) ]; // Klaus Brockhaus, Feb 08 2011

Union[With[{no=30},Join[5Range[0,no]+4,7Range[0,no]+5,11Range[0,no]+6]]]  (* Harvey P. Dale, Feb 18 2011 *)

a(n) = a(n-145) + 385 = a(n-1) + a(n-145) - a(n-146).

Rewritten by Charles R Greathouse IV and Klaus Brockhaus, Feb 08 2011

A304157 a(n) is the first Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference.

Original entry on oeis.org

24, 68, 112, 156, 200, 244, 288, 332, 376, 420, 464, 508, 552, 596, 640, 684, 728, 772, 816, 860, 904, 948, 992, 1036, 1080, 1124, 1168, 1212, 1256, 1300, 1344, 1388, 1432, 1476, 1520, 1564, 1608, 1652, 1696, 1740, 1784, 1828, 1872, 1916, 1960, 2004, 2048

Offset: 1

Emeric Deutsch, May 07 2018

The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of the linear phenylene G[n] is M(G[n];x,y) = 6*x^2*y^2 + 4*(n - 1)*x^2*y^3 + 4(n - 1)*x^3*y^3.
a(n) is the first Zagreb index of the angular phenylene shown in the Bodroza-Pantic et al. reference (Fig. 1 (b)). - Emeric Deutsch, May 24 2018

			From _Andrew Howroyd_, May 09 2018: (Start)
Illustration of the first two graphs:
       o              o         o
     /   \          /   \     /   \
    o     o        o     o---o     o
    |     |        |     |   |     |
    o     o        o     o---o     o
     \   /          \   /     \   /
       o              o         o
In general, the graph consists of a chain of n linked hexagons.
.
Case n=1: There are 6 vertices of degree 2, so a(1) = 6*2^2 = 24.
Case n=2: There are 8 vertices of degree 2 and 4 of degree 3, so a(2) = 8*2^2 + 4*3^3 = 32 + 36 = 68.
In general, there will be 2n + 4 vertices of degree 2 and 4n - 4 of degree 3.
(End)

Colin Barker, Table of n, a(n) for n = 1..1000
O. Bodroza-Pantic, I. Gutman, and S. J. Cyvin, Fibonacci numbers and algebraic structure count of some non-benzenoid conjugated polymers, The Fibonacci Quarterly, 35, 1, 1997, 75-83.
M. R. Darafsheh, Computation of topological indices of some graphs, Acta Appl. Math., 110, 2010, 1225-1235.
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
P. Gayathri and U. Priyanka, Degree based topological indices of linear phenylene, Internat. J. of Innovative Research in Science, Engineering and Technology,6, 8, 2017, 16986-16997.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017461, A224454, A224455, A304158.

Maple
```
seq(44*n - 20, n = 1 .. 40);
```

Vec(4*x*(6 + 5*x) / (1 - x)^2 + O(x^60)) \\ Colin Barker, May 07 2018

a(n) = 44*n - 20.
a(n) = 4 * A017461(n-1).
From Colin Barker, May 07 2018: (Start)
G.f.: 4*x*(6 + 5*x) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>2.
(End)

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

Original entry on oeis.org

1, 5, 2, 13, 8, 3, 25, 18, 11, 4, 41, 32, 23, 14, 5, 61, 50, 39, 28, 17, 6, 85, 72, 59, 46, 33, 20, 7, 113, 98, 83, 68, 53, 38, 23, 8, 145, 128, 111, 94, 77, 60, 43, 26, 9, 181, 162, 143, 124, 105, 86, 67, 48, 29, 10, 221, 200, 179, 158, 137, 116, 95, 74, 53, 32, 11

Offset: 1

Stefano Spezia, Dec 20 2019

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).

			n\k|   0   1   2   3   4   5
---+------------------------
1  |   1
2  |   5   2
3  |  13   8   3
4  |  25  18  11   4
5  |  41  32  23  14   5
6  |  61  50  39  28  17   6
...
For n = 3 the matrix M is
      1, 2, 3
      2, 4, 6
      3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A005408, A330287.

Cf. A000027: diagonal; A001105: 2nd column; A001844: 1st column; A016789: 1st subdiagonal; A016885: 2nd subdiagonal; A017029: 3rd subdiagonal; A017221: 4th subdiagonal; A017461: 5th subdiagonal; A081436: row sums; A132209: 3rd column; A164284: 7th subdiagonal; A269044: 6th subdiagonal.

Flatten[Table[1+k-2n-2k*n+2n^2,{n,1,11},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]

O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]

cp's OEIS Frontend

A017471 a(n) = (11*n + 6)^11.

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A017472 a(n) = (11*n + 6)^12.

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A076394 a(n) = p(11n+6)/11 where p(n) = number of partitions of n (A000041).

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A213256 p(11n+6) where p(k) = number of partitions of k = A000041(k).

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A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

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A141694 a(n) = 22*n + 12.

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A182719 Numbers of the form 5k + 4, 7k + 5, or 11k + 6.

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.