A027620 -id:A027620 - cp's OEIS frontend

A081437 Diagonal in array of n-gonal numbers A081422.

1, 10, 33, 76, 145, 246, 385, 568, 801, 1090, 1441, 1860, 2353, 2926, 3585, 4336, 5185, 6138, 7201, 8380, 9681, 11110, 12673, 14376, 16225, 18226, 20385, 22708, 25201, 27870, 30721, 33760, 36993, 40426, 44065, 47916, 51985, 56278, 60801, 65560

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.
For q a prime power, a(q-1) = q^3 + q^2 - q is the number of pairs of commuting nilpotent 2*2 matrices with coefficients in GF(q). (Proof: the zero matrix commutes with all q^2 nilpotent matrices, there are q^2-1 nonzero nilpotent matrices, all conjugate, each commuting with q nilpotent matrices.) - Mark Wildon, Jun 20 2017
Also the cyclomatic number (= circuit rank) of the n+1 X n+1 rook graph. - Eric W. Weisstein, Jun 20 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Rook Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081435, A081436, A081437, A081438.

Equals A027620(n-1) + 1.

List([0..40], n-> (n+1)^3+n*(n+1)); # G. C. Greubel, Aug 14 2019

[n^3+4*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

a:=n->sum(n*k, k=0..n):seq(a(n)+sum(n*k, k=2..n), n=1..40); # Zerinvary Lajos, Jun 10 2008
a:=n->sum(-2+sum(2+sum(2, j=1..n),j=1..n),j=1..n):seq(a(n)/2,n=1..40); # Zerinvary Lajos, Dec 06 2008

Table[n^3 + 4 n^2 + 4n + 1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 10, 33, 76}, 40] (* Harvey P. Dale, Jan 24 2012 *)
CoefficientList[Series[(1 + 5 x - 7 x^2 + x^3)/(1 - x)^5, {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (n+1)^3+n*(n+1)) \\ G. C. Greubel, Aug 14 2019

[(n+1)^3+n*(n+1) for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = n^3 + 4*n^2 + 4*n + 1.
G.f.: (1 +5*x -7*x^2 +x^3)/(1-x)^5.
a(0)=1, a(1)=10, a(2)=33, a(3)=76; for n>3, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Harvey P. Dale, Jan 24 2012
E.g.f.: (1 +9*x +7*x^2 +x^3)*exp(x). - G. C. Greubel, Aug 14 2019

A027621 a(n) = n + (n+1)^2 + (n+2)^3 + (n+3)^4.

Original entry on oeis.org

90, 288, 700, 1440, 2646, 4480, 7128, 10800, 15730, 22176, 30420, 40768, 53550, 69120, 87856, 110160, 136458, 167200, 202860, 243936, 290950, 344448, 405000, 473200, 549666, 635040, 729988, 835200, 951390, 1079296, 1219680

Offset: 0

Patrick De Geest

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Patrick De Geest, Palindromic Quasi_Under_Squares of the form n+(n+1)^2
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000027, A027620, A027622, A028387.

[n + (n+1)^2 + (n+2)^3 + (n+3)^4: n in [0..40]]; // Vincenzo Librandi, Aug 05 2011

seq( (n+3)^2*(n^2 + 7*n + 10), n=0..40); # G. C. Greubel, Aug 05 2022

Table[Total[Table[(n+i)^(i+1),{i,0,3}]],{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{90,288,700,1440,2646},40] (* Harvey P. Dale, Jun 08 2017 *)

[i+(i+1)^2+(i+2)^3+(i+3)^4 for i in range(0,40)] # Zerinvary Lajos, Jul 03 2008

G.f.: 16/(1-x) + 16/(1-x)^2 + 16/(1-x)^3 + 18/(1-x)^4 + 24/(1-x)^5. - R. J. Mathar, Feb 22 2008
a(n) = (n+3)^2*(n^2 + 7*n + 10). - Bruno Berselli, Aug 05 2011
E.g.f.: (90 + 198*x + 107*x^2 + 19*x^3 + x^4)*exp(x). - G. C. Greubel, Aug 05 2022

A027622 a(n) = n + (n+1)^2 + (n+2)^3 + (n+3)^4 + (n+4)^5.

Original entry on oeis.org

1114, 3413, 8476, 18247, 35414, 63529, 107128, 171851, 264562, 393469, 568244, 800143, 1102126, 1488977, 1977424, 2586259, 3336458, 4251301, 5356492, 6680279, 8253574, 10110073, 12286376, 14822107, 17760034, 21146189, 25029988, 29464351, 34505822, 40214689

Offset: 0

Patrick De Geest

Colin Barker, Table of n, a(n) for n = 0..1000
Patrick De Geest, Palindromic Quasi_Under_Squares of the form n+(n+1)^2
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000027, A027620, A027621, A028387.

[n+(n+1)^2+(n+2)^3+(n+3)^4+(n+4)^5: n in [0..30]]; // Vincenzo Librandi, Dec 28 2010

seq( add((n+j)^(j+1), j=0..4), n=0..30); # G. C. Greubel, Aug 05 2022

Table[n +(n+1)^2 +(n+2)^3 +(n+3)^4 +(n+4)^5, {n, 0, 29}] (* Alonso del Arte, Nov 22 2016 *)
Table[ReleaseHold@ Total@ MapIndexed[#1^First@ #2 &, Rest@ FactorList[ Pochhammer[Hold@ n, 5]][[All, 1]]], {n, 0, 29}] (* or *)
CoefficientList[Series[(1114 -3271x +4708x^2 -3694x^3 +1522x^4 -259x^5)/(1-x)^6, {x, 0, 29}], x] (* Michael De Vlieger, Dec 05 2016 *)
Table[Total[Table[(n+k)^(k+1),{k,0,4}]],{n,0,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1}, {1114,3413,8476,18247,35414,63529}, 30] (* Harvey P. Dale, Aug 04 2022 *)

Vec((1114-3271*x+4708*x^2-3694*x^3+1522*x^4-259*x^5) / (1-x)^6 + O(x^30)) \\ Colin Barker, Dec 05 2016

[sum((n+j)^(j+1) for j in (0..4)) for n in (0..30)] # G. C. Greubel, Aug 05 2022

From Colin Barker, Dec 05 2016: (Start)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
G.f.: (1114-3271*x+4708*x^2-3694*x^3+1522*x^4-259*x^5) / (1-x)^6.
(End)
E.g.f.: (1114 +2299*x +1382*x^2 +324*x^3 +31*x^4 +x^5)*exp(x). - G. C. Greubel, Aug 05 2022

A379587 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)^2/(k - 1), with k >= 2.

Original entry on oeis.org

0, 1, 0, 9, 2, 0, 49, 32, 3, 0, 225, 338, 75, 4, 0, 961, 3200, 1323, 144, 5, 0, 3969, 29282, 21675, 3844, 245, 6, 0, 16129, 264992, 348843, 97344, 9245, 384, 7, 0, 65025, 2389298, 5589675, 2439844, 335405, 19494, 567, 8, 0, 261121, 21516800, 89467563, 61027344, 12090125, 960000, 37303, 800, 9, 0

Offset: 0

Stefano Spezia, Dec 26 2024

			The array begins as:
    0,     0,      0,       0,        0,        0, ...
    1,     2,      3,       4,        5,        6, ...
    9,    32,     75,     144,      245,      384, ...
   49,   338,   1323,    3844,     9245,    19494, ...
  225,  3200,  21675,   97344,   335405,   960000, ...
  961, 29282, 348843, 2439844, 12090125, 47073606, ...
  ...

Cf. A027620, A060867 (k=2), A060868 (k=3), A060869 (k=4), A060870 (k=5), A060871 (k=7), A361475, A379588 (antidiagonal sums).

A[n_,k_]:=(k^n-1)^2/(k-1); Table[A[n-k+2,k],{n,0,9},{k,2,n+2}]//Flatten

G.f. of column k: (1 - k)*x*(1 + k*x)/((1 - x)*(1 - k*x)*(1 - k^2*x)).
E.g.f. of column k: exp(x)*(1 - 2*exp((k-1)*x) + exp((k^2-1)*x))/(k - 1).
A(2, n) = A027620(n-2) for n > 1.

A152619 n*(n+2)^2.

Original entry on oeis.org

0, 9, 32, 75, 144, 245, 384, 567, 800, 1089, 1440, 1859, 2352, 2925, 3584, 4335, 5184, 6137, 7200, 8379, 9680, 11109, 12672, 14375, 16224, 18225, 20384, 22707, 25200, 27869, 30720, 33759, 36992, 40425, 44064, 47915, 51984, 56277, 60800, 65559

Offset: 0

Philippe Deléham, Dec 09 2008

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Table[n(n+2)^2,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)
CoefficientList[Series[x (9-4x+x^2)/(1-x)^4,{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,9,32,75},50] (* Harvey P. Dale, Aug 25 2023 *)

a(n) = A027620(n-1), n>0.

G.f.: x(9-4x+x^2)/(1-x)^4. [From R. J. Mathar, Dec 10 2008]

A348462 Size of largest bipartite biregular Moore graph of diameter 6 and degrees n and 2.

Original entry on oeis.org

12, 35, 78, 147, 248

Offset: 2

N. J. A. Sloane, Oct 31 2021

a(7) >= 387, a(8) = 570, a(9) = 803, a(10) = 1092.

G. Araujo-Pardo, C. Dalfó, M. Á. Fiol and N. López, Bipartite biregular Moore graphs, arXiv:2103.11443 [math.CO], 2021.
G. Araujo-Pardo, C. Dalfó, M. Á. Fiol and N. López, Bipartite biregular Moore graphs, Discrete Math., 334 (2021), # 112582.

Cf. A027620, A081437, A348461, A348463.

Empirical observation: For the terms a(2)-a(6) and a(8)-a(10) a(n) = A081437(n-1) + 2. It is unknown whether this is also valid for n = 7 and for n > 10. - Hugo Pfoertner, Oct 31 2021

a(n) <= A027620(n-2) + 3 = A081437(n-1) + 2 (the Moore bound). - Pontus von Brömssen, Oct 31 2021

A172176 Triangle T(n, k) = 1 + (n + k - nk)(2n - k - n(n-k)), read by rows.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, -8, 0, 0, -8, -31, -4, 5, -4, -31, -74, -10, 22, 22, -10, -74, -143, -18, 57, 82, 57, -18, -143, -244, -28, 116, 188, 188, 116, -28, -244, -383, -40, 205, 352, 401, 352, 205, -40, -383, -566, -54, 330, 586, 714, 714, 586, 330, -54, -566

Offset: 0

Roger L. Bagula, Jan 28 2010

			Triangle begins as:
     1;
     2,   2;
     1,   2,   1;
    -8,   0,   0,  -8;
   -31,  -4,   5,  -4,  -31;
   -74, -10,  22,  22,  -10,  -74;
  -143, -18,  57,  82,   57,  -18, -143;
  -244, -28, 116, 188,  188,  116,  -28, -244;
  -383, -40, 205, 352,  401,  352,  205,  -40, -383;
  -566, -54, 330, 586,  714,  714,  586,  330,  -54, -566;
  -799, -70, 497, 902, 1145, 1226, 1145,  902,  497,  -70, -799;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A027620, A028552, A033445.

[1 + (n-(n-1)*k)*(n-(n-1)*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2022

A172176:= proc(n,m) 1+(n+m-n*m)*(2*n-m-n*(n-m)); end proc:
seq(seq(A172176(n,m), m=0..n), n=0..12);

T[n_, k_]= 1 + (n-(n-1)*k)*(n-(n-1)*(n-k));
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

def A172176(n,k): return 1 + (n-(n-1)*k)*(n-(n-1)*(n-k))
flatten([[A172176(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2022

T(n, k) = 1 + (n-(n-1)*k)*(n-(n-1)*(n-k)).
T(n, n-k) = T(n, k).
T(n, 0) = 1 - A027620(n-3).
T(n, 1) = -A028552(n-3).
T(n, 2) = A033445(n-2).
Sum_{k=0..n} T(n, k) = (n+1)*(n^4 - 9*n^3 + 15*n^2 - n + 6)/6.

A325173 Perfect squares of the form a + b^2 + c^3, where a,b,c are consecutive numbers.

Original entry on oeis.org

9, 144, 1089, 5184, 18225, 51984, 127449, 278784, 558009, 1040400, 1830609, 3069504, 4941729, 7683984, 11594025, 17040384, 24472809, 34433424, 47568609, 64641600, 86545809, 114318864, 149157369, 192432384, 245705625, 310746384, 389549169, 484352064, 597655809, 732243600

Offset: 1

Philip Mizzi, Sep 05 2019

			9 = 0 + 1^2 + 2^3. 0,1,2 are consecutive numbers and 9 is a perfect square. Hence, 9 is a member of the sequence.
18225 = 24 + 25^2 + 26^3. 24,25,26 are consecutive numbers and 18225 is a perfect square. Hence 18225 is a member of the sequence.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Intersection of A000290 and A027620.

Cf. A005563 (the indices n that give these squares), A054602.

a(n) = n^2*(2 + n^2)^2 \\ David A. Corneth, Sep 11 2019

Vec(9*x*(1 + x)*(1 + 8*x + 22*x^2 + 8*x^3 + x^4) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Sep 11 2019

a(n) = A000290(A054602(n)). - Michel Marcus, Sep 05 2019
From Colin Barker, Sep 05 2019: (Start)
G.f.: 9*x*(1 + x)*(1 + 8*x + 22*x^2 + 8*x^3 + x^4) / (1 - x)^7.
a(n) = n^2*(2 + n^2)^2.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)
E.g.f.: exp(x)*x*(9 + 63*x + 114*x^2 + 69*x^3 + 15*x^4 + x^5). - Conjectured by Stefano Spezia, Sep 05 2019 after Colin Barker
From Chai Wah Wu, Sep 10 2019: (Start)
Above conjectures are true. Proof: k + (k+1)^2 + (k+2)^3 = (k+1)*(k+3)^2 and thus is a perfect square if and only if k+1 = n^2 is a perfect square. This implies that (k+1)*(k+3)^2 = n^2*(n^2+2)^2.
(End)

cp's OEIS Frontend

A081437 Diagonal in array of n-gonal numbers A081422.

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A027621 a(n) = n + (n+1)^2 + (n+2)^3 + (n+3)^4.

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A027622 a(n) = n + (n+1)^2 + (n+2)^3 + (n+3)^4 + (n+4)^5.

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A379587 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)^2/(k - 1), with k >= 2.

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A152619 n*(n+2)^2.

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A348462 Size of largest bipartite biregular Moore graph of diameter 6 and degrees n and 2.

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A172176 Triangle T(n, k) = 1 + (n + k - n*k)*(2*n - k - n*(n-k)), read by rows.

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A325173 Perfect squares of the form a + b^2 + c^3, where a,b,c are consecutive numbers.

A172176 Triangle T(n, k) = 1 + (n + k - nk)(2n - k - n(n-k)), read by rows.