A037270 -id:A037270 - cp's OEIS frontend

A097726 Pell equation solutions (5a(n))^2 - 26b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

1, 103, 10505, 1071407, 109273009, 11144775511, 1136657829113, 115927953794015, 11823514629160417, 1205882564220568519, 122988198035868828521, 12543590317094399940623, 1279323224145592925115025, 130478425272533383961791927, 13307520054574259571177661529

Offset: 0

Wolfdieter Lang, Aug 31 2004

a(-1) = -1. - Artur Jasinski, Feb 10 2010

5*a(n) gives the x-values in the solution to the Pell equation x^2 - 26*y^2 = -1. - Colin Barker, Aug 24 2013

			(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 = -1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (102,-1).

Cf. A097725 for S(n, 102).
Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121. - Artur Jasinski, Feb 10 2010
Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (* Artur Jasinski, Feb 10 2010 *)
CoefficientList[Series[(1 + x)/(1 - 102 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 13 2014 *)
LinearRecurrence[{102,-1},{1,103},20] (* Harvey P. Dale, Aug 20 2017 *)

x='x+O('x^99); Vec((1+x)/(1-102*x+x^2)) \\ Altug Alkan, Apr 05 2018

G.f.: (1 + x)/(1 - 102*x + x^2).
a(n) = S(n, 2*51) + S(n-1, 2*51) = S(2*n, 2*sqrt(26)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 5*i)/(5*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=103. - Philippe Deléham, Nov 18 2008
a(n) = (1/5)*sinh((2*n-1)*arcsinh(5)), n >= 1. - Artur Jasinski, Feb 10 2010

More terms from Harvey P. Dale, Aug 20 2017

A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 9, 1, 1, 20, 45, 16, 1, 1, 35, 165, 136, 25, 1, 1, 56, 495, 816, 325, 36, 1, 1, 84, 1287, 3876, 2925, 666, 49, 1, 1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1, 1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1, 1, 220, 12870, 170544

Offset: 1

Paul D. Hanna, Jul 15 2012

This is also the array A(n,k) read upwards antidiagonals, where the entry in row n and column k counts the vertex-labeled digraphs with n arcs and k vertices, allowing multi-edges and multi-loops (labeled analog to A138107). The binomial formula counts the weak compositions of distributing n arcs over the k^2 positions in the adjacency matrix. - R. J. Mathar, Aug 03 2017

			Triangle begins:
1;
1, 1;
1, 4, 1;
1, 10, 9, 1;
1, 20, 45, 16, 1;
1, 35, 165, 136, 25, 1;
1, 56, 495, 816, 325, 36, 1;
1, 84, 1287, 3876, 2925, 666, 49, 1;
1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1;
1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1;
1, 220, 12870, 170544, 593775, 658008, 270725, 45760, 3321, 100, 1; ...

Paul D. Hanna, Rows n = 0..45, flattened.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 80.

Cf. A214400 (central terms), A178325 (row sums), A054688, A000290 (1st subdiagonal), A037270 (2nd subdiagonal).

Cf. A230049.

A214398 := proc(n,k)
    binomial(k^2+n-k-1,n-k) ;
end proc:
seq(seq(A214398(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Aug 03 2017

nmax = 11;
T[n_, k_] := SeriesCoefficient[1/(1-x)^(k^2), {x, 0, n-k}];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten

T(n,k)=binomial(k^2+n-k-1,n-k)
for(n=1,11,for(k=1,n,print1(T(n,k),", "));print(""))

T(n,k) = binomial(k^2+n-k-1, n-k).
Row sums form A178325.
Central terms form A214400.
T(n,n-2) = A037270(n-2). - R. J. Mathar, Aug 03 2017
T(n,n-3) = (n^2-6*n+11)*(n^2-6*n+10)*(n-3)^2 /6. - R. J. Mathar, Aug 03 2017

A229417 T(n,k) = number of n X n 0..k zero-diagonal arrays with corresponding row and column sums equal.

Original entry on oeis.org

1, 1, 2, 1, 3, 10, 1, 4, 45, 152, 1, 5, 136, 4743, 7736, 1, 6, 325, 59008, 3801411, 1375952, 1, 7, 666, 426425, 345706336, 23938685973, 877901648, 1, 8, 1225, 2164680, 11782824375, 28256240134144, 1215663478473627, 2046320373120, 1, 9, 2080

Offset: 1

R. H. Hardin, Sep 22 2013

Table starts
.........1................1....................1................1............1
.........2................3....................4................5............6
........10...............45..................136..............325..........666
.......152.............4743................59008...........426425......2164680
......7736..........3801411............345706336......11782824375.213067487016
...1375952......23938685973.......28256240134144.7093199984236625
.877901648.1215663478473627.33097994593655140864

			Some solutions for n=4 k=4
..0..0..2..0....0..1..0..4....0..0..1..3....0..1..1..4....0..1..1..0
..1..0..2..1....2..0..4..0....1..0..2..3....4..0..2..3....0..0..1..2
..1..2..0..4....2..4..0..2....2..3..0..1....1..4..0..1....0..0..0..4
..0..2..3..0....1..1..4..0....1..3..3..0....1..4..3..0....2..2..2..0

R. H. Hardin, Table of n, a(n) for n = 1..43

Columns 1..3 are A007080, A229415, A229416.
Rows 3..6 are A037270(n+1), A229418, A229419, A229420.
Cf. A229870.

Empirical for row n:
n=1: a(n) = 1
n=2: a(n) = n + 1
n=3: a(n) = (1/2)*n^4 + 2*n^3 + (7/2)*n^2 + 3*n + 1
n=4: [polynomial of degree 9]
Row n is an Ehrhart polynomial of degree (n-1)^2 for the polytope of x(i,j), i,j = 1..n for j <> i, with 0 <= x(i,j) <= 1 and Sum_i x(i,j) = Sum_i x(j,i). - Robert Israel, Mar 30 2023
T(n,k) = A229870(n,k) / (k + 1)^n. - Andrew Howroyd, Mar 30 2023

A337953 Number of achiral colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

Original entry on oeis.org

1, 33328, 32524281, 4312863360, 191243490675, 4239501280272, 58236754527707, 563536359633920, 4172726943804861, 25016666666700400, 126431377927701253, 554909560378102656, 2163457078062360639, 7625429483925609552, 24638829565429941975

Offset: 1

Robert A. Russell, Oct 03 2020

An achiral coloring is identical to its reflection. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.
There are 60 elements in the automorphism group of the regular dodecahedron/icosahedron that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^15
  Edge rotation*        15     x_1^4x_2^13     Asterisk indicates that the
  Vertex rotation*      20     x_6^5           operation is followed by an
  Small face rotation*  12     x_10^3          inversion.
  Large face rotation*  12     x_10^3

Index entries for linear recurrences with constant coefficients, signature (18, -153, 816, -3060, 8568, -18564, 31824, -43758, 48620, -43758, 31824, -18564, 8568, -3060, 816, -153, 18, -1).

Cf. A282670 (oriented), A337963 (unoriented), A337964 (chiral).
Other elements: A337960 (dodecahedron vertices, icosahedron faces), A337962 (dodecahedron faces, icosahedron vertices).
Cf. A037270 (tetrahedron), A331351 (cube/octahedron).

Table[(15n^17+n^15+20n^5+24n^3)/60,{n,30}]

a(n) = n^3 * (15*n^14 + n^12 + 20*n^2 + 24) / 60.
a(n) = 1*C(n,1) + 33326*C(n,2) + 32424300*C(n,3) + 4182966200*C(n,4) + 170004083410*C(n,5) + 3156083300916*C(n,6) + 32426546302332*C(n,7) + 205938803790720*C(n,8) + 864860752435680*C(n,9) + 2503126577952000*C(n,10) + 5110943178781440*C(n,11) + 7428048096268800*C(n,12) + 7644417350169600*C(n,13) + 5446616304729600*C(n,14) + 2556525184012800*C(n,15) + 711374856192000*C(n,16) + 88921857024000*C(n,17), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A337963(n) - A282670(n) = A282670(n) - 2*A337964(n) = A337963(n) - A337964(n).

A071237 a(n) = n(n+1)(n^2+1)/2.

Original entry on oeis.org

0, 2, 15, 60, 170, 390, 777, 1400, 2340, 3690, 5555, 8052, 11310, 15470, 20685, 27120, 34952, 44370, 55575, 68780, 84210, 102102, 122705, 146280, 173100, 203450, 237627, 275940, 318710, 366270, 418965, 477152, 541200, 611490, 688415, 772380, 863802, 963110

Offset: 0

N. J. A. Sloane, Jun 12 2002

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A027445, A037270.

[n*(n+1)*(n^2+1)/2: n in [0..40] ]; // Vincenzo Librandi, May 23 2011

Table[(n^4 + n^3 + n^2 + n)/2, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,15,60,170},40] (* Harvey P. Dale, Feb 08 2025 *)

def A071237(n): return (n^2+1)*binomial(n+1,2)
[A071237(n) for n in range(51)] # G. C. Greubel, Aug 05 2024

From Arkadiusz Wesolowski, Apr 01 2012: (Start)
a(n) = A000217(n)*A002522(n).
a(0) = 0, a(1) = 2; for n >= 2, a(n) = ceiling(n^5/(2*n-2)) - 1.
G.f.: x*(2 + 5*x*(1 + x))/(1 - x)^5. (End)
a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5) for n>4, a(0)=0, a(1)=2, a(2)=15, a(3)=60, a(4)=170. - Yosu Yurramendi, Sep 03 2013
E.g.f.: (1/2)*x*(4 + 11*x + 7*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 05 2024

A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .

Original entry on oeis.org

1, 1, 8, 1, 1, 8, 27, 8, 1, 1, 8, 27, 64, 27, 8, 1, 1, 8, 27, 64, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 729

Offset: 0

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,8,1,1,8,27,8,1,..., analogous to A004737.

T(n,k) = min(n,k)^3. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

			Triangle starts
  1;
  1, 8, 1;
  1, 8, 27, 8, 1;
  1, 8, 27, 64, 27, 8, 1;
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...8...8...8...8...8...
  1...8..27..27..27..27...
  1...8..27..64..64..64...
  1...8..27..64.125.125...
  1...8..27..64.125.216...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  1,8,1;
  1,8,27,8,1;
  1,8,27,64,27,8,1;
  1,8,27,64,125,64,27,8,1;
  1,8,27,64,125,216,125,64,27,8,1;
  . . .
Row number k contains 2*k-1 numbers 1,8,...,(k-1)^3,k^3,(k-1)^3,...,8,1. (End)

Harvey P. Dale, Table of n, a(n) for n = 0..9999
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A004737, A037270 (row sums), A133820, A124258, A133824, A003983.

Table[Join[Range[n]^3,Range[n-1,1,-1]^3],{n,10}]//Flatten (* Harvey P. Dale, May 29 2019 *)

O.g.f.: (1+qx)(1+4qx+q^2x^2)/((1-x)(1-qx)^3(1-q^2x)) = 1 + x(1 + 8q + q^2) + x^2(1 + 8q + 27q^2 + 8q^3 + q^4) + ... .
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = (A004737(n))^3.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^3. (End)

A329636 Numbers that are sums of consecutive centered cube numbers (A005898).

Original entry on oeis.org

1, 9, 10, 35, 44, 45, 91, 126, 135, 136, 189, 280, 315, 324, 325, 341, 530, 559, 621, 656, 665, 666, 855, 900, 1089, 1180, 1215, 1224, 1225, 1241, 1414, 1729, 1755, 1944, 2035, 2070, 2079, 2080, 2096, 2331, 2655, 2970, 2996, 3059, 3185, 3276, 3311, 3320, 3321, 3825, 3925

Offset: 1

Ilya Gutkovskiy, Nov 18 2019

nonn

Eric Weisstein's World of Mathematics, Centered Cube Number

Cf. A005898, A037270, A217843, A322611, A329634, A329635.

A182427 Triangular numbers that can be represented as a sum of a nonzero square number and a nonzero triangular number.

Original entry on oeis.org

10, 15, 28, 45, 55, 91, 136, 190, 210, 231, 253, 325, 378, 406, 435, 496, 561, 595, 666, 703, 741, 820, 861, 903, 946, 990, 1081, 1128, 1176, 1225, 1378, 1431, 1540, 1596, 1711, 1770, 1830, 1891, 2080, 2145, 2211, 2278, 2346, 2415, 2485, 2556, 2701, 2926, 3160, 3321

Offset: 1

Ivan N. Ianakiev, Apr 28 2012

nonn

Theorem (by Ivan N. Ianakiev): There are infinitely many such numbers. Proof: Any triangular number of the form A000217(n^2) for n>1 is such a number, as A000217(n^2) = A000217(n^2-1) + A000290(n), for n>=1. Observation: Other numbers not of the form A000217(n^2), for example 15 and 28, are also in A182427. - Ivan N. Ianakiev, May 30 2012

For any integer k>1, all triangular numbers with indices of the form 3*k-2 (A060544) are terms as (3*k-2)*(3*k-1)/2 = (2*k-1)^2 + (k-1)*k/2. - Ivan N. Ianakiev, Nov 25 2015

			10, 15, 28 are in the sequence because 10 = 2^2 + 3*4/2 = 3^2 + 1*2/2, 15 = 3^2 + 3*4/2, 28 = 5^2 + 2*3/2.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000217, A000290, A037270.

isok(t) = {for (k=1, sqrtint(t), my(tt = t - k^2); if ((tt) && ispolygonal(tt, 3), return (1)););}
lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); if (isok(t), print1(t, ", ")););} \\ Michel Marcus, Nov 25 2015

A322844 a(n) = (1/12)n^2(3(1 + n^2) - 2(2 + n^2)*(n mod 2)).

Original entry on oeis.org

0, 0, 5, 6, 68, 50, 333, 196, 1040, 540, 2525, 1210, 5220, 2366, 9653, 4200, 16448, 6936, 26325, 10830, 40100, 16170, 58685, 23276, 83088, 32500, 114413, 44226, 153860, 58870, 202725, 76880, 262400, 98736, 334373, 124950, 420228, 156066, 521645, 192660, 640400, 235340

Offset: 0

Stefano Spezia, Dec 28 2018

Conjectures: (Start)
For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix M(n) defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even (see A317614). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix M(n), or the absolute value of the sum of all principal minors of M(n) of size 2.
For k > 2, the trace of the k-th exterior power of the matrix M(n) is equal to zero.
(End)

Stefano Spezia, Table of n, a(n) for n = 0..10000
Wikipedia, Characteristic polynomial
Wikipedia, Exterior algebra
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A317614 (trace of matrix M(n)).

Cf. A002415, A037270, A074147 (antidiagonals of M matrices), A241016 (row sums of M matrices), A317617 (column sums of M matrices), A322277 (permanent of matrix M(n)), A323723 (subdiagonal sum of M matrices), A323724 (superdiagonal sum of M matrices), A325516 (k-superdiagonal sum of M matrices), A325655 (k-subdiagonal sum of M matrices).

Flat(List([0..50], n->(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n mod 2))));

[IsEven(n) select (1/4)*n^2*(1 + n^2) else (1/12)*(- 1 + n)*n^2*(1 + n): n in [0..50]];

a:=n->(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*modp(n,2)): seq(a(n), n=0..50);

a[n_]:=(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*Mod[n,2]); Array[a,50,0]
LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{0,0,5,6,68,50,333,196,1040,540},50] (* Harvey P. Dale, Aug 23 2025 *)

a(n):=(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*mod(n,2))$ makelist(a(n), n, 0, 50);

a(n) = (1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n % 2));

a(n) = abs(polcoeff(charpoly(matrix(n, n, i, j, if (i %2, j + n*(i-1), n*i - j + 1))), n-2)); \\ Michel Marcus, Feb 06 2019

[int(n**2*(3*(1 + n**2) - 2*(2 + n**2)*pow(n, 1, 2))/12) for n in range(0,50)]

O.g.f.: -x^2*(5 + 6*x + 43*x^2 + 20*x^3 + 43*x^4 + 6*x^5 + 5*x^6)/((-1 + x)^5*(1 + x)^5).
E.g.f.: (1/(12*x^2))*exp(-x)*(24 - 60*exp(x) + 21*x + 9*x^2 + 2*x^3 + x^4 + exp(2*x)*(36 - 33*x + 15*x^2 - 4*x^3 + 2*x^4)).
a(n) = (1/4)*n^2*(1 + n^2) for n even.
a(n) = (1/2)*A037270(n) for n even.
a(n) = (1/12)*(-1 + n)*n^2*(1 + n) for n odd.
a(n) = A002415(n) for n odd.
a(2*n+1) = 5*a(2*n-1) - 10*a(2*n-3) + 10*a(2*n-5) - 5*a(2*n-7) + a(2*n-9), for n > 4.
a(2*n) = 5*a(2*n-2) - 10*a(2*n-4) + 10*a(2*n-6) - 5*a(2*n-8) + a(2*n-10), for n > 4.
O.g.f. for a(2*n+1): -x*(2*(3 + 10*x + 3*x^2))/(-1 + x)^5.
O.g.f. for a(2*n): x*(-5 - 43*x - 43*x^2 - 5*x^3)/(-1 + x)^5.
E.g.f. for a(2*n+1): (1/12)*(6*x*cosh(sqrt(x)) + sqrt(x)*(6 + x)*sinh(sqrt(x))).
E.g.f. for a(2*n): (1/4)*(x*(8 + x)*cosh(sqrt(x)) + 2*sqrt(x)*(1 + 3*x)*sinh(sqrt(x))).
Sum_{k>=1} 1/a(2*k) = (1/6)*(12 + Pi^2 - 6*Pi*coth(Pi/2)) = 0.21955691692893092525407699347398665248691900...
Sum_{k>=1} 1/a(2*k+1) = 3*(5 - Pi^2/2) = 0.1955933983659620717482635001857732970...
Sum_{k>=2} 1/a(k) = 17 - (4*Pi^2)/3 - Pi*coth(Pi/2) = 0.415150315294892997002340493659759949516369894...

A341736 a(n) is the label of the square of the n-th element in the semigroup S = {(0,0), (i,j): i >= j >= 1}.

Original entry on oeis.org

0, 1, 7, 10, 37, 40, 45, 121, 124, 129, 136, 301, 304, 309, 316, 325, 631, 634, 639, 646, 655, 666, 1177, 1180, 1185, 1192, 1201, 1212, 1225, 2017, 2020, 2025, 2032, 2041, 2052, 2065, 2080, 3241, 3244, 3249, 3256, 3265, 3276, 3289, 3304, 3321, 4951, 4954, 4959

Offset: 0

Alois P. Heinz, Feb 17 2021

nonn

The product in S is computed componentwise.

For the labeling of the elements in S and further information see A341317.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Main diagonal of A341317 and of A341318.

Cf. A000217, A000290, A037270, A123578, A161680.

t:= n-> n*(n-1)/2:
f:= n-> ceil((sqrt(1+8*n)-1)/2):
g:= n-> (x-> [x, n-t(x)])(f(n)):
a:= n-> (h-> t(h[1]^2)+h[2]^2)(g(n)):
seq(a(n), n=0..60);

t[n_] := n*(n - 1)/2;
f[n_] := Ceiling[(Sqrt[1 + 8*n] - 1)/2];
g[n_] := Function[x, {x, n - t[x]}][f[n]];
a[n_] := Function[h, t[h[[1]]^2] + h[[2]]^2][g[n]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

a(n) = A341317(n,n) = A341318(n,n).

a(A000217(n)) = A037270(n) = A000217(A000290(n)).

cp's OEIS Frontend

A097726 Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A229417 T(n,k) = number of n X n 0..k zero-diagonal arrays with corresponding row and column sums equal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A337953 Number of achiral colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A071237 a(n) = n*(n+1)*(n^2+1)/2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A329636 Numbers that are sums of consecutive centered cube numbers (A005898).

Views

Author

Keywords

Links

Crossrefs

A182427 Triangular numbers that can be represented as a sum of a nonzero square number and a nonzero triangular number.

Views

Author

Keywords

Comments

A097726 Pell equation solutions (5a(n))^2 - 26b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

A071237 a(n) = n(n+1)(n^2+1)/2.

A322844 a(n) = (1/12)n^2(3(1 + n^2) - 2(2 + n^2)*(n mod 2)).