A002412 -id:A002412 - cp's OEIS frontend

A193068 Generating primitive Pythagorean triangles by using (n, n+1) gives perimeters for each n. This sequence lists the sum of these perimeters for each n triangles.

12, 42, 98, 188, 320, 502, 742, 1048, 1428, 1890, 2442, 3092, 3848, 4718, 5710, 6832, 8092, 9498, 11058, 12780, 14672, 16742, 18998, 21448, 24100, 26962, 30042, 33348, 36888, 40670, 44702, 48992, 53548, 58378, 63490, 68892, 74592, 80598, 86918, 93560

Offset: 1

J. M. Bergot, Jul 15 2011

Partial sums of A002939 starting at A002939(2). - R. J. Mathar, Aug 23 2011

			The perimeters of the first five triangles produced by pairs (1,2), (2,3), (3,4), (4,5), (5,6) are in order 12, 30, 56, 90, 132 with sum 320.
From the formula, a(5) = 5*(4*5^2 + 15*5 + 17)/3 = 320.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A083374 (sum of areas for the first n triangles), A002412.

I:=[12, 42, 98, 188]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 04 2012

CoefficientList[Series[(2*(6-3*x+x^2))/((x-1)^4),{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)
LinearRecurrence[{4,-6,4,-1},{12,42,98,188},40] (* Harvey P. Dale, Oct 29 2022 *)

a(n) = n*(4*n^2 + 15*n + 17)/3.
G.f.: ( 2*x*(6-3*x+x^2) ) / ( (x-1)^4 ). - R. J. Mathar, Aug 23 2011
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Vincenzo Librandi, Jul 04 2012
a(n) = 2*(A002412(n+1) - 1). - Hugo Pfoertner, Oct 22 2024

A266387 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

Original entry on oeis.org

0, 0, 0, 0, 0, 7, 42, 147, 392, 882, 1764, 3234, 5544, 9009, 14014, 21021, 30576, 43316, 59976, 81396, 108528, 142443, 184338, 235543, 297528, 371910, 460460, 565110, 687960, 831285, 997542, 1189377, 1409632, 1661352, 1947792, 2272424, 2638944, 3051279

Offset: 1

Philippe A.J.G. Chevalier, Dec 28 2015

The sequence was discovered by enumerating all orbits of Aut(Z^7) and sorting the orbits as function of the infinity norm of the representative integer lattice points. This sequence is one of the 30 sequences that are obtained by classifying the orbits in a table with the rows being the infinity norm and the columns being the 30 cardinalities (1, 14, 84, 128, 168, 280, 448, 560, 672, 840, 896, 1680, 2240, 2688, 3360, 4480, 5376, 6720, 8960, 13440, 17920, 20160, 26880, 40320, 53760, 80640, 107520, 161280, 322560, 645120) generated by signed permutations of integer lattice points of Z^7.

The continued fraction expansion of this sequence is finite and simplifies to the g.f. 7*x^6/(1-x)^6 (see Mathematica). - Benedict W. J. Irwin, Feb 09 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Other sequences that give the number of orbits of Aut(Z^7) as function of the infinity norm for different cardinalities of these orbits: A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

Join[{0, 0, 0, 0, 0},Table[Abs[SeriesCoefficient[Series[7/(x+6/(x - 5/2/(x + ContinuedFractionK[If[Mod[k, 2] ==0, (7 + k/2)/(6 + 2 k), ((k + 1)/2 - 5)/(2 (k - 1) +6)], x, {k, 0, 8}]))), {x, Infinity, 101}],2 n + 1]], {n, 0, 50}]] - (* Benedict W. J. Irwin, Feb 09 2016 *)

concat(vector(5), Vec(7*x^6/(1-x)^6 + O(x^50))) \\ Colin Barker, May 04 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)/120.
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>6.
G.f.: 7*x^6 / (1-x)^6.
(End)

A266395 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 161280.

Original entry on oeis.org

0, 0, 0, 0, 15, 75, 225, 525, 1050, 1890, 3150, 4950, 7425, 10725, 15015, 20475, 27300, 35700, 45900, 58140, 72675, 89775, 109725, 132825, 159390, 189750, 224250, 263250, 307125, 356265, 411075, 471975, 539400, 613800, 695640, 785400, 883575, 990675, 1107225

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(4), Vec(15*x^5/(1-x)^5 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = 5*(n-1)*(n-2)*(n-3)*(n-4)/8 = 15*A000332(n-1).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5.
G.f.: 15*x^5 / (1-x)^5.
(End)

A266396 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 80640.

Original entry on oeis.org

0, 0, 0, 10, 41, 105, 215, 385, 630, 966, 1410, 1980, 2695, 3575, 4641, 5915, 7420, 9180, 11220, 13566, 16245, 19285, 22715, 26565, 30866, 35650, 40950, 46800, 53235, 60291, 68005, 76415, 85560, 95480, 106216, 117810, 130305, 143745, 158175, 173641, 190190

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

LinearRecurrence[{5,-10,10,-5,1},{0,0,0,10,41},50] (* Harvey P. Dale, Nov 18 2024 *)

concat(vector(3), Vec(x^4*(10-9*x)/(1-x)^5 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (n^4+30*n^3-205*n^2+390*n-216)/24.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5.
G.f.: x^4*(10-9*x) / (1-x)^5.
(End)

A266397 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 26880.

Original entry on oeis.org

0, 0, 9, 31, 70, 130, 215, 329, 476, 660, 885, 1155, 1474, 1846, 2275, 2765, 3320, 3944, 4641, 5415, 6270, 7210, 8239, 9361, 10580, 11900, 13325, 14859, 16506, 18270, 20155, 22165, 24304, 26576, 28985, 31535, 34230, 37074, 40071, 43225, 46540, 50020, 53669

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(2), Vec(x^3*(9-5*x)/(1-x)^4 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (4*n^3+3*n^2-37*n+30)/6.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x^3*(9-5*x) / (1-x)^4.
(End)

A279662 a(n) = (2/3)^nGamma(n+3/4)Gamma(n+1)*Gamma(n+2)/Gamma(3/4).

Original entry on oeis.org

1, 1, 7, 154, 7700, 731500, 117771500, 29678418000, 11040371496000, 5796195035400000, 4144279450311000000, 3920488359994206000000, 4790836775912919732000000, 7411424492337286825404000000, 14266992147749277138902700000000, 33670101468688294047810372000000000

Offset: 0

Ilya Gutkovskiy, Dec 16 2016

nonn

Hexagonal pyramidal factorial numbers.

More generally, the m-gonal pyramidal factorial numbers is 6^(-n)*(m-2)^n*Gamma(n+1)*Gamma(n+2)*Gamma(n+3/(m-2))/Gamma(3/(m-2)), m>2.

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for sequences related to factorial numbers

Cf. A002412.
Cf. A000680 (hexagonal factorial numbers).
Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279663 (heptagonal pyramidal factorial numbers).

[Round((2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2) / Gamma(3/4)): n in [0..20]]; // Vincenzo Librandi, Dec 17 2016

FullSimplify[Table[(2/3)^n Gamma[n + 3/4] Gamma[n + 1] Gamma[n + 2]/Gamma[3/4], {n, 0, 15}]]

a(n) = Product_{k=1..n} k*(k + 1)*(4*k - 1)/6, a(0)=1.
a(n) = Product_{k=1..n} A002412(k), a(0)=1.
a(n) ~ (2*Pi)^(3/2)*(2/3)^n*n^(3*n+9/4)/(Gamma(3/4)*exp(3*n)).

A376923 T(n, k) = T(n - 1, k) + 2^(n - 1)*T(n - 2, k - 1), if k > 0 and T(n, 0) = 2^n.

Original entry on oeis.org

1, 2, 0, 4, 2, 0, 8, 10, 0, 0, 16, 42, 16, 0, 0, 32, 170, 176, 0, 0, 0, 64, 682, 1520, 512, 0, 0, 0, 128, 2730, 12400, 11776, 0, 0, 0, 0, 256, 10922, 99696, 206336, 65536, 0, 0, 0, 0, 512, 43690, 798576, 3380736, 3080192, 0, 0, 0, 0, 0, 1024, 174762, 6390640, 54425088, 108724224, 33554432, 0, 0, 0, 0, 0

Offset: 0

Thomas Scheuerle, Oct 17 2024

This is the case r = 2 of the more general recurrence: T(n, k, r) = T(n-1, k, r) + r^(n-1)*T(n - 2, k - 1, r), if k > 0 and T(n, 0, r) = 1 + (r^n - 1)/(r - 1) if r > 1. Consider the sequence b(n) = Sum_{k=0..n-1} b(n - k - 1)*T(n - 1, k, r)*(-1)^k, with b(0) = 1. The sequence b(n) will have an ordinary generating function which can be represented as the continued fraction expansion: 1/(1 - x/(1 - r^0*x/(1 - r^1*x/(1 - r^2*x/(1 - r^3*x/(...)))))). In short b(n) will have the ordinary generating function 1/(1-G(x)*x), where G(x) is the generating function of the Carlitz-Riordan q-Catalan numbers for q = r. The Hankel determinant of b(0)..b(2*n) will be r^A016061(n). The Hankel determinant of b(1)..b(2*n+1) will be r^A002412(n).

			Triangle begins:
n\k  0 |  1 |  2 | 3 | 4 | 5
[0]  1,
[1]  2,   0
[2]  4,   2,   0
[3]  8,  10,   0,  0
[4] 16,  42,  16,  0,  0
[5] 32, 170, 176,  0,  0,  0

Cf. A002412, A015083, A016061, A377132.

T(n, k) = if(n < 0, return(0), return(if(k == 0, return(2^n), T(n-1,k) + 2^(n-1)*T(n-2,k-1))))

Column k has o.g.f.: x^(2*k)*2^(k^2)/((1 - 2^(k+1)*x)*Product_{m=1..k}(1 - 2^(m-1)*x)).

A377132 The expansion of 1/(1 - G(x)*x), where G(x) is the ordinary generating function of the Carlitz-Riordan q-Catalan numbers for q = 2 (A015083).

Original entry on oeis.org

1, 1, 2, 6, 28, 228, 3592, 113880, 7267952, 929696784, 237968445472, 121835841547872, 124758916812038592, 255505766282965942848, 1046551115668335283290240, 8573345713494489300568753536, 140465691975467799273799959144192, 4602779760325164559879331800453222656, 301647773810532495378626041621616755442176, 39537576990478498231890121766124629197694682624

Offset: 0

Thomas Scheuerle, Oct 19 2024

nonn

Cf. A002412, A015083, A016061, A376923.

A376923(n, k) = if(n < 0, return(0),return(if(k == 0, return(2^n), T(n-1, k) + 2^(n-1)*T(n-2, k-1) )))
a(n) = if(n==0, 1, sum(k=0, n-1, a(n-k-1)*A376923(n-1, k)*(-1)^k))

O.g.f.: Continued fraction expansion 1/(1 - x/(1 - 2^0*x/(1 - 2^1*x/(1 - 2^2*x/(1 - 2^3*x/(...)))))).
a(n) = Sum_{k=0..n-1} a(n - k - 1)*A376923(n - 1, k)*(-1)^k, with a(0) = 1.
The Hankel determinant of a(0)..a(2*n) is 2^A016061(n). The Hankel determinant of a(1)..a(2*n+1) is 2^A002412(n).

A128116 A128064 * A122432 (unsigned).

Original entry on oeis.org

1, 5, 2, 12, 7, 3, 22, 15, 9, 4, 35, 26, 18, 11, 5, 51, 40, 30, 21, 13, 6, 70, 57, 45, 34, 24, 15, 7, 92, 77, 63, 50, 38, 27, 17, 8, 117, 100, 84, 69, 55, 42, 30, 19, 9, 145, 126, 108, 91, 75, 60, 46, 33, 21, 10

Offset: 1

Gary W. Adamson, Feb 14 2007

Left border of the triangle = A000326, the pentagonal numbers: (1, 5, 12, 22, 35, ...).

Row sums = A002412: (1, 7, 22, 50, 95, ...).

			First few rows of the triangle:
   1;
   5,  2;
  12,  7,  3;
  22, 15,  9,  4;
  35, 26, 18, 11,  5;
  51, 40, 30, 21, 13,  6;
  70, 57, 45, 34, 24, 15,  7;
  ...

Cf. A128064, A122432, A000326, A002412.

A128064 * A122432 (unsigned), where the unsigned version of A122432 = (1; 3, 1; 6, 3, 1; 10, 6, 3, 1; ...).

A267522 a(n) = 4(n + 1)(n + 2)(4n + 3)/3.

Original entry on oeis.org

8, 56, 176, 400, 760, 1288, 2016, 2976, 4200, 5720, 7568, 9776, 12376, 15400, 18880, 22848, 27336, 32376, 38000, 44240, 51128, 58696, 66976, 76000, 85800, 96408, 107856, 120176, 133400, 147560, 162688, 178816, 195976, 214200, 233520, 253968, 275576, 298376, 322400

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

Partial sums of A152750.

			a(0) = (0 + 2)*(1 + 3) = 8;
a(1) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) = 56;
a(2) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) = 176;
a(3) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) + (6 + 8)*(7 + 9) = 400, etc

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002412, A152750, A268684.

Table[(4 (n + 1)) (n + 2) ((4 n + 3)/3), {n, 0, 38}]
LinearRecurrence[{4, -6, 4, -1}, {8, 56, 176, 400}, 39]

a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3; \\ Michel Marcus, Apr 10 2016

x='x+O('x^99); Vec(8*(1+3*x)/(1-x)^4) \\ Altug Alkan, Apr 10 2016

G.f.: 8*(1 + 3*x)/(1 - x)^4.
E.g.f.: (4/3)*exp(x)*(6 + 36*x + 27*x^2 + 4*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 4*A268684(n + 1).
Sum_{n>=0} 1/a(n) = -3*(2*Pi - 12*log(2) + 1)/20 = 0.15518712893...
a(n) = 8*A002412(n+1). - Yasser Arath Chavez Reyes, Feb 23 2024

cp's OEIS Frontend

A193068 Generating primitive Pythagorean triangles by using (n, n+1) gives perimeters for each n. This sequence lists the sum of these perimeters for each n triangles.

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A266387 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

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A266395 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 161280.

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A266396 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 80640.

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A266397 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 26880.

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A279662 a(n) = (2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2)/Gamma(3/4).

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A376923 T(n, k) = T(n - 1, k) + 2^(n - 1)*T(n - 2, k - 1), if k > 0 and T(n, 0) = 2^n.

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A377132 The expansion of 1/(1 - G(x)*x), where G(x) is the ordinary generating function of the Carlitz-Riordan q-Catalan numbers for q = 2 (A015083).

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A279662 a(n) = (2/3)^nGamma(n+3/4)Gamma(n+1)*Gamma(n+2)/Gamma(3/4).

A267522 a(n) = 4(n + 1)(n + 2)(4n + 3)/3.