A158482 -id:A158482 - cp's OEIS frontend

A195020 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The edges of the spiral have length A195019.

0, 3, 7, 13, 21, 30, 42, 54, 70, 85, 105, 123, 147, 168, 196, 220, 252, 279, 315, 345, 385, 418, 462, 498, 546, 585, 637, 679, 735, 780, 840, 888, 952, 1003, 1071, 1125, 1197, 1254, 1330, 1390, 1470, 1533, 1617, 1683, 1771, 1840, 1932, 2004, 2100

Offset: 0

Omar E. Pol, Sep 07 2011 - Sep 12 2011

Zero together with the partial sums of A195019.
The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives A008587. The vertices on the main diagonal are the numbers A024966 = (3+4)*A000217 = 7*A000217, where both 3 and 4 are the first two edges in the spiral. The distance "a" between nearest edges that are perpendicular to the initial edge of the spiral is 3, while the distance "b" between nearest edges that are parallel to the initial edge is 4, so the distance "c" between nearest vertices on the same axis is 5 because from the Pythagorean theorem we can write c = (a^2+b^2)^(1/2) = sqrt(3^2+4^2) = sqrt(9+16) = sqrt(25) = 5.
Let an array have m(0,n)=m(n,0)=n*(n-1)/2 and m(n,n)=n*(n+1)/2. The first n+1 terms in row(n) are the numbers in the closed interval m(0,n) to m(n,n). The terms in column(n) are the same from m(n,0) to m(n,n). The first few antidiagonals are 0; 0,0; 1,1,1; 3,2,2,3; 6,4,3,4,6; 10,7,5,5,7,10. a(n) is the difference between the sum of the terms in the n+1 X n+1 matrices and those in the n X n matrices. - J. M. Bergot, Jul 05 2013 [The first five rows are: 0,0,1,3,6; 0,1,2,4,7; 1,2,3,5,8; 3,4,5,6,9; 6,7,8,9,10]

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ron Knott, Pythagorean triangles and triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A024966, A008585, A008586, A001106, A022264, A024966, A033572, A144555, A158482, A158485, A195018, A195032, A195034, A195036.

[(2*n*(7*n+13)+(2*n-5)*(-1)^n+5)/16: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

With[{r = Range[50]}, Join[{0}, Accumulate[Riffle[3*r, 4*r]]]] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 3, 7, 13, 21}, 100] (* Paolo Xausa, Feb 09 2024 *)

From Bruno Berselli, Oct 13 2011: (Start)
G.f.: x*(3+4*x)/((1+x)^2*(1-x)^3).
a(n) = (1/2)*A004526(n+2)*A047335(n+1) = (2*n*(7*n+13) + (2*n-5)*(-1)^n+5)/16.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) - a(n-2) = A047355(n+1). (End)

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.

Original entry on oeis.org

1, 5, 10, 17, 26, 36, 49, 62, 79, 95, 116, 135, 160, 182, 211, 236, 269, 297, 334, 365, 406, 440, 485, 522, 571, 611, 664, 707, 764, 810, 871, 920, 985, 1037, 1106, 1161, 1234, 1292, 1369, 1430, 1511, 1575, 1660, 1727, 1816, 1886, 1979, 2052, 2149, 2225, 2326

Offset: 0

Bruno Berselli, Oct 20 2011 - based on remarks and sequences by Omar E. Pol

For an origin of this sequence, see the numerical spiral illustrated in the Links section.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A195020 (vertices of the numerical spiral in link).

Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A195021, A195023-A195030, A195320, A198017 [incomplete list].

[(14*n*(n+3)+(2*n-5)*(-1)^n+21)/16: n in [0..50]];

Table[(14*n*(n + 3) + (2*n - 5)*(-1)^n + 21)/16, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,10,17,26},60] (* Harvey P. Dale, Jun 19 2020 *)

for(n=0, 50, print1((14*n*(n+3)+(2*n-5)*(-1)^n+21)/16", "));

O.g.f.: (1 + 4*x + 3*x^2 - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: (1/16)*((21 + 56*x + 14*x^2)*exp(x) - (5 + 2*x)*exp(-x)). - G. C. Greubel, Aug 19 2017
a(n) = A195020(n) + n + 1.
a(n)   - a(-n-1)  = A047336(n+1).
a(n+1) - a(-n)    = A113804(n+1).
a(n+2) - a(n)     = A047385(n+3).
a(n+4) - a(n)     = A113803(n+4).
a(2*n) + a(2*n-1) = A069127(n+1).
a(2*n) - a(2*n-1) = A016813(n).
a(2*n+1) - a(2*n) = A016777(n+1).
a(n+2) + 2*a(n+1) + a(n) = A024966(n+2).

A198017 a(n) = n(7n + 11)/2 + 1.

Original entry on oeis.org

1, 10, 26, 49, 79, 116, 160, 211, 269, 334, 406, 485, 571, 664, 764, 871, 985, 1106, 1234, 1369, 1511, 1660, 1816, 1979, 2149, 2326, 2510, 2701, 2899, 3104, 3316, 3535, 3761, 3994, 4234, 4481, 4735, 4996, 5264, 5539, 5821, 6110, 6406, 6709, 7019, 7336, 7660, 7991

Offset: 0

Bruno Berselli, Oct 21 2011 - based on remarks and sequences by Omar E. Pol

First bisection of A193053 (see also the numerical spiral illustrated in the Links section).

The inverse binomial transform yields 1, 9, 7, 0, 0 (0 continued).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A195020 (vertices of the numerical spiral in link).
Cf. A017005 (first differences).
Cf. A069099, A001106.
Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A193053, A195021, A195023-A195030, A195320 [incomplete list].

Magma
```
[n*(7*n+11)/2+1: n in [0..47]];
```

Table[(n(7n+11))/2+1,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{1,10,26},60] (* Harvey P. Dale, Mar 03 2013 *)

for(n=0, 47, print1(n*(7*n+11)/2+1", "));

G.f.: (1 + 7*x - x^2)/(1-x)^3.
a(n) = A195020(2*n) + 2*n + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = 2*a(n-1) - a(n-2) + 7.
From Elmo R. Oliveira, Dec 24 2024: (Start)
E.g.f.: exp(x)*(2 + 18*x + 7*x^2)/2.
a(n) = n + A001106(n+1). (End)

A158481 a(n) = 49*n^2 + 7.

Original entry on oeis.org

56, 203, 448, 791, 1232, 1771, 2408, 3143, 3976, 4907, 5936, 7063, 8288, 9611, 11032, 12551, 14168, 15883, 17696, 19607, 21616, 23723, 25928, 28231, 30632, 33131, 35728, 38423, 41216, 44107, 47096, 50183, 53368, 56651, 60032, 63511, 67088, 70763, 74536, 78407

Offset: 1

Vincenzo Librandi, Mar 20 2009

The identity (14*n^2 + 1)^2 - (49*n^2 + 7)*(2*n)^2 = 1 can be written as A158482(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158482, A247541.

I:=[56, 203, 448];[n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

LinearRecurrence[{3,-3,1},{56,203,448},40]

PARI
```
a(n)=49*n^2+7.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 7*x*(8+5*x+x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(7))*Pi/sqrt(7) - 1)/14.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(7))*Pi/sqrt(7))/14. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 7*(exp(x)*(7*x^2 + 7*x + 1) - 1).
a(n) = 7*A247541(n). (End)

A158660 a(n) = 56*n^2 + 1.

Original entry on oeis.org

1, 57, 225, 505, 897, 1401, 2017, 2745, 3585, 4537, 5601, 6777, 8065, 9465, 10977, 12601, 14337, 16185, 18145, 20217, 22401, 24697, 27105, 29625, 32257, 35001, 37857, 40825, 43905, 47097, 50401, 53817, 57345, 60985, 64737, 68601, 72577, 76665, 80865, 85177, 89601

Offset: 0

Vincenzo Librandi, Mar 23 2009

The identity (56*n^2 + 1)^2 - (784*n^2 + 28)*(2*n)^2 = 1 can be written as a(n)^2 - A158659(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158482, A158657, A158659.

I:=[1, 57, 225]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 17 2012

LinearRecurrence[{3, -3, 1}, {1, 57, 225}, 50] (* Vincenzo Librandi, Feb 17 2012 *)
56 Range[0,40]^2+1 (* Harvey P. Dale, Jun 14 2022 *)

for(n=0, 40, print1(56*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 17 2012

G.f.: -(1 + 54*x + 57*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 20 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(14)))*Pi/(2*sqrt(14)) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(14)))*Pi/(2*sqrt(14)) + 1)/2. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(1 + 56*x + 56*x^2).
a(n) = A158482(2*n) for n > 0. (End)

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009

A010018 a(0) = 1, a(n) = 28*n^2 + 2 for n>0.

Original entry on oeis.org

1, 30, 114, 254, 450, 702, 1010, 1374, 1794, 2270, 2802, 3390, 4034, 4734, 5490, 6302, 7170, 8094, 9074, 10110, 11202, 12350, 13554, 14814, 16130, 17502, 18930, 20414, 21954, 23550, 25202, 26910, 28674, 30494, 32370, 34302, 36290, 38334, 40434, 42590, 44802

Offset: 0

N. J. A. Sloane

First bisection of A005919. - Bruno Berselli, Feb 07 2012
a(n) = the second level of difference between the sum of the terms in the n+1 X n+1 matrices and those in the n X n matrices starting at n=1 for an array constructed by using the terms in A016813 as the antidiagonals; the first few antidiagonals are 1; 5,9; 13,17,21; 25,29,33,37. - J. M. Bergot, Jul 05 2013
[More formally: (sum[m(n+1),j {j=0..n+1}]+sum[m(i,n+1) {i=0..n}]) - (sum[m(n,j) {j=0...n}] + sum[m(i,n) {i=0..n-1}])=a(n)]
[The first five rows begin: 1,9,21,37,57; 5,17,33,53,77; 13,29,49,73,101;25,45,69,97,129; 41,65,93,125,161]

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

Join[{1}, 28 Range[40]^2 + 2] (* Bruno Berselli, Feb 07 2012 *)
LinearRecurrence[{3, -3, 1}, {1, 30, 114, 254}, 40] (* Robert G. Wilson v, Jul 06 2013 *)

G.f.: (1+x)*(1+26*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 07 2012
E.g.f.: (x*(x+1)*28+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(14)/56*Pi*coth(Pi/sqrt 14) = 1.05615979263340... - R. J. Mathar, May 07 2024
a(n) = 2*A158482(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A195314(n)+A195314(n+1). - R. J. Mathar, May 07 2024

cp's OEIS Frontend

A195020 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The edges of the spiral have length A195019.

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A193053 a(n) = (14*n*(n+3) + (2*n-5)*(-1)^n + 21)/16.

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Magma

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A198017 a(n) = n*(7*n + 11)/2 + 1.

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Magma

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A158481 a(n) = 49*n^2 + 7.

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Magma

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A158660 a(n) = 56*n^2 + 1.

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A010018 a(0) = 1, a(n) = 28*n^2 + 2 for n>0.

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Mathematica

Formula

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.

A198017 a(n) = n(7n + 11)/2 + 1.