A047335 -id:A047335 - 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)

A274406 Numbers m such that 9 divides m*(m + 1).

Original entry on oeis.org

0, 8, 9, 17, 18, 26, 27, 35, 36, 44, 45, 53, 54, 62, 63, 71, 72, 80, 81, 89, 90, 98, 99, 107, 108, 116, 117, 125, 126, 134, 135, 143, 144, 152, 153, 161, 162, 170, 171, 179, 180, 188, 189, 197, 198, 206, 207, 215, 216, 224, 225, 233, 234, 242, 243, 251, 252, 260, 261, 269

Offset: 1

Bruno Berselli, Jun 20 2016

Equivalently, numbers congruent to 0 or 8 mod 9.

Terms of A007494 with indices in A047264. Also, terms of A060464 with indices in A047335.

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

Cf. A008591 (first bisection), A010689 (first differences), A017257 (second bisection).
Cf. similar sequences in which m*(m+1) is divisible by k: A014601 (k=4), A047208 (k=5), A007494 (k=3 and 6), A047335 (k=7), A047521 (k=8), this sequence (k=9).
Cf. A301451: numbers congruent to {1, 7} mod 9; A193910: numbers congruent to {2, 6} mod 9.

[n: n in [0..300] | IsDivisibleBy(n*(n+1),9)];

Select[Range[0, 300], Divisible[# (# + 1), 9] &]

for(n=0, 300, if(n*(n+1)%9==0, print1(n", ")))

[n for n in range(300) if 9.divides(n*(n+1))]

G.f.: x^2*(8 + x)/((1 + x)*(1 - x)^2).
a(n) = (18*n + 7*(-1)^n - 11)/4. Therefore: a(2*m) = 9*m-1, a(2*m+1) = 9*m. It follows that a(j)+a(k) and a(j)*a(k) belong to the sequence if j and k are not both even.
a(n) = -A090570(-n+2).
a(n) = a(n-1) + a(n-2) - a(n-3).
a(2*r+1) + a(2*r+s+1) = a(4*r+s+1) and a(2*r) + a(2*r+2*s+1) = a(4*r+2*s). A particular case provided by these identities: a(n) = a(n - 2*floor(n/6)) + a(2*floor(n/6) + 1).
E.g.f.: 1 + ((9*x - 2)*cosh(x) + 9*(x - 1)*sinh(x))/2. - Stefano Spezia, Apr 24 2021

A219191 Numbers of the form k(7k+1), where k = 0,-1,1,-2,2,-3,3,...

Original entry on oeis.org

0, 6, 8, 26, 30, 60, 66, 108, 116, 170, 180, 246, 258, 336, 350, 440, 456, 558, 576, 690, 710, 836, 858, 996, 1020, 1170, 1196, 1358, 1386, 1560, 1590, 1776, 1808, 2006, 2040, 2250, 2286, 2508, 2546, 2780, 2820, 3066, 3108, 3366, 3410, 3680, 3726, 4008

Offset: 1

Bruno Berselli, Nov 14 2012

Equivalently, numbers m such that 28*m+1 is a square.
Also, integer values of h*(h+1)/7.
Let F(r) = Product_{n >= 1} 1 - q^(14*n-r). The sequence terms are the exponents in the expansion of F(0)*F(6)*F(8) = 1 - q^6 - q^8 + q^26 + q^30 - q^60 - q^66 + + - - ... (by the triple product identity).- Peter Bala, Dec 25 2024

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

Cf. numbers of the form k*(i*k+1) with k in A001057: i=0, A001057; i=1, A110660; i=2, A000217; i=3, A152749; i=4, A074378; i=5, A219190; i=6, A036498; i=7, this sequence; i=8, A154260.
Cf. A113801 (square roots of 28*a(n)+1, see the comment).
Cf. similar sequences listed in A219257.
Subsequence of A011860.

k:=7; f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]];

I:=[0,6,8,26,30]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A := proc (q) local n; for n from 0 to q do if type(sqrt(28*n+1), integer) then print(n) fi; od; end: A(4100); # Peter Bala, Dec 25 2024

Rest[Flatten[{# (7 # - 1), # (7 # + 1)} & /@ Range[0, 25]]]
CoefficientList[Series[2 x (3 + x + 3 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,6,8,26,30},50] (* Harvey P. Dale, Sep 14 2022 *)

G.f.: 2*x^2*(3+x+3*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+5)/8.
a(n) = 2*A057570(n) = (1/7)*A047335(n)*A047274(n+1).
Sum_{n>=2} 1/a(n) = 7 - cot(Pi/7)*Pi. - Amiram Eldar, Mar 17 2022

A176355 Periodic sequence: Repeat 6, 1.

Original entry on oeis.org

6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6

Offset: 0

Klaus Brockhaus, Apr 15 2010

Interleaving of A010722 and A000012.
Also continued fraction expansion of 3+sqrt(15).
Also decimal expansion of 61/99.
Essentially first differences of A047335.
Binomial transform of 6 followed by A166577 without initial terms 1, 4.
Inverse binomial transform of A005009 preceded by 6.

			0.6161616161616161616161616161616161616161...

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010722 (all 6's sequence), A000012 (all 1's sequence), A092294 (decimal expansion of 3+sqrt(15)), A010687 (repeat 1, 6), A047335 (congruent to 0 or 6 mod 7), A166577, A005009 (7*2^n).

Magma
```
&cat[ [6, 1]: n in [0..52] ];
```
Magma
```
[(7+5*(-1)^n)/2: n in [0..104]];
```

PadRight[{},120,{6,1}] (* Harvey P. Dale, Apr 12 2018 *)

G.f.: (6 + x)/(1 - x^2).
a(n) = (7 + 5*(-1)^n)/2.
a(n) = a(n-2) for n>1, a(0)=6, a(1)=1.
a(n) = -a(n-1)+7 for n>0, a(0)=6.
a(n) = 6*((n+1) mod 2) + (n mod 2).
a(n) = A010687(n+1).
a(n) = 13^n mod 7. - Vincenzo Librandi, Jun 01 2016
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 6, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(1+5/2^s). (End)
E.g.f.: 6*cosh(x) + sinh(x). - Stefano Spezia, Feb 09 2025

A199626 G.f.: (1+x)^(2g)(1+x^3)^(3g)/((1-x^2)(1-x^4))-x^(2g)(1+x)^4/((1-x^2)*(1-x^4)) for g=0.

Original entry on oeis.org

0, -4, -6, -8, -7, -12, -13, -16, -14, -20, -20, -24, -21, -28, -27, -32, -28, -36, -34, -40, -35, -44, -41, -48, -42, -52, -48, -56, -49, -60, -55, -64, -56, -68, -62, -72, -63, -76, -69, -80, -70, -84, -76, -88, -77, -92, -83, -96, -84, -100, -90, -104, -91, -108, -97, -112, -98, -116, -104, -120

Offset: 0

N. J. A. Sloane, Nov 08 2011

Colin Barker, Table of n, a(n) for n = 0..1000
Bott, Raoul, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982), no. 2, 331-358; reprinted in Vol. 48 (October, 2011). See Eq. (4.30).
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A047335, A008586 (signed bisections).

g:=0; m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4))));  // Bruno Berselli, Nov 08 2011

f:=g->(1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4));
s:=g->seriestolist(series(f(g),x,60));
s(0);

LinearRecurrence[{0,1,0,1,0,-1},{0,-4,-6,-8,-7,-12},80] (* Harvey P. Dale, Jul 30 2019 *)

a(n)=if(n%2,,(n+5*I^n+3)/4)-2*n-2 \\ Charles R Greathouse IV, Nov 08 2011

concat(0, Vec(-x*(x+2)*(x^2+2*x+2)/((x-1)^2*(x+1)^2*(x^2+1)) + O(x^100))) \\ Colin Barker, Jul 10 2015

a(n) = ((-1)^n+1)*(n+5*i^n+3)/8-2*(n+1). - Bruno Berselli, Nov 08 2011

G.f.: -x*(x+2)*(x^2+2*x+2) / ((x-1)^2*(x+1)^2*(x^2+1)). - Colin Barker, Jul 10 2015

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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A274406 Numbers m such that 9 divides m*(m + 1).

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A219191 Numbers of the form k*(7*k+1), where k = 0,-1,1,-2,2,-3,3,...

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A176355 Periodic sequence: Repeat 6, 1.

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A199626 G.f.: (1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4)) for g=0.

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A219191 Numbers of the form k(7k+1), where k = 0,-1,1,-2,2,-3,3,...

A199626 G.f.: (1+x)^(2g)(1+x^3)^(3g)/((1-x^2)(1-x^4))-x^(2g)(1+x)^4/((1-x^2)*(1-x^4)) for g=0.