A224419 -id:A224419 - cp's OEIS frontend

A147875 Second heptagonal numbers: a(n) = n(5n+3)/2.

0, 4, 13, 27, 46, 70, 99, 133, 172, 216, 265, 319, 378, 442, 511, 585, 664, 748, 837, 931, 1030, 1134, 1243, 1357, 1476, 1600, 1729, 1863, 2002, 2146, 2295, 2449, 2608, 2772, 2941, 3115, 3294, 3478, 3667, 3861, 4060, 4264, 4473, 4687, 4906, 5130, 5359, 5593

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Zero followed by partial sums of A016897.
Apparently = every 2nd term of A111710 and A085787.
Bisection of A085787. Sequence found by reading the line from 0, in the direction 0, 13, ... and the line from 4, in the direction 4, 27, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012
Numbers of the form m^2 + k*m*(m+1)/2: in this case is k=3. See also A254963. - Bruno Berselli, Feb 11 2015

			G.f. = 4*x + 13*x^2 + 27*x^3 + 46*x^4 + 70*x^5 + 99*x^6 + 133*x^7 + ... - _Michael Somos_, Jan 25 2019

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Sliced Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016897, A111710, A000217, A085787, A224419 (positions of squares).
Second n-gonal numbers: A005449, A014105, A045944, A179986, A033954, A062728, A135705.
Cf. A000566.
Cf. A003215, A000096, A000290.

List([0..50], n-> n*(5*n+3)/2); # G. C. Greubel, Jul 04 2019

[n*(5*n+3)/2: n in [0..50]]; // G. C. Greubel, Jul 04 2019

Table[(n(5n+3))/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 4, 13}, 50] (* Harvey P. Dale, May 15 2013 *)

a(n)=n*(5*n+3)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(5*n+3)/2 for n in (0..50)] # G. C. Greubel, Jul 04 2019

G.f.: x*(4+x)/(1-x)^3.
a(n) = Sum_{k=0..n-1} A016897(k).
a(n) - a(n-1) = 5*n -1. - Vincenzo Librandi, Nov 26 2010
G.f.: U(0) where U(k) = 1 + 2*(2*k+3)/(k + 2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3) + (2*k+2)*(2*k+3)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 14 2012
E.g.f.: U(0) where U(k) = 1 + 2*(2*k+3)/(k + 2 - 2*x*(k+2)^2*(k+3)/(2*x*(k+2)*(k+3) + (2*k+2)^2*(2*k+3)/U(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 14 2012
a(n) = A130520(5n+3). - Philippe Deléham, Mar 26 2013
a(n) = A131242(10n+7)/2. - Philippe Deléham, Mar 27 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=13. - Harvey P. Dale, May 15 2013
Sum_{n>=1} 1/a(n) = 10/9 + sqrt(1 - 2/sqrt(5))*Pi/3 - 5*log(5)/6 + sqrt(5)*log((1 + sqrt(5))/2)/3 = 0.4688420784500060750083432... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(n) + A000217(2*n). - Bruno Berselli, Jul 01 2016
From Ilya Gutkovskiy, Jul 01 2016: (Start)
E.g.f.: x*(8 + 5*x)*exp(x)/2.
Dirichlet g.f.: (5*zeta(s-2) + 3*zeta(s-1))/2. (End)
a(n) = A000566(-n) for all n in Z. - Michael Somos, Jan 25 2019
From Leo Tavares, Feb 14 2022: (Start)
a(n) = A003215(n) - A000217(n+1). See Sliced Hexagons illustration in links.
a(n) = A000096(n) + 2*A000290(n). (End)

Edited by Klaus Brockhaus and R. J. Mathar, Nov 20 2008

New name from Bruno Berselli, Jan 13 2011

A225785 Numbers n such that triangular(n) + triangular(2*n) is a triangular number.

Original entry on oeis.org

0, 12, 84, 3960, 27144, 1275204, 8740380, 410611824, 2814375312, 132215732220, 906220110180, 42573055163112, 291800061102744, 13708391546789940, 93958713454973484, 4414059505011197664, 30254413932440359200, 1421313452222058857964

Offset: 1

Alex Ratushnyak, May 16 2013

Equivalently, numbers n such that oblong(n) + oblong(2*n) is an oblong number, where oblong(n) = A002378(n) = n*(n+1).
Also, x values in the equation A147875(x) = A000217(y) - see Ralf Stephan in Program lines. - Bruno Berselli, May 18 2013
Also, numbers m such that 2*m+1 and 10*m+1 are both squares. - Bruno Berselli, Mar 03 2016

			12*13/2 + 24*25/2 = 27*28/2, so 12 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..798
Index entries for linear recurrences with constant coefficients, signature (1,322,-322,-1,1).

Cf. A000217, A002378, A082183.
Cf. A224419 (numbers n such that triangular(n) + triangular(2*n) is a square).
Cf. A011916 (numbers n such that triangular(2*n) - triangular(n) is a triangular number).
Cf. A225786 (numbers n such that oblong(2*n) + oblong(n) is a square).
Cf. A225839 (triangular numbers of the form triangular(x) + triangular(2*x)).

#include 
#include 
int main() {
  unsigned long long i, s, t;
  for (i = 0; i< (1ULL<<31); i++) {
    s = 2*i*(2*i+1) + i*(i+1);
    t = sqrt(s);
    if (s==t*(t+1)) printf("%llu, ", i);
  }
  return 0;
}

CoefficientList[Series[12 x (1 + 6 x + x^2)/((1 - x) (1 - 18 x + x^2) (1 + 18 x + x^2)), {x, 0, 20}], x] (* Bruno Berselli, May 18 2013 *)
LinearRecurrence[{1,322,-322,-1,1},{0,12,84,3960,27144},20] (* Harvey P. Dale, Apr 08 2021 *)

for(n=1,10^9,t=n*(5*n+3)/2;x=sqrtint(2*t);if(t==x*(x+1)/2,print(n))) /* Ralf Stephan, May 17 2013 */

G.f.: 12*x*(1+6*x+x^2)/((1-x)*(1-18*x+x^2)(1+18*x+x^2)). [Bruno Berselli, May 18 2013]
a(n) = (1/20)*((3+(-1)^n*sqrt(5))*(2-sqrt(5))^(4*floor(n/2))+(3-(-1)^n*sqrt(5))*(2+sqrt(5))^(4*floor(n/2))-6). [Bruno Berselli, May 18 2013]
a(2*n) = (Fibonacci(6*n-3)^2 + Lucas(6*n-3)*Fibonacci(6*n-1))/2. - Greg Dresden, Sep 24 2023

More terms from Bruno Berselli, May 18 2013

A225786 Numbers k such that oblong(2k) + oblong(k) is a square, where oblong(k) = A002378(k) = k(k+1).

Original entry on oeis.org

0, 48, 15552, 5007792, 1612493568, 519217921200, 167186558132928, 53833552500881712, 17334236718725778432, 5581570389877199773488, 1797248331303739601284800, 578708381109414274413932208, 186342301468900092621684886272

Offset: 1

Alex Ratushnyak, May 16 2013

Numbers k such that k*(5*k+3) is a perfect square. Apparently a(n) = 323*a(n-1) -323*a(n-2) +a(n-3). - R. J. Mathar, May 18 2013

			48*49 + 96*97 = 108^2, so 48 is in the sequence.

Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Cf. A002378.
Cf. A098301 (numbers n such that oblong(2*n) - oblong(n) is a square).
Cf. A224419 (triangular(2*n) + triangular(n) is a square).
Cf. A220186 (triangular(2*n) - triangular(n) is a square).
Cf. A225785 (oblong(2*n) + oblong(n) is an oblong number).

#include 
#include 
int main() {
  unsigned long long i, s, t;
  for (i = 0; i< (1ULL<<31); i++) {
    s = 2*i*(2*i+1) + i*(i+1);
    t = sqrt(s);
    if (s==t*t) printf("%llu, ", i);
  }
  return 0;
}

LinearRecurrence[{323, -323, 1}, {0, 48, 15552}, 15] (* Bruno Berselli, May 18 2013 *)

G.f.: 48*x*(1+x)/((1-x)*(1-322*x+x^2)). - Bruno Berselli, May 18 2013

a(n) = (3/20)*((2-sqrt(5))^(4n-4)+(2+sqrt(5))^(4n-4)-2). - Bruno Berselli, May 18 2013

a(6) from Ralf Stephan, May 17 2013

More terms from Bruno Berselli, May 18 2013

cp's OEIS Frontend

A147875 Second heptagonal numbers: a(n) = n*(5*n+3)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A225785 Numbers n such that triangular(n) + triangular(2*n) is a triangular number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Mathematica

PARI

Formula

Extensions

A225786 Numbers k such that oblong(2*k) + oblong(k) is a square, where oblong(k) = A002378(k) = k*(k+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Mathematica

Formula

Extensions

A147875 Second heptagonal numbers: a(n) = n(5n+3)/2.

A225786 Numbers k such that oblong(2k) + oblong(k) is a square, where oblong(k) = A002378(k) = k(k+1).