A135768 -id:A135768 - cp's OEIS frontend

A135771 Terms in A136112 which are not in A135768.

5, 23, 51, 71, 72, 99, 123, 239, 263, 311, 359, 479, 599, 699, 743, 863, 911, 1031, 1103, 1151, 1431, 1563, 1583, 1823, 1851, 1863, 2111, 2543, 2663, 3023, 3119, 3191, 3291, 3671, 3719, 3863, 4131, 4203, 4271, 4463, 4671, 4703, 5039, 5231, 5351, 5391, 5399

Offset: 1

R. J. Mathar and M. F. Hasler, Feb 07 2008

nonn

Pentagonal-Indices of terms in A136113 which are not in A135769.

A135768 resp. A135769 are subsequences of A136112 resp. A136113; the present sequence gives the indices of the elements of the former which are not in the latter: A136113(A135771(k)), k=1,2,3,... are the pentagonal numbers P(m) which are not the difference of two pentagonal numbers P(n)-P(q) with n,q>m, but only with n>m>q. A136112(A135771(k)) are the corresponding indices of the pentagonal numbers.

			The first terms of this sequence correspond to the following elements of A136113:
P_5 = P_7 - P_5,
P_23 = P_24 - P_7,
P_51 = P_66 - P_42,
P_71 = P_74 - P_21,
P_72 = P_80 - P_35,
P_99 = P_104 - P_32,
P_123 = P_144 - P_75,
P_239 = P_249 - P_70,
P_263 = P_274 - P_77,
P_311 = P_324 - P_91,
P_359 = P_374 - P_10.

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A000326, A136112-A136118, A135768-A135769.

P(n)=n*(3*n-1)/2
isPent(t)=P(sqrtint((t*2)\3)+1)==t
{for( i=1,999, for( j=1,i-1, isPent(P(i)+P(j))|next; for( k=i+1,(P(i)-1)\3, isPent(P(i)+P(k))&next(3)); print1(i", "); next(2)))}

Equals the difference set A136112 \ A135768.

A135769 Pentagonal numbers > 0 which are not the difference of two other pentagonal numbers > 0.

Original entry on oeis.org

1, 5, 12, 51, 92, 117, 176, 330, 477, 852, 1080, 4347, 9801, 13776, 24512, 27270, 39285, 69876, 88452, 124272, 137562, 220992, 268182, 315792, 354051, 403782, 523626, 612162, 629532, 699392, 796797, 919242, 1119312, 1158522, 1190376

Offset: 1

R. J. Mathar and M. F. Hasler, Feb 07 2008

nonn

A subsequence of A136113, obtained by omitting A136113(A135771(k)), k=1,2,3,... ; i.e. those which are not the difference of two larger pentagonal numbers, but the difference of a larger and a smaller pentagonal number.

The definition ("...two other...") excludes the case P(n) = P(m)-P(n), cf. comment by R. J. Mathar in A000326.

			See A135768 for a list of P(n) which are in A136113 but not in A135769.

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A000326, A136112-A136118, a(n) = A000326(A135768(n)), A135771 = A136112 \ A135768.

P(n)=n*(3*n-1)/2
isPent(t)=P(sqrtint((t*2)\3)+1)==t
for( i=1,999,for( j=1,(P(i)-1)\3, isPent(P(i)+P(j))&j!=i&next(2)); print1(P(i)","))

P(n)=n*(3*n-1)/2 <=> n*(n-1/3) = (2/3)*P(n), thus m = P(n) <=> m = P([sqrt(2m/3)]+1)

and m = P(n) <=> 24m+1 = (6n-1)^2, useful for investigating the possibility of writing P(n)=P(n')+P(n"): this is possible whenever (6n-1)^2=(6n'-1)^2+(6n"-1)^2-1.

A137694 Numbers k such that 6k^2-2k = 3n^2-n for some integer n>0.

Original entry on oeis.org

5, 5577, 6435661, 7426747025, 8570459630997, 9890302987423321, 11413401077026881245, 13171054952586033533217, 15199386001883205670450981, 17540078275118266757666898665, 20241235130100477955141930608237, 23358367800057676441967030255006641

Offset: 1

M. F. Hasler, Feb 08 2008

Also indices of pentagonal numbers which are half of some other pentagonal number: see A137693 for more details, comments and links.

Colin Barker, Table of n, a(n) for n = 1..250
Dario Alpern, Quadratic two integer variable equation solver
Index entries for linear recurrences with constant coefficients, signature (1155,-1155,1).

Cf. A000326, A136112-A136118, A135768-A135769, A137693.

vector(20,i, (v=if(i>1,[577,408;816,577]*v-[164;232], [5;7]))[1,1])

a(n) = f^{2n-2}(5,7)[1], where f(x,y) = (577x + 408y - 164, 816x + 577y - 232).
a(n) = (5,7,1,5,7,1,...) (mod 10).
G.f.: -x*(5-198*x+x^2) / ( (x-1)*(x^2-1154*x+1) ). - R. J. Mathar, Apr 17 2011

A137693 Numbers n such that 3n^2-n = 6k^2-2k for some integer k>0.

Original entry on oeis.org

7, 7887, 9101399, 10503006367, 12120460245927, 13987000620793199, 16140986595935105527, 18626684544708490984767, 21495177823607002661315399, 24805416581757936362666985487, 28625429240170834955515039936407, 33033720537740561780727993419627999

Offset: 1

M. F. Hasler, Feb 08 2008

Also indices of pentagonal numbers which are twice some other pentagonal number.
Note that A000326(n) = 2 A000326(k) <=> n(3n-1)=2k(3k-1), which is easily solved by standard Pell-type techniques (cf. link to D. Alpern's quadratic solver). Here we consider only positive solutions.
Inspired by a recent comment on A000326 by R. J. Mathar.

Colin Barker, Table of n, a(n) for n = 1..250
Dario Alpern, Quadratic two integer variable equation solver
Index entries for linear recurrences with constant coefficients, signature (1155,-1155,1).

Cf. A000326, A136112-A136118, A135768-A135769, A135771, A137694.

CoefficientList[Series[x (-7+198x+x^2)/((x-1)(x^2-1154x+1)),{x,0,20}],x] (* or *) Join[{0},LinearRecurrence[{1155,-1155,1},{7,7887,9101399}, 20]] (* Harvey P. Dale, Jun 21 2011 *)

vector(20,i, (v=if(i>1,[577,408; 816,577]*v-[164;232], [5;7]))[2,1])

a(n) = f^{2n-2}(5,7)[2], where f(x,y) = (577x + 408y - 164, 816x + 577y - 232)
a(n) = (7,7,9,7,7,9,...) mod 10
G.f. x*(-7+198*x+x^2) / ( (x-1)*(x^2-1154*x+1) ). - R. J. Mathar, Apr 17 2011
a(0)=0, a(1)=7, a(2)=7887, a(3)=9101399, a(n)=1155*a(n-1)-1155*a(n-2)+ a(n-3). - Harvey P. Dale, Jun 21 2011

More terms from Harvey P. Dale, Jun 21 2011

A330657 Number of ways the n-th pentagonal number A000326(n) can be written as the difference of two positive pentagonal numbers.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 2, 1, 1, 3, 1, 0, 2, 3, 0, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 4, 1, 0, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 1, 3, 2, 1, 6, 1, 1, 1, 3, 2

Offset: 1

Bradley Klee, Mar 01 2020

nonn

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(3*n-1) + k*(3*k-1) - m*(3*m-1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A137694. Solutions of the form [n,m-1,m] appear only when n=3*z+1, z > 0. The n for which a(n)=0 are listed in A135768.

			Isosceles case, n=5: 2*5*(3*5-1) - 7*(3*7-1) = 0.

N. J. A. Sloane et al., "sum of 2 triangular numbers is a triangular number", math-fun mailing list, Feb. 19-29, 2020.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
B. Klee, Pentathagorean triples: two easy proofs, seqfan mailing list, Mar. 20, 2020.
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A046079, A309507, A000326.

PentaTriples[PNn_] := Sort[Select[{PNn,
      (-PNn + 3 PNn^2 + # - 3 #^2)/(6 #),
      (-PNn + 3 PNn^2 + # + 3 #^2)/(6 #)
      } & /@ Divisors[PNn*(3*PNn - 1)],
   And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[PentaTriples[#]] & /@ Range[100]
a[n_] := Length@FindInstance[n > 0 && y > 0 && z > 0 &&
     n (3 n - 1) + y (3 y - 1) == z (3 z - 1), {y, z}, Integers, 10^9];
a /@ Range[100]

A199639 Indices of hexagonal numbers that are not the difference of two positive hexagonal numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 16, 19, 22, 24, 28, 31, 32, 43, 48, 64, 76, 79, 96, 103, 112, 128, 139, 142, 163, 166, 184, 192, 199, 211, 223, 256, 262, 268, 271, 283, 304, 307, 316, 331, 352, 367, 376, 379, 384, 412, 439, 448, 454, 463, 496, 499, 512, 526, 547

Offset: 1

T. D. Noe, Dec 06 2011

nonn

Cf. A000384 (hexagonal numbers), A135768 (pentagonal case).

Select[Range[600], Reduce[#*(2*#-1) == x*(2*x-1) - y*(2*y-1) && x > 0 && y > 0, {x, y}, Integers] == False &]

cp's OEIS Frontend

A135771 Terms in A136112 which are not in A135768.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A135769 Pentagonal numbers > 0 which are not the difference of two other pentagonal numbers > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A137694 Numbers k such that 6k^2-2k = 3n^2-n for some integer n>0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A137693 Numbers n such that 3n^2-n = 6k^2-2k for some integer k>0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A330657 Number of ways the n-th pentagonal number A000326(n) can be written as the difference of two positive pentagonal numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A199639 Indices of hexagonal numbers that are not the difference of two positive hexagonal numbers.

Views

Author

Keywords

Crossrefs

Programs

Mathematica