A135769 -id:A135769 - cp's OEIS frontend

A135768 Indices of pentagonal numbers > 0 which are not the difference of 2 other pentagonal numbers > 0.

1, 2, 3, 6, 8, 9, 11, 15, 18, 24, 27, 54, 81, 96, 128, 135, 162, 216, 243, 288, 303, 384, 423, 459, 486, 519, 591, 639, 648, 683, 729, 783, 864, 879, 891, 1215, 1458, 1719, 1944, 2031, 2043, 2048, 2151, 2187, 2463, 2799, 3231, 3456, 3543, 3879, 3903, 4023

Offset: 1

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

nonn

A subsequence of A136112, obtained by omitting A136112(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 (or equal) pentagonal number. Sequence A135769 has the pentagonal numbers corresponding to these indices.

			Indices of the following numbers are not here but in A136112:
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_105

Donovan Johnson and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..200 from Donovan Johnson

Cf. A000326, A136112-A136118, A135769(n) = A000326(a(n)), A135771 = A136112 \ A135768.

Select[Range[100], Reduce[# (3 # - 1) == x (3 x - 1) - y (3 y - 1) && x > 0 && y > 0, {x, y}, Integers] == False &] (* T. D. Noe, Dec 05 2011 *)

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))&next(2)); print1(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 write P(n)=P(n')+P(n"): this is possible whenever (6n-1)^2 = (6n'-1)^2 + (6n"-1)^2.

Extended by T. D. Noe, Dec 05 2011

A135771 Terms in A136112 which are not in A135768.

Original entry on oeis.org

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.

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

cp's OEIS Frontend

A135768 Indices of pentagonal numbers > 0 which are not the difference of 2 other pentagonal numbers > 0.

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A135771 Terms in A136112 which are not in A135768.

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

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

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Formula

Extensions