A135771 -id:A135771 - 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

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.

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

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

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

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula

Extensions