A188892 -id:A188892

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A051682

11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.

+10
70

0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606, 715, 833, 960, 1096, 1241, 1395, 1558, 1730, 1911, 2101, 2300, 2508, 2725, 2951, 3186, 3430, 3683, 3945, 4216, 4496, 4785, 5083, 5390, 5706, 6031, 6365, 6708, 7060, 7421, 7791, 8170

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

From Floor van Lamoen, Jul 21 2001: (Start)

Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the line from 0 in the direction 0,1,...

The spiral begins:

/ \

16 14

/ \

17 3 13

/ / \ \

18 4 2 12

/ \ \

5 0---1 11

/ \

6---7---8---9--10

. (End)

(1), (4+7), (7+10+13), (10+13+16+19), ... - Jon Perry, Sep 10 2004

This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - T. D. Noe, Apr 13 2011

Sequence found by reading the line from 0, in the direction 0, 11, ... and the parallel line from 1, in the direction 1, 30, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Jul 18 2012

Starting with offset 1, the sequence is the binomial transform of (1, 10, 9, 0, 0, 0, ...). - Gary W. Adamson, Aug 01 2015

REFERENCES

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = n*(9*n-7)/2.

G.f.: x*(1+8*x)/(1-x)^3.

Row sums of triangle A131432. - Gary W. Adamson, Jul 10 2007

a(n) = 9*n + a(n-1) - 8 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=1, a(2)=11. - Harvey P. Dale, May 07 2012

a(n) = A218470(9n). - Philippe Deléham, Mar 27 2013

a(9*a(n)+37*n+1) = a(9*a(n)+37*n) + a(9*n+1). - Vladimir Shevelev, Jan 24 2014

a(n+y) - a(n-y-1) = (a(n+x) - a(n-x-1))*(2*y+1)/(2*x+1), 0 <= x < n, y <= x, a(0)=0. - Gionata Neri, May 03 2015

a(n) = A000217(n-1) + A000217(3*n-2) - A000217(n-2). - Charlie Marion, Dec 21 2019

Product_{n>=2} (1 - 1/a(n)) = 9/11. - Amiram Eldar, Jan 21 2021

E.g.f.: exp(x)*x*(2 + 9*x)/2. - Stefano Spezia, Dec 25 2022

MATHEMATICA

Table[n (9n-7)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 11}, 51] (* Harvey P. Dale, May 07 2012 *)

PROG

(PARI) a(n)=(9*n-7)*n/2 \\ Charles R Greathouse IV, Jun 16 2011

(Magma) [n*(9*n-7)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 01 2015

CROSSREFS

First differences of A007586.

Cf. A093644 ((9, 1) Pascal, column m=2). Partial sums of A017173.

Cf. A000217, A004188, A131432, A188892, A195160, A218470.

KEYWORD

nonn,easy

AUTHOR

Barry E. Williams

STATUS

approved

A051870

18-gonal (or octadecagonal) numbers: a(n) = n*(8*n-7).

+10
25

0, 1, 18, 51, 100, 165, 246, 343, 456, 585, 730, 891, 1068, 1261, 1470, 1695, 1936, 2193, 2466, 2755, 3060, 3381, 3718, 4071, 4440, 4825, 5226, 5643, 6076, 6525, 6990, 7471, 7968, 8481, 9010, 9555, 10116, 10693, 11286, 11895, 12520

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Also, sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Apr 26 2008

This sequence does not contain any triangular numbers other than 0 and 1. See A188892. - T. D. Noe, Apr 13 2011

Also sequence found by reading the line from 0, in the direction 0, 18, ... and the parallel line from 1, in the direction 1, 51, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, Jul 18 2012

Partial sums of 16n + 1 (with offset 0), compare A005570. - Jeremy Gardiner, Aug 04 2012

All x values for Diophantine equation 32*x + 49 = y^2 are given by this sequence and A139278. - Bruno Berselli, Nov 11 2014

This is also a star enneagonal number: a(n) = A001106(n) + 9*A000217(n-1). - Luciano Ancora, Mar 30 2015

REFERENCES

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing, 2012, page 6.

LINKS

Jeremy Gardiner, Table of n, a(n) for n = 0..999

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Wikipedia, Polygonal number.

Index to sequences related to polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: x*(1+15*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(n) = 16*n + a(n-1) - 15, with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 06 2010

a(16*a(n)+121*n+1) = a(16*a(n)+121*n) + a(16*n+1). - Vladimir Shevelev, Jan 24 2014

E.g.f.: (8*x^2 + x)*exp(x). - G. C. Greubel, Jul 18 2017

Sum_{n>=1} 1/a(n) = ((1+sqrt(2))*Pi + 2*sqrt(2)*arccoth(sqrt(2)) + 8*log(2))/14. - Amiram Eldar, Oct 20 2020

Product_{n>=2} (1 - 1/a(n)) = 8/9. - Amiram Eldar, Jan 22 2021

MAPLE

A051870 := proc(n) n*(8*n-7) ; end proc: seq(A051870(n), n=0..30) ; # R. J. Mathar, Feb 05 2011

MATHEMATICA

Table[n (8 n - 7), {n, 0, 40}] (* Bruno Berselli, Nov 11 2014 *)

PROG

(PARI) a(n)=n*(8*n-7) \\ Charles R Greathouse IV, Jul 19 2011

CROSSREFS

Cf. A014634, A014635, A033585, A033586, A033587, A035008, A069129, A085250, A129271-A129278, A139278.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 15 1999

STATUS

approved

A188896

Numbers n such that there is no square n-gonal number greater than 1.

+10
12

10, 20, 52, 164, 340, 580, 884, 1252, 1684, 2180, 2740, 4052, 4804, 5620, 6500, 7444, 8452, 9524, 10660, 11860, 13124, 14452, 15844, 17300, 18820, 20404, 22052, 25540, 27380, 29284, 31252, 33284, 35380, 37540, 39764, 42052, 44404, 46820, 49300, 51844

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

It is easy to find squares that are triangular, pentagonal, hexagonal, etc. So it is somewhat surprising that there are no square 10-gonal numbers other than 0 and 1. For these n, the equation 2*x^2 = (n-2)*y^2 - (n-4)*y has no integer solutions x>1 and y>1.

Chu shows how to transform the equation into a generalized Pell equation. When n has the form 2k^2+2 (A005893), then the Pell equation has only a finite number of solutions and it is simple to select the n that produce no integer solutions greater than 1.

The general case is in A188950.

LINKS

Muniru A Asiru, Table of n, a(n) for n = 1..244

Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781-802.

MATHEMATICA

P[n_, k_]:=1/2n(n(k-2)+4-k); data1=2#^2+2&/@Range[2, 161]; data2=Head[Reduce[m^2==P[n, #] && 1<m && 1<n && !m==n, {m, n}, Integers]]&/@data1; data3=Flatten[Position[data2, Symbol]]; data1[[#]]&/@data3 (* Ant King, Mar 01 2012 *)

CROSSREFS

Cf. A001107 (10-gonal numbers), A051872 (20-gonal numbers), A188892, A100252, A188950, A005893.

Subsequence of A271624. - Muniru A Asiru, Oct 16 2016

KEYWORD

nonn

AUTHOR

T. D. Noe, Apr 13 2011

STATUS

approved

A188891

Least triangular n-gonal number greater than 1, or 0 if none exists.

+10
5

3, 36, 210, 6, 55, 21, 325, 10, 0, 105, 36, 1275, 15, 45, 231, 0, 946, 276, 21, 11935, 66, 136, 351, 1596, 78, 28, 1225, 595, 820, 58653, 190, 325, 1335795, 36, 6670, 0, 561, 4005, 120, 1128, 1485, 203841, 45, 666, 6903, 465, 4950, 20910, 741, 153, 10731, 8911, 55, 1953

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,1

COMMENTS

See A188893 and A188894 for the corresponding indices of these terms. Note that a(n) is zero for n = 11, 18, 38 (numbers in A188892). Although the Mathematica program searches only the first 20000 triangular numbers for n-gonal numbers, the Reduce function can show that there are no triangular n-gonal numbers (other than 0 and 1) for these n.

LINKS

Table of n, a(n) for n=3..56.

Eric W. Weisstein, MathWorld: Polygonal Number

Eric W. Weisstein, MathWorld: Triangular Number

MATHEMATICA

NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[tr = k*(k+1)/2; i = NgonIndex[n, tr]; k < 20000 && ! IntegerQ[i], k++]; If[k==20000, tr=0]; tr, {n, 3, 50}]

Table[SelectFirst[PolygonalNumber[n, Range[2, 1000]], OddQ[Sqrt[8#+1]]&], {n, 3, 100}]/.Missing["NotFound"]->0 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 10 2019 *)

CROSSREFS

Cf. A000217 (triangular numbers), A100252 (similar sequence for squares).

KEYWORD

nonn

AUTHOR

T. D. Noe, Apr 13 2011

STATUS

approved

A188950

Pairs of numbers (n,k) such that there is no n-gonal k-gonal number greater than 1, sorted by the sum n+k and then n.

+10
3

3, 11, 4, 10, 6, 11, 5, 14, 3, 18, 4, 20, 6, 18, 7, 22, 11, 18, 10, 20, 6, 27, 5, 29, 8, 26, 11, 27, 9, 30, 3, 38, 14, 29, 6, 38, 10, 34, 18, 27, 11, 38, 7, 47, 12, 42, 20, 34, 5, 50, 4, 52, 18, 38, 6, 51, 13, 46, 11, 51, 8, 56, 14, 50, 27, 38, 15, 54, 22, 47

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

These are n and k such that the generalized Pell equation (k-2)*x^2 - (k-4)*x = (n-2)*y^2 - (n-4)*y has no solution in integers x>1 and y>1. The paper by Chu shows how to solve these equations. A necessary condition for a pair to be in this sequence is (n-2)(k-2) is a square. These (n,k) pairs indicate where the zeros are in triangle A189216, which gives the least n-gonal k-gonal number greater than 1. For triangular (n=3) and square (n=4) numbers, see A188892 and A188896 for lists of k.

LINKS

Table of n, a(n) for n=1..70.

Wenchang Chu, Regular polygonal numbers and generalized Pell equations, Int. Math. Forum 2 (2007), 781-802.

Eric W. Weisstein, MathWorld: Polygonal Number

EXAMPLE

The pairs begin (3,11), (4,10), (6,11), (5,14), (3,18), (4,20), (6,18).

MATHEMATICA

maxSum=100; Reap[Do[k=s-n; If[k>n && IntegerQ[Sqrt[(n-2)*(k-2)]] && FindInstance[(k-2)*x^2 - (k-4)*x == (n-2)*y^2 - (n-4)*y && x>1 && y>1, {x, y}, Integers] == {}, Sow[{n, k}]], {s, 7, maxSum}, {n, 3, s-3}]][[2, 1]]

CROSSREFS

Cf. A188892, A188896.

KEYWORD

nonn,tabf

AUTHOR

T. D. Noe, Apr 20 2011

STATUS

approved

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