A069099 -id:A069099 - cp's OEIS frontend

A128919 Numbers simultaneously heptagonal and centered heptagonal.

1, 148, 21022, 2984983, 423846571, 60183228106, 8545594544488, 1213414242089197, 172296276782121493, 24464857888819162816, 3473837523935538998386

Offset: 0

Steven Schlicker, Apr 24 2007

			a(1)=148 because 148 is the seventh centered heptagonal number and the eighth heptagonal number.

S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
Index entries for linear recurrences with constant coefficients, signature (143,-143,1)

Cf. A000566, A069099.

CP := n -> 1+1/2*7*(n^2-n): N:=10: u:=6: v:=1: x:=7: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+35*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp),CP(s)]: end do: k_pcp;

Nest[Append[#,142Last[#]-#[[-2]]+7]&,{1,148},20]  (* Harvey P. Dale, Apr 17 2011 *)

x(n) + y(n)*sqrt(35) = (7+sqrt(35))*(6+sqrt(35))^n s(n) = (y(n)+1)/2 a(n) = (1/2)*(2+7*(s(n)^2-s(n))).
From Richard Choulet, Oct 01 2007: (Start)
a(n+2) = 142*a(n+1)-a(n)+7.
a(n+1) = 71*a(n)+3.5+1.5*(2240*a(n)^2+224*a(n)-63)^0.5.
G.f.: z*(1+5*z+z^2)/((1-z)*(1-142*z+z^2)). (End)

A209294 a(n) = (7n^2 - 7n + 4)/2.

Original entry on oeis.org

2, 9, 23, 44, 72, 107, 149, 198, 254, 317, 387, 464, 548, 639, 737, 842, 954, 1073, 1199, 1332, 1472, 1619, 1773, 1934, 2102, 2277, 2459, 2648, 2844, 3047, 3257, 3474, 3698, 3929, 4167, 4412, 4664, 4923, 5189, 5462

Offset: 1

Marco Piazzalunga, Jan 17 2013

a(n) is the sum of the n-th centered triangular number and n-th centered square number.

Difference of consecutive terms gives A008589 (multiples of 7).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005448, A001844, A008589.

[(7*n^2 - 7*n + 4)/2: n in [1..30]]; // G. C. Greubel, Jan 04 2018

Table[(7*n^2 - 7*n + 4)/2, {n, 1, 50}] (* G. C. Greubel, Jan 04 2018 *)
LinearRecurrence[{3,-3,1},{2,9,23},40] (* Harvey P. Dale, Nov 02 2020 *)

a(n)=7*n*(n-1)/2+2 \\ Charles R Greathouse IV, Jan 17 2013

a(n) = (7*n^2 - 7*n + 4) = 7*T(n) + 2 with T = A000217.
G.f.: x*(2+3*x+2*x^2)/(1-x)^3. - Bruno Berselli, Jan 18 2013
a(n) = a(-n+1) = 3*a(n-1)-3*a(n-2)+a(n-3).  - Bruno Berselli, Jan 18 2013
a(n) = 1 + A069099(n). - Omar E. Pol, Apr 27 2017
E.g.f.: ((7*x^2 + 4)*exp(x) - 4)/2. - G. C. Greubel, Jan 04 2018

A274588 Values of n such that 2n-1 and 7n-1 are both triangular numbers.

Original entry on oeis.org

1, 8, 638, 6931, 572671, 6223778, 514257668, 5588945461, 461802812941, 5018866799948, 414698411763098, 4506936797407591, 372398711960448811, 4047224225205216518, 334413628642071268928, 3634402847297487025321, 300303066121868039048281

Offset: 1

Colin Barker, Jun 29 2016

			8 is in the sequence because 2*8-1 = 15, 7*8-1 = 55, and 15 and 55 are both triangular numbers.

Colin Barker, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (1,898,-898,-1,1).

Cf. A069099, A174114, A274587.

CoefficientList[Series[(1 + 7 x - 268 x^2 + 7 x^3 + x^4)/((1 - x) (1 - 30 x + x^2) (1 + 30 x + x^2)), {x, 0, 16}], x] (* Michael De Vlieger, Jun 30 2016 *)
LinearRecurrence[{1,898,-898,-1,1},{1,8,638,6931,572671},20] (* Harvey P. Dale, Apr 10 2023 *)

isok(n) = ispolygonal(2*n-1, 3) && ispolygonal(7*n-1, 3)

Vec((1+7*x-268*x^2+7*x^3+x^4)/((1-x)*(1-30*x+x^2)*(1+30*x+x^2)) + O(x^20))

Intersection of A069099 and A174114.

G.f.: (1+7*x-268*x^2+7*x^3+x^4) / ((1-x)*(1-30*x+x^2)*(1+30*x+x^2)).

A338492 Least number of centered heptagonal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers

Cf. A069099, A101412, A234533, A338480, A338482, A338484, A338491.

A360183 Centered heptagonal numbers which are sphenic numbers.

Original entry on oeis.org

638, 4922, 6322, 11978, 15478, 16906, 19426, 21022, 23822, 25586, 28666, 35351, 39698, 48322, 53383, 55126, 70078, 80333, 83546, 92422, 98197, 105358, 107801, 132406, 147806, 156563, 162541, 171718, 182743, 209231, 210946, 233878, 248578, 263726, 269522, 281303

Offset: 1

Massimo Kofler, Jan 29 2023

nonn

			A069099(14) =  638 = (7*14^2 - 7*14 + 2)/2 = 2 * 11 * 29.
A069099(38) = 4922 = (7*38^2 - 7*38 + 2)/2 = 2 * 23 * 107.
A069099(43) = 6322 = (7*43^2 - 7*43 + 2)/2 = 2 * 29 * 109.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A069099 and A007304.

select(t -> ifactors(t)[2][..,2]=[1,1,1], [(7*i^2-7*i+2)/2 $ i=1..1000]); # Robert Israel, Feb 28 2023

Select[Table[(7*n^2 - 7*n + 2)/2, {n, 1, 300}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* Amiram Eldar, Jan 29 2023 *)

A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

Original entry on oeis.org

1, 4, 11, 26, 55, 105, 184, 301, 466, 690, 985, 1364, 1841, 2431, 3150, 4015, 5044, 6256, 7671, 9310, 11195, 13349, 15796, 18561, 21670, 25150, 29029, 33336, 38101, 43355, 49130, 55459, 62376, 69916, 78115, 87010, 96639, 107041, 118256, 130325

Offset: 1

Dan Graybill (clopen(AT)comcast.net), Aug 12 2008

Consider the array of triangular, square and centered polygonal numbers (irregular variant of A086272 and A086273):
 3   6  10  15  21  28  36  45  55  A000217
 4   9  16  25  36  49  64  81 100  A000290
 6  16  31  51  76 106 141 181 226  A005891
 7  19  37  61  91 127 169 217 271  A003215
 8  22  43  71 106 148 197 253 316  A069099
 9  25  49  81 121 169 225 289 361  A016754
10  28  55  91 136 190 253 325 406  A060544
11  31  61 101 151 211 281 361 451  A062786
12  34  67 111 166 232 309 397 496  A069125
13  37  73 121 181 253 337 433 541  A003154
14  40  79 131 196 274 365 469 586  A069126
15  43  85 141 211 295 393 505 631  A069127
etc. The sequence contains the antidiagonal sums of this array. - R. J. Mathar, Jun 05 2011
For comparison, the antidiagonal sums of A086270 are essentially A006522 starting at the 4th term. - R. J. Mathar, Sep 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217.

a(n) = (n-1)*(n^3+11*n^2-38*n+120)/24, n>1. - R. J. Mathar, Sep 12 2008

G.f.: x*(1-x+x^2+x^3-x^5)/(1-x)^5. - Alexander R. Povolotsky, Jun 06 2011

A144975 Centered heptagonal twin prime numbers.

Original entry on oeis.org

43, 71, 197, 463, 1933, 5741, 8233, 9283, 11173, 14561, 34651, 41203, 57793, 68111, 84631, 104147, 139301, 168631, 207523, 244861, 307693, 333103, 357281, 415381, 465011, 475273, 506731, 592663, 595547, 607153, 729373, 742211, 781397, 876751

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 27 2008

nonn

			43 is a term since it is centered heptagonal and the greater member of the twin primes pair (41, 43).
71 is a term since it is centered heptagonal and the lesser member of the twin primes pair (71, 73).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Centered heptagonal twin prime numbers

Cf. A001097, A069099.

[a:k in [1..510]|IsPrime(a) and (IsPrime(a-2) or IsPrime(a+2)) where a is (7*k^2-7*k+2) div 2]; // Marius A. Burtea, Jan 30 2020

TwinPrimeQ[n_]:=If[PrimeQ[n],If[PrimeQ[n-2]||PrimeQ[n+2],True,False],False](*TwinPrimeQ*) lst={};Do[p=(7*n^2-7*n+2)/2;If[TwinPrimeQ[p],AppendTo[lst,p]],{n,2*6!}];lst

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 7, 1, 1, 5, 10, 13, 11, 1, 1, 6, 13, 19, 21, 16, 1, 1, 7, 16, 25, 31, 31, 22, 1, 1, 8, 19, 31, 41, 46, 43, 29, 1, 1, 9, 22, 37, 51, 61, 64, 57, 37, 1, 1, 10, 25, 43, 61, 76, 85, 85, 73, 46, 1, 1, 11, 28, 49, 71, 91, 106, 113, 109, 91

Offset: 0

Kival Ngaokrajang, Jul 07 2014

T(n,k) is the total number of boxes, when we start with 1 center box (n = 0) then expand 1 box on k-arms for each n iteration. See illustration in links.
It seems that column C(k) = centered k-gonal numbers, and row R(n) = A000217(n)*k + 1.
The triangle under the main diagonal is A121722.
Column N (CN) is the Narayana transform (A001263) of (1, N, 0, 0, 0, ...). Example: C2 (1, 3, 7, 13, ...) is the Narayana transform of (1, 2, 0, 0, 0, ...). - Gary W. Adamson, Oct 01 2015

			Table begins:
       C0  C1  C2  C3  C4  C5
  n/k  0   1   2   3   4   5   ...
R0 0   1   1   1   1   1   1   ...
R1 1   1   2   3   4   5   6   ...
R2 2   1   4   7   10  13  16  ...
R3 3   1   7   13  19  25  31  ...
R4 4   1   11  21  31  41  51  ...
R5 5   1   16  31  46  61  76  ...
R6 6   1   22  43  64  85  106 ...
R7 7   1   29  57  85  113 141 ...
R8 8   1   37  73  109 145 181 ...
R9 9   1   46  91  136 181 226 ...
  ...  ... ... ... ... ... ... ...
C1 = A000124, C2 = A002061, C3 = A005448, C4 = A001844, C5 = A005891, C6 = A003215, C7 = A069099, C8 = A016754, C9 = A060544, C10 = A062786, C11 = A069125, C12  =  A003154.
R1 = A000027, R2 = A016777, R3 = A016921, R4 = A017281, R5 = 15*k + 1, R6 = A215146, R7 = A161714.

Kival Ngaokrajang, Illustration for n = 0..3, k = 1..4

Cf. A000217, A121722, A000124, A002061, A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A000027, A016777, A016921, A017281, A215146, A161714.

T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

A274682 Numbers n such that 8*n-1 is a triangular number.

Original entry on oeis.org

2, 7, 29, 44, 88, 113, 179, 214, 302, 347, 457, 512, 644, 709, 863, 938, 1114, 1199, 1397, 1492, 1712, 1817, 2059, 2174, 2438, 2563, 2849, 2984, 3292, 3437, 3767, 3922, 4274, 4439, 4813, 4988, 5384, 5569, 5987, 6182, 6622, 6827, 7289, 7504, 7988, 8213, 8719

Offset: 1

Colin Barker, Jul 02 2016

			2 is in the sequence since 8*2 - 1 = 15, and 15 = 1 + 2 + 3 + 4 + 5 is a triangular number. - _Michael B. Porter_, Jul 03 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000124 (n-1), A174114 (2*n-1), A213399 (4*n-1), A069099 (7*n-1).

Table[(5 + 3 (-1)^n - 2 (8 + 3 (-1)^n) n + 16 n^2)/4, {n, 47}] (* or *)
Rest@ CoefficientList[Series[x (2 + 5 x + 18 x^2 + 5 x^3 + 2 x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 47}], x] (* Michael De Vlieger, Jul 02 2016 *)

PARI
```
isok(n) = ispolygonal(8*n-1, 3)
```

select(n->ispolygonal(8*n-1, 3), vector(10000, n, n-1))

Vec(x*(2+5*x+18*x^2+5*x^3+2*x^4)/((1-x)^3*(1+x)^2) + O(x^100))

a(n) = (5+3*(-1)^n-2*(8+3*(-1)^n)*n+16*n^2)/4.
a(n) = (8*n^2-11*n+4)/2 for n even.
a(n) = (8*n^2-5*n+1)/2 for n odd.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.
G.f.: x*(2+5*x+18*x^2+5*x^3+2*x^4) / ((1-x)^3*(1+x)^2).

A279946 Numbers that are both dodecagonal and centered heptagonal.

Original entry on oeis.org

1, 10396, 326656, 2619897841, 82318050361, 660219495802336, 20744313326831116, 166376633378560463881, 5227608446905776928921, 41927244364003774523222476, 1317367783816405284315203776, 10565749434051302554022550018121, 331979316252074156011094205115681

Offset: 1

Ann Skoryk, Dec 23 2016

From Jon E. Schoenfield, Dec 24 2016: (Start)
Intersection of dodecagonal numbers A051624 and centered heptagonal numbers A069099. A051624(j) = j(5j - 4), A069099(k) = (7*k^2 - 7^k + 2)/2, and the table below gives indices j and k at which A051624(j) = A069099(k):
.
  n               a(n)         j         k
  =  =================  ========  ========
  1                  1         1      0, 1
  2              10396        46        55
  3             326656       256       306
  4         2619897841     22891     27360
  5        82318050361    128311    153361
  6    660219495802336  11491036  13734415
  7  20744313326831116  64411666  76986666
  ... (End)

			From _Jon E. Schoenfield_, Dec 24 2016: (Start)
10396 is both the 46th dodecagonal number and the 55th centered heptagonal number: A051624(46) = 46(5*46 - 4) = 10396 and A069099(55) = (7*55^2 - 7*55 + 2)/2 = 10396.
A051624(256) = 256(5*256 - 4) = 326656 = (7*306^2 - 7*306 + 2)/2 = A069099(306). (End)

F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88-89.

Colin Barker, Table of n, a(n) for n = 1..350
Wikipedia, Polygonal number
Index entries for linear recurrences with constant coefficients, signature (1,252002,-252002,-1,1).

Cf. dodecagonal numbers A051624, centered heptagonal numbers A069099.

LinearRecurrence[{1,252002,-252002,-1,1},{1,10396,326656,2619897841,82318050361},20] (* Harvey P. Dale, Jul 06 2021 *)

Vec(x*(1 + 10395*x + 64258*x^2 + 10395*x^3 + x^4) / ((1 - x)*(1 - 502*x + x^2)*(1 + 502*x + x^2)) + O(x^20)) \\ Colin Barker, Dec 24 2016

Empirical: a(1)=1, a(2)=10396, a(3)=326656, a(4)=2619897841, a(n) = 252002*a(n-2) - a(n-4) + 85050 for n > 4. - Jon E. Schoenfield, Dec 24 2016

G.f.: x*(1 + 10395*x + 64258*x^2 + 10395*x^3 + x^4) / ((1 - x)*(1 - 502*x + x^2)*(1 + 502*x + x^2)). - Colin Barker, Dec 24 2016

cp's OEIS Frontend

A128919 Numbers simultaneously heptagonal and centered heptagonal.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A209294 a(n) = (7*n^2 - 7*n + 4)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A274588 Values of n such that 2*n-1 and 7*n-1 are both triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A338492 Least number of centered heptagonal numbers needed to represent n.

Views

Author

Keywords

Links

Crossrefs

A360183 Centered heptagonal numbers which are sphenic numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A141534 Derived from the centered polygonal numbers: start with the first triangular number, then the sum of the first square number and the second triangular number, then the sum of first pentagonal number, the second square number and the third triangular number, and so on and so on...

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A144975 Centered heptagonal twin prime numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A274682 Numbers n such that 8*n-1 is a triangular number.

A209294 a(n) = (7n^2 - 7n + 4)/2.

A274588 Values of n such that 2n-1 and 7n-1 are both triangular numbers.