cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A203627 Numbers which are both 9-gonal (nonagonal) and 10-gonal (decagonal).

Original entry on oeis.org

1, 1212751, 977965238701, 788633124418157851, 635955328796073362530201, 512835649051022518566661395751, 413551693065406705688396809494274501, 333488912390817262631483541451235285166451, 268926125929366270527488184087670639619302551601
Offset: 1

Views

Author

Ant King, Jan 06 2012

Keywords

Comments

As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity, a(n)/a(n-1)) = (2*sqrt(2)+sqrt(7))^8 = 403201+107760*sqrt(14).

Examples

			The second number that is both nonagonal and decagonal is A001106(589) = A001107(551) = 1212751. Hence a(2) = 1212751.

Links

Crossrefs

Cf. A203628, A203629, A001107, A001106.

Programs

  • Mathematica
    LinearRecurrence[{806403, -806403, 1}, {1, 1212751, 977965238701}, 9]

Formula

G.f.: x*(1+406348*x+451*x^2) / ((1-x)*(1-806402*x+x^2)).
a(n) = 806402*a(n-1)-a(n-2)+406800.
a(n) = 806403*a(n-1)-806403*a(n-2)+a(n-3).
a(n) = 1/448*((15+2*sqrt(14))*(2*sqrt(2)+sqrt(7))^(8*n-6)+(15-2*sqrt(14))*(2*sqrt(2)-sqrt(7))^(8*n-6)-226).
a(n) = floor(1/448*(15+2*sqrt(14))*(2*sqrt(2)+sqrt(7))^(8*n-6)).

A139268 Twice nonagonal numbers (or twice 9-gonal numbers): a(n) = n*(7*n-5).

Original entry on oeis.org

0, 2, 18, 48, 92, 150, 222, 308, 408, 522, 650, 792, 948, 1118, 1302, 1500, 1712, 1938, 2178, 2432, 2700, 2982, 3278, 3588, 3912, 4250, 4602, 4968, 5348, 5742, 6150, 6572, 7008, 7458, 7922, 8400, 8892, 9398, 9918, 10452, 11000
Offset: 0

Views

Author

Omar E. Pol, May 15 2008

Keywords

Links

Crossrefs

Cf. A001106, A051868.
Cf. numbers of the form n*(n*k - k + 4)/2 listed in A226488 (this sequence is the case k=14). - Bruno Berselli, Jun 10 2013

Programs

Formula

a(n) = 2*A001106(n) = 7*n^2 - 5*n = n*(7*n-5).
a(n) = 14*n + a(n-1) - 12, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 2*x*(1 + 6*x)/(1 - x)^3. - Philippe Deléham, Apr 03 2013
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(2 + 7*x).
a(n) = n + A051868(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A226451 a(n) = n*(7*n^2-12*n+7)/2.

Original entry on oeis.org

0, 1, 11, 51, 142, 305, 561, 931, 1436, 2097, 2935, 3971, 5226, 6721, 8477, 10515, 12856, 15521, 18531, 21907, 25670, 29841, 34441, 39491, 45012, 51025, 57551, 64611, 72226, 80417, 89205, 98611, 108656, 119361, 130747, 142835, 155646, 169201, 183521
Offset: 0

Views

Author

Bruno Berselli, Jun 07 2013

Keywords

Comments

See the comment in A226449.

Links

Crossrefs

Cf. A001106.
Similar sequences of the type b(m)+m*b(m-1), where b is a polygonal number: A006003, A069778, A143690, A204674, A212133, A226449, A226450.

Programs

  • Magma
    [n*(7*n^2-12*n+7)/2: n in [0..40]];
  • Magma
    I:=[0,1,11,51]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 18 2013
  • Mathematica
    Table[n (7 n^2 - 12 n + 7)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 7 x + 13 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: x*(1+7*x+13*x^2)/(1-x)^4.
a(n) = A001106(n) + n*A001106(n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Oct 15 2023

A227321 a(n) is the least r>=3 such that the difference between the nearest r-gonal number >= n and n is an r-gonal number.

Original entry on oeis.org

3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 5, 3, 8, 3, 3, 4, 5, 3, 11, 3, 3, 3, 5, 4, 3, 10, 3, 3, 11, 3, 17, 4, 3, 5, 3, 3, 7, 14, 3, 4, 15, 3, 23, 3, 3, 5, 11, 4, 3, 5, 5, 3, 19, 3, 3, 3, 8, 5, 21, 3, 32, 14, 3, 4, 3, 3, 15, 3, 5, 5, 25, 3, 38, 7, 3, 6, 3, 3, 13, 4, 3
Offset: 0

Views

Author

Vladimir Shevelev, Jul 30 2013

Keywords

Comments

The n-th r-gonal numbers is n((n-1)r-2(n-2))/2, such that 3-gonal numbers are triangular numbers, 4-gonal numbers are squares, etc.

Links

Crossrefs

Cf. A000217 (r=3), A000290 (r=4), A000326 (r=5), A000384 (r=6), A000566 (r=7), A000567 (r=8), A001106-7 (r=9,10), A051682 (r=11), A051624 (r=12), A051865-A051876 (r=13-24).

Programs

  • Mathematica
    rGonalQ[r_,0]:=True; rGonalQ[r_,n_]:=IntegerQ[(Sqrt[((8r-16)n+(r-4)^2)]+r-4)/(2r-4)]; nthrGonal[r_,n_]:=(n (r-2)(n-1))/2+n; nextrGonal[r_,n_]:=nthrGonal[r,Ceiling[(Sqrt[((8r-16)n+(r-4)^2)]+r-4)/(2r-4)]]; (* next r-gonal number greater than or equal to n *) Table[NestWhile[#+1&,3,!rGonalQ[#,nextrGonal[#,n]-n]&],{n,0,99}] (* Peter J. C. Moses, Aug 03 2013 *)

Formula

If n is prime, then n == 1 or 2 mod (a(n)-2). If n >= 13 is the greater of a pair of twin primes (A006512), then a(n) = (n+3)/2. - Vladimir Shevelev, Aug 07 2013

Extensions

More terms from Peter J. C. Moses, Jul 30 2013

A236257 a(n) = 2*n^2 - 7*n + 9.

Original entry on oeis.org

9, 4, 3, 6, 13, 24, 39, 58, 81, 108, 139, 174, 213, 256, 303, 354, 409, 468, 531, 598, 669, 744, 823, 906, 993, 1084, 1179, 1278, 1381, 1488, 1599, 1714, 1833, 1956, 2083, 2214, 2349, 2488, 2631, 2778, 2929, 3084, 3243, 3406, 3573, 3744, 3919, 4098, 4281, 4468
Offset: 0

Views

Author

Vladimir Shevelev, Jan 21 2014

Keywords

Comments

If zero polygonal numbers are ignored, then for n >= 3, the a(n)-th n-gonal number is a sum of the (a(n)-1)-th n-gonal number and the (2*n-3)-th n-gonal number.

Examples

			a(7)=58. This means that the 58th heptagonal number 8323 (cf. A000566) is a sum of two heptagonal numbers. We have 8323 = 8037 + 286 with indices in A000566 58,57,11.

Links

Crossrefs

Cf. A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051624, A051682, A051865-A051876, A152948.

Programs

  • Mathematica
    Table[2 n^2 - 7 n + 9, {n, 0, 48}] (* Michael De Vlieger, Apr 19 2015 *)
LinearRecurrence[{3,-3,1},{9,4,3},50] (* Harvey P. Dale, Nov 24 2017 *)
  • PARI
    Vec(-(18*x^2-23*x+9)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jan 21 2014

Formula

From Colin Barker, Jan 21 2014: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(18*x^2 - 23*x + 9)/(x-1)^3. (End)
E.g.f.: exp(x)*(9 - 5*x + 2*x^2). - Elmo R. Oliveira, Nov 13 2024

A307829 Base numbers of decagonal (10-gonal) palindromic numbers.

Original entry on oeis.org

0, 1, 8, 84, 139, 2528, 42293, 198891, 712178, 714154, 826684, 3628625, 12999736, 84439174, 135593913, 136500523, 2527472528, 2637951184, 3960451966, 4094127596, 4415308953, 5192254461
Offset: 1

Views

Author

Robert Price, Apr 30 2019

Keywords

Links

Crossrefs

Cf. A001106, A082723, A055560.

Programs

  • Mathematica
    Select[Range[0, 10^5], PalindromeQ[PolygonalNumber[10, #]] &]

A028991 Odd 9-gonal (or enneagonal) numbers.

Original entry on oeis.org

1, 9, 75, 111, 261, 325, 559, 651, 969, 1089, 1491, 1639, 2125, 2301, 2871, 3075, 3729, 3961, 4699, 4959, 5781, 6069, 6975, 7291, 8281, 8625, 9699, 10071, 11229, 11629, 12871, 13299, 14625, 15081, 16491, 16975, 18469, 18981, 20559, 21099, 22761, 23329, 25075
Offset: 0

Views

Author

Patrick De Geest

Keywords

Links

Crossrefs

Cf. A001106, A028992.

Programs

  • Mathematica
    Select[PolygonalNumber[9,Range[100]],OddQ] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,9,75,111,261},50] (* Harvey P. Dale, Dec 27 2020 *)
  • PARI
    Vec(-(19*x^4+20*x^3+64*x^2+8*x+1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, May 30 2015

Formula

a(n) = (28*n^2 + 4*n + 1 + (14*n+1)*(-1)^n)/2.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>4. - Colin Barker, May 30 2015
G.f.: -(19*x^4+20*x^3+64*x^2+8*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, May 30 2015

A028992 Even 9-gonal (or enneagonal) numbers.

Original entry on oeis.org

0, 24, 46, 154, 204, 396, 474, 750, 856, 1216, 1350, 1794, 1956, 2484, 2674, 3286, 3504, 4200, 4446, 5226, 5500, 6364, 6666, 7614, 7944, 8976, 9334, 10450, 10836, 12036, 12450, 13734, 14176, 15544, 16014, 17466, 17964, 19500, 20026, 21646, 22200, 23904
Offset: 0

Views

Author

Patrick De Geest

Keywords

Links

Crossrefs

Cf. A001106.
Cf. A014642, A028991, A028994.

Programs

  • Magma
    [(1/2)*(28*(n-1)^2 + 60*(n-1) + 33 + (14*(n-1)+15)*(-1)^(n-1)): n in [0..40]]; // Vincenzo Librandi, Aug 19 2011
  • PARI
    concat(0, Vec(-2*x*(3*x^3+30*x^2+11*x+12)/((x-1)^3*(x+1)^2) + O(x^100))) \\ Colin Barker, May 30 2015

Formula

a(n) = (1/2)*(28*(n-1)^2 + 60*(n-1) + 33 + (14*(n-1)+15)*(-1)^(n-1)).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>4. - Colin Barker, May 30 2015
G.f.: -2*x*(3*x^3+30*x^2+11*x+12) / ((x-1)^3*(x+1)^2). - Colin Barker, May 30 2015

Extensions

0 inserted, offset and formula corrected by Omar E. Pol, Aug 19 2011

A117053 Enneagonal numbers for which both the sum of the digits and the product of the digits are also enneagonal numbers.

Original entry on oeis.org

0, 1, 9, 1350, 10071, 39804, 46806, 66309, 80484, 175056, 204369, 226950, 235950, 260169, 305916, 450186, 460284, 473064, 556206, 570246, 581604, 676500, 704481, 733029, 822075, 835701, 930606, 1015476, 1065084, 1155750, 1208634, 1305096
Offset: 0

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 15 2006

Keywords

Comments

An enneagopnal number is also called a nonagonal number. - Harvey P. Dale, Apr 13 2020

Examples

			39804 is in the sequence because (1) it is an enneagonal number,(2)the sum of its digits 3+9+8+0+4=24 is an enneagonal number and (3)the product of its digits 3*9*8*0*4=0 is also an enneagonal number.

Links

Crossrefs

Cf. A001106.

Programs

  • Mathematica
    With[{nnn=Table[(n(7n-5))/2,{n,0,1000}]},Select[nnn,MemberQ[nnn,Total[ IntegerDigits[ #]]]&&MemberQ[nnn,Times@@IntegerDigits[#]]&]] (* Harvey P. Dale, Apr 13 2020 *)

A131875 Triangle, A000012 * A131844 as infinite lower triangular matrices.

Original entry on oeis.org

1, 5, 4, 12, 5, 7, 22, 6, 8, 10, 35, 7, 9, 11, 13, 51, 8, 10, 12, 14, 16, 70, 9, 11, 13, 15, 17, 19, 92, 10, 12, 14, 16, 18, 20, 22, 117, 11, 13, 15, 17, 19, 21, 23, 25, 145, 12, 14, 16, 18, 20, 22, 24, 26, 28
Offset: 0

Views

Author

Gary W. Adamson, Jul 22 2007

Keywords

Comments

Left column = pentagonal numbers, A000326: (1, 5, 12, 22, ...).
Row sums = A001106: (1, 9, 24, 46, 75, 111, ...).

Examples

			First few rows of the triangle:
   1;
   5,  4;
  12,  5,  7;
  22,  6,  8, 10;
  35,  7,  9, 11, 13;
  51,  8, 10, 12, 14, 16;
  70,  9, 11, 13, 15, 17, 19;
  92, 10, 12, 14, 16, 18, 20, 22;
  ...

Crossrefs

Cf. A131844, A001106, A000326.
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