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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A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2 + j^2).

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2017

Keywords

Examples

			Square array begins:
   1,   1,   1,   1,   1,   1, ...
  -2,  -1,  -1,  -1,  -1,  -1, ...
   0,  -1,   0,   0,   0,   0, ...
   0,   0,  -1,   0,   0,   0, ...
   2,   0,   0,  -1,   0,   0, ...
   0,   1,   0,   0,  -1,   0, ...

Links

Crossrefs

Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.
Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

Formula

G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).

A333641 11-gonal (or hendecagonal) square numbers.

Original entry on oeis.org

0, 1, 196, 29241, 1755625, 261468900, 38941102225, 2337990844401, 348201795147556, 51858411008887561, 3113535139359330841, 463705205422871375236, 69060571958250748760481, 4146338334574433921200225, 617522713934165528806340100, 91968930524758079223806760025
Offset: 1

Views

Author

Bernard Schott, Mar 31 2020

Keywords

Comments

The 11-gonal square numbers correspond to the nonnegative integer solutions of the Diophantine equation k*(9*k-7)/2 = m^2, equivalent to (18*k-7)^2 - 72*m^2 = 49. Substituting x = 18*k-7 and y = m gives the Pell equation x^2-72*y^2 = 49. The integer solutions (x,y) = (-7,0), (11,1), (119,14), (1451,171), (11243,1325), ... correspond to the following solutions (k,m) = (0,0), (1,1), (7,14), (81,171), (625,1325), ...

Examples

			1755625 is a term because 625*(9*625-7)/2 = 1325^2 = 1755625; that means that 1755625 is the 625th 11-gonal number and the square of 1325.

Links

Crossrefs

Intersection of A000290 (squares) and A051682 (11-gonals).
Cf. A106525.
Cf. A001110 (square triangulars), A036353 (square pentagonals), A046177 (square hexagonals), A036354 (square heptagonals), A036428 (square octagonals), A036411 (square 9-gonals), A188896 (only {0,1} are square 10-gonals), this sequence (square 11-gonals), A342709 (square 12-gonals).

Programs

  • Maple
    for k from 0 to 8000000 do
d:= k*(9*k-7)/2;
if issqr(d) then print(k,sqrt(d),d); else fi; od:
  • Mathematica
    Last /@ Solve[(18*x - 7)^2 - 72*y^2 == 49 && x >= 0 && y >= 0 && y < 10^16, {x, y}, Integers] /. Rule -> (#2^2 &) (* Amiram Eldar, Mar 31 2020 *)
  • PARI
    concat(0, Vec(-x*(1 + 195*x + 29045*x^2 + 394670*x^3 + 29045*x^4 + 195*x^5 + x^6)/(-1 + x + 1331714*x^3 - 1331714*x^4 - x^6 + x^7) + O(x^20))) \\ Jinyuan Wang, Mar 31 2020

Formula

a(n) = k*(9*k-7)/2 for n > 1, where k = (A106525(4*n-6) + 7)/18. - Jinyuan Wang, Mar 31 2020

Extensions

More terms from Amiram Eldar, Mar 31 2020

A354448 11-gonal numbers which are products of two distinct primes.

Original entry on oeis.org

58, 95, 141, 415, 1241, 2101, 2951, 3683, 6031, 7421, 16531, 24383, 35333, 39433, 42001, 50191, 53083, 66551, 83981, 95411, 123421, 146791, 173951, 182911, 190241, 229051, 296321, 307981, 336883, 409361, 442583, 451091, 477101, 500833, 546883, 588431, 669131
Offset: 1

Views

Author

Massimo Kofler, May 30 2022

Keywords

Comments

A squarefree subsequence of 11-gonal numbers, i.e., numbers of the form k*(9*k-7)/2.
Numbers of the form p*(9*p-7)/2 where p and (9*p-7)/2 are prime, and numbers of the form p*(18*p-7) where p and 18*p-7 are prime. - Robert Israel, Mar 03 2025

Examples

			    58 =  4*(9*4  - 7)/2 =  2*29;
   141 =  6*(9*6  - 7)/2 =  3*47;
   415 = 10*(9*10 - 7)/2 =  5*83;
  3683 = 29*(9*29 - 7)/2 = 29*127.

Links

Crossrefs

Intersection of A051682 and A006881.

Programs

  • Maple
    N:= 10^6: # for terms <= N
x1:= floor(fsolve(x*(9*x-7)/2=N)[2]):
A:= map(p -> p*(9*p-7)/2, select(p -> isprime(p) and isprime((9*p-7)/2), [seq(i,i=3..x1,2)])):
x2:= floor(fsolve(x*(18*x-7)=N)[2]):
B:= map(p -> p*(18*p-7), select(p -> isprime(p) and isprime(18*p-7),
  [2, seq(i,i=3..x2,2)])):
sort([op(A),op(B)]); # Robert Israel, Mar 03 2025
  • Mathematica
    Select[Table[n*(9*n - 7)/2, {n, 1, 400}], FactorInteger[#][[;; , 2]] == {1, 1} &] (* Amiram Eldar, May 30 2022 *)
  • Python
    from sympy import factorint
from itertools import count, islice
def agen():
    for h in (k*(9*k - 7)//2 for k in count(1)):
        f = factorint(h, multiple=True)
        if len(f) == len(set(f)) == 2: yield h
print(list(islice(agen(), 37))) # Michael S. Branicky, May 30 2022

A373615 Positive integers that cannot be written as a sum of a practical number and an 11-gonal number.

Original entry on oeis.org

10, 11, 14, 22, 26, 44, 45, 52, 63, 68, 69, 87, 92, 93, 116, 117, 152, 164, 172, 182, 188, 194, 233, 242, 243, 247, 248, 259, 279, 327, 374, 377, 402, 434, 482, 517, 579, 629, 712, 752, 759, 777, 824, 852, 934, 997, 1209, 1238, 1287, 1314, 1454, 1602, 1804
Offset: 1

Views

Author

Duc Van Khanh Tran, Jun 11 2024

Keywords

Comments

Somu and Tran (2024) conjectured that there are finitely many such integers. It was also conjectured that 105712 is the largest such integer. This conjecture was checked up to 10^8.

Links

Crossrefs

Cf. A005153, A051682.

A375583 a(n) is the number of ways n can be written as a sum of a practical number and two 11-gonal numbers.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 2, 3, 2, 1, 2, 2, 3, 2, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 4, 3, 5, 2, 3, 2, 3, 2, 3, 2, 3, 2, 6, 4, 2, 1, 2, 3, 3, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 4, 5, 5, 5, 2, 4, 4, 6, 4, 3, 1, 4, 5, 4, 4, 2, 3, 4, 4, 6, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 6, 7, 4, 4, 1, 4, 4, 6, 6
Offset: 1

Views

Author

Duc Van Khanh Tran, Aug 19 2024

Keywords

Comments

Somu and Tran (2024) proved that a(n) > 0 for sufficiently large n.
Conjecture (checked up to 10^8): a(n) = 0 if and only if n = 11.

Links

Crossrefs

Cf. A005153, A051682.

A096966 Triangle (read by rows) in which the number of entries in a row only increases by 1 every other row, the first column and the 'diagonal' is set to all 1's and a(i,j) = a(i-1,j) + a(i-1,j-1) + a(i-2,j-1) + a(i-3,j-1) for other entries.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 7, 1, 1, 10, 13, 1, 13, 34, 1, 1, 16, 64, 49, 1, 19, 103, 160, 1, 1, 22, 151, 361, 211, 1, 25, 208, 679, 781, 1, 1, 28, 274, 1141, 1981, 994, 1, 31, 349, 1774, 4162, 3967, 1, 1, 34, 433, 2605, 7756, 10891, 4963, 1, 37, 526, 3661, 13276, 24790, 20815, 1
Offset: 0

Views

Author

Gerald McGarvey, Aug 18 2004

Keywords

Comments

The 2nd column is A016777 (3n+1), the 3rd column is A081271 (Vertical of triangular spiral in A051682.)

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0
Offset: 1

Views

Author

Robert G. Wilson v, Apr 27 2020

Keywords

Comments

\c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
r\
_0 0 1 0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195 A067998
_1 0 1 1 0 -2 -5 -9 -14 -20 -27 -35 -44 -54 -65 -77 -90 A080956
_2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A001477
_3 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 A000217
_4 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 A000290
_5 0 1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 A000326
_6 0 1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 A000384
_7 0 1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 A000566
_8 0 1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 A000567
_9 0 1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 A001106
10 0 1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 A001107
11 0 1 11 30 58 95 141 196 260 333 415 506 606 715 833 960 A051682
12 0 1 12 33 64 105 156 217 288 369 460 561 672 793 924 1065 A051624
13 0 1 13 36 70 115 171 238 316 405 505 616 738 871 1015 1170 A051865
14 0 1 14 39 76 125 186 259 344 441 550 671 804 949 1106 1275 A051866
15 0 1 15 42 82 135 201 280 372 477 595 726 870 1027 1197 1380 A051867
...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

Links

Crossrefs

Cf. A317302 (the same array) but read by ascending antidiagonals.
Sub-arrays: A089000, A139600, A206735;
Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;
Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
Number of times k>1 appears: A129654, First occurrence of k: A063778.

Programs

  • Mathematica
    Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

Formula

P(r, c) = (r - 2)(c(c-1)/2) + c.

A354086 11-gonal numbers which are products of four distinct primes.

Original entry on oeis.org

4785, 8170, 11526, 14421, 27105, 30710, 38595, 59110, 60146, 77946, 94105, 107570, 118990, 120458, 121935, 132526, 140361, 141955, 156706, 158390, 161785, 181101, 199606, 203415, 213095, 215058, 217030, 221001, 243485, 249806, 267058, 287155, 298635, 303290
Offset: 1

Views

Author

Massimo Kofler, Jun 08 2022

Keywords

Comments

A squarefree subsequence of 11-gonal numbers, i.e., numbers of the form k*(9*k-7)/2.

Examples

			4785 = 33*(9*33-7)/2 = 3*5*11*29.
30710 = 83*(9*83-7)/2 = 2*5*37*83.
140361 = 177*(9*177-7)/2 = 3*13*59*61.
303290 = 260*(9*260-7)/2 = 2*5*13*2333.

Crossrefs

Intersection of A051682 and A046386.

Programs

  • Maple
    q:= n-> is(map(x-> x[2], ifactors(n)[2])=[1$4]):
select(q, [n*(9*n-7)/2$n=1..300])[];  # Alois P. Heinz, Jun 15 2022
  • Mathematica
    Select[Table[n*(9*n - 7)/2, {n, 1, 300}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Jun 08 2022 *)

A354446 11-gonal numbers which are products of three distinct primes.

Original entry on oeis.org

30, 506, 606, 715, 1558, 1730, 3945, 5083, 6365, 8558, 9361, 11986, 12455, 14935, 15458, 17081, 19371, 19966, 21183, 25726, 29971, 32215, 32981, 37766, 45551, 46461, 51146, 54065, 57065, 58083, 62245, 68758, 74433, 75595, 76766, 80333, 86458, 88971, 90241
Offset: 1

Views

Author

Massimo Kofler, Jun 01 2022

Keywords

Comments

A squarefree subsequence of 11-gonal numbers.

Examples

			     30 =   3*(9*3   - 7)/2 =  2 *  3 *   5;
    506 =  11*(9*11  - 7)/2 =  2 * 11 *  23;
   3945 =  30*(9*30  - 7)/2 =  3 *  5 * 263;
  80333 = 134*(9*134 - 7)/2 = 11 * 67 * 109.

Crossrefs

Intersection of A051682 and A007304.

Programs

  • Maple
    q:= n-> is(map(x-> x[2], ifactors(n)[2])=[1$3]):
select(q, [n*(9*n-7)/2$n=1..200])[];  # Alois P. Heinz, Jun 15 2022
  • Mathematica
    Select[Table[n*(9*n-7)/2, {n, 1, 150}], FactorInteger[#][[;; , 2]]=={1, 1, 1} &] (* Amiram Eldar, Jun 01 2022 *)
Previous Showing 61-69 of 69 results.

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