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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A323178 a(n) = 1 + 100*n^2 for n >= 0.

Original entry on oeis.org

1, 101, 401, 901, 1601, 2501, 3601, 4901, 6401, 8101, 10001, 12101, 14401, 16901, 19601, 22501, 25601, 28901, 32401, 36101, 40001, 44101, 48401, 52901, 57601, 62501, 67601, 72901, 78401, 84101, 90001, 96101, 102401, 108901
Offset: 0

Views

Author

Paul Curtz, Jan 06 2019

Keywords

Comments

Terms of A261327 ending in 1 (01 for n > 0.)
a(n) mod 9 = period 9: repeat [1, 2, 5, 1, 8, 8, 1, 5, 2] = A275704(n+3).
(Analogous sequence: b(n) = 29 + 100*n*(n+1) = A261327(A017329) = 29, 229, 629, ... .)

Crossrefs

Cf. A000290, A008602 (20*n), A261327, A275704.
Subsequence of A017281.

Programs

  • Mathematica
    a[n_] := 1 + 100*n^2 ; Array[a, 50, 0] (* or *)
CoefficientList[Series[(-1 - 98 x - 101 x^2)/(-1 + x)^3, {x, 0, 50}], x] (* or *)
CoefficientList[Series[E^x (1 + 100 x + 100 x^2), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* Stefano Spezia, Jan 06 2019 *)

Formula

a(n) = A261327(A008602(n)).
Recurrence: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2 with initial values a(0) = 1, a(1) = 101 and a(2) = 401.
From Stefano Spezia, Jan 06 2019: (Start)
O.g.f.: (-1 - 98*x - 101*x^2)/(-1 + x)^3.
E.g.f.: exp(x)*(1 + 100*x + 100*x^2).
(End)

Extensions

Corrected and extended (recurrence formula) by Werner Schulte, Feb 18 2019

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.

A384735 Numbers that are prime or end in a prime number (of any length).

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 15, 17, 19, 22, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 42, 43, 45, 47, 52, 53, 55, 57, 59, 61, 62, 63, 65, 67, 71, 72, 73, 75, 77, 79, 82, 83, 85, 87, 89, 92, 93, 95, 97, 101, 102, 103, 105, 107, 109, 111, 112, 113, 115, 117
Offset: 1

Views

Author

Mohd Anwar Jamal Faiz, Jun 08 2025

Keywords

Comments

If k is a term, so is m*10^A055642(k) + k for all m > 0. - Michael S. Branicky, Jun 10 2025

Examples

			2, 3, 5, 7 and 11 are terms since they are prime.
15 is a term since it ends in the prime 5.
111 is a term since it ends in the prime 11.

Links

Crossrefs

Cf. A000040, A033664, A055642.
Contains A017293, A017305, A017329, A017353.

Programs

  • Maple
    q:= n-> isprime(n) or (k-> k>1 and q(n mod 10^(k-1)))(length(n)):
select(q, [$1..150])[];  # Alois P. Heinz, Jun 08 2025
  • Mathematica
    q[n_] := AnyTrue[Range[1, IntegerLength[n]-1], PrimeQ[Mod[n, 10^#]] &]; Select[Range[120], PrimeQ[#] || q[#] &] (* Amiram Eldar, Jun 10 2025 *)
  • PARI
    isok(x) = my(y=x, nb=0); while(y>1, y/=10; nb++; if (isprime(x%(10^nb)), return(1))); \\ Michel Marcus, Jun 10 2025
  • Python
    from sympy import isprime
def ok(n):
    s = str(n)
    return any(isprime(int(s[i:])) for i in range(len(s)))
print([k for k in range(118) if ok(k)])
Previous Showing 61-63 of 63 results.

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