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.

Showing 1-10 of 11 results. Next

A049628 a(n) = Sum_{i=0..n} T(i,n-i) where T is A049627.

Original entry on oeis.org

1, 4, 8, 14, 22, 34, 48, 68, 92, 122, 156, 200, 248, 308, 374, 448, 530, 628, 732, 854, 984, 1126, 1278, 1452, 1634, 1836, 2050, 2282, 2526, 2798, 3078, 3388, 3714, 4060, 4422, 4808, 5206, 5640, 6092, 6568, 7060, 7592, 8136, 8722, 9328, 9958, 10610, 11308
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049627.
Partial sums of A225531.

Programs

  • PARI
    T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1)); \\ A049627
a(n) = sum(i=0, n, T(i,n-i)); \\ Michel Marcus, Aug 06 2021

Extensions

More terms from Sean A. Irvine, Aug 05 2021

A049630 a(n) = Sum_{i=0..floor(n/2)} T(2i,n-2i) where T is A049627.

Original entry on oeis.org

1, 2, 4, 7, 10, 17, 22, 34, 42, 61, 72, 100, 116, 154, 176, 224, 250, 314, 348, 427, 470, 563, 612, 726, 786, 918, 988, 1141, 1220, 1399, 1492, 1694, 1802, 2030, 2148, 2404, 2534, 2820, 2968, 3284, 3444, 3796, 3976, 4361, 4562, 4979, 5192, 5654, 5890, 6389
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049627.

Programs

  • PARI
    T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1)); \\ A049627
a(n) = sum(i=0, n\2, T(2*i,n-2*i)); \\ Michel Marcus, Aug 06 2021

Extensions

More terms from Sean A. Irvine, Aug 05 2021

A049631 a(n) = Sum_{i=0..floor(n/2)} T(2i+1,n-2i-1) where T is A049627.

Original entry on oeis.org

0, 2, 4, 7, 12, 17, 26, 34, 50, 61, 84, 100, 132, 154, 198, 224, 280, 314, 384, 427, 514, 563, 666, 726, 848, 918, 1062, 1141, 1306, 1399, 1586, 1694, 1912, 2030, 2274, 2404, 2672, 2820, 3124, 3284, 3616, 3796, 4160, 4361, 4766, 4979, 5418, 5654, 6132, 6389
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049627.

Programs

  • PARI
    T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1)); \\ A049627
a(n) = sum(i=0, n\2, T(2*i+1, n-2*i-1)); \\ Michel Marcus, Aug 06 2021

Extensions

More terms and name corrected by Sean A. Irvine, Aug 05 2021

A049632 a(n) = T(n,n), array T as in A049627.

Original entry on oeis.org

1, 4, 6, 10, 14, 22, 26, 38, 46, 58, 66, 86, 94, 118, 130, 146, 162, 194, 206, 242, 258, 282, 302, 346, 362, 402, 426, 462, 486, 542, 558, 618, 650, 690, 722, 770, 794, 866, 902, 950, 982, 1062, 1086, 1170, 1210, 1258, 1302, 1394, 1426, 1510, 1550, 1614, 1662
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049627, A049643, A005728.

Programs

  • PARI
    T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1)); \\ A049627
a(n) = T(n, n); \\ Michel Marcus, Aug 06 2021

Formula

a(n) = 2*A005728(n) if n>=1. - R. J. Mathar, Feb 05 2008

Extensions

More terms from Sean A. Irvine, Aug 05 2021

A049633 a(n) = T(n,n+1), array T as in A049627.

Original entry on oeis.org

2, 5, 8, 12, 18, 24, 32, 42, 52, 62, 76, 90, 106, 124, 138, 154, 178, 200, 224, 250, 270, 292, 324, 354, 382, 414, 444, 474, 514, 550, 588, 634, 670, 706, 746, 782, 830, 884, 926, 966, 1022, 1074, 1128, 1190, 1234, 1280, 1348, 1410, 1468, 1530, 1582, 1638
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049627.

Programs

  • PARI
    T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1)); \\ A049627
a(n) = T(n, n+1); \\ Michel Marcus, Aug 06 2021

Extensions

More terms from Sean A. Irvine, Aug 05 2021

A049634 a(n) = T(n,n+2), array T as in A049627.

Original entry on oeis.org

2, 6, 9, 15, 19, 29, 35, 47, 55, 71, 79, 101, 111, 131, 145, 169, 183, 217, 231, 261, 279, 313, 331, 373, 393, 431, 455, 501, 521, 579, 603, 653, 685, 729, 757, 817, 847, 907, 941, 1005, 1033, 1115, 1147, 1213, 1255, 1325, 1363, 1451, 1487, 1561, 1605, 1689
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049627.

Programs

  • PARI
    T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1)); \\ A049627
a(n) = T(n, n+2); \\ Michel Marcus, Aug 06 2021

Extensions

More terms from Sean A. Irvine, Aug 05 2021

A049635 a(n) = T(n,n+3), array T as in A049627.

Original entry on oeis.org

2, 7, 11, 16, 23, 32, 39, 50, 63, 74, 89, 106, 117, 138, 159, 174, 199, 224, 241, 270, 299, 320, 349, 384, 409, 442, 481, 508, 549, 594, 621, 668, 707, 740, 791, 834, 869, 922, 979, 1016, 1073, 1134, 1169, 1234, 1299, 1340, 1403, 1470, 1517, 1584, 1655, 1706
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049627.

Programs

  • PARI
    T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1)); \\ A049627
a(n) = T(n, n+3); \\ Michel Marcus, Aug 06 2021

Extensions

More terms from Sean A. Irvine, Aug 05 2021

A049638 a(n) = T(n,4), array T as in A049627.

Original entry on oeis.org

2, 7, 9, 12, 14, 18, 19, 23, 25, 28, 30, 34, 35, 39, 41, 44, 46, 50, 51, 55, 57, 60, 62, 66, 67, 71, 73, 76, 78, 82, 83, 87, 89, 92, 94, 98, 99, 103, 105, 108, 110, 114, 115, 119, 121, 124, 126, 130, 131, 135, 137, 140, 142, 146, 147, 151, 153, 156, 158, 162
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A049627.

Programs

  • PARI
    T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1)); \\ A049627
a(n) = T(n, 4); \\ Michel Marcus, Aug 06 2021

Formula

Conjectured g.f.: 2 -x*(-7-9*x-5*x^2+2*x^3+3*x^4) / ( (1+x)*(1+x+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 26 2015

Extensions

More terms from Sean A. Irvine, Aug 06 2021

A199408 Triangle T(n,k) = n + k - gcd(n,k) read by rows, 1 <= n, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 4, 6, 4, 5, 6, 7, 8, 5, 6, 6, 6, 8, 10, 6, 7, 8, 9, 10, 11, 12, 7, 8, 8, 10, 8, 12, 12, 14, 8, 9, 10, 9, 12, 13, 12, 15, 16, 9, 10, 10, 12, 12, 10, 14, 16, 16, 18, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 11, 12, 12, 12, 12, 16, 12
Offset: 1

Views

Author

Brian Hopkins, Nov 05 2011

Keywords

Comments

A diagonal of an n by k rectangle drawn on a square grid passes through T(n,k) squares: the diagonal enters n squares crossing horizontal segments and enters k squares crossing vertical segments. Gcd(n,k) counts the squares entered at a lattice point, which have been over-counted.

Examples

			T(6,4) = 6 + 4 - 2 = 8.
Triangular array begins
  1
  2  2
  3  4  3
  4  4  6  4
  5  6  7  8  5
  6  6  6  8 10  6
  7  8  9 10 11 12  7
  8  8 10  8 12 12 14  8

References

  • M. Ollerton, Mathematics Teacher's Handbook, Continuum, 2009, pp. 14-15.

Links

Crossrefs

Cf. A049627, A074712. Third column A061800.

Programs

Formula

T(d*a,d*b) = d*T(a,b).

A049636 Congruent to 0 or 2 mod 3, but not equal to 0 or 3.

Original entry on oeis.org

2, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

a(n) = T(n, 2), array T as in A049627.

Links

Crossrefs

Cf. A007494.

Programs

  • Mathematica
    Join[{2},Select[Range[4,100],MemberQ[{0,2},Mod[#,3]]&]] (* or *) With[{nn=100},Join[{2},Riffle[Range[5,nn,3],Range[6,nn,3]]]] (* Harvey P. Dale, Jul 14 2012 *)
  • PARI
    x='x+O('x^20); Vec(-(x+2)*(x^2-x-1)/((1-x)^2*(1+x))) \\ G. C. Greubel, Dec 15 2017

Formula

O.g.f.: -(x+2)(x^2-x-1)/((1-x)^2*(1+x)). - R. J. Mathar, Aug 13 2008

Extensions

Terms a(38) onward added by G. C. Greubel, Dec 15 2017
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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