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 44 results. Next

A058351 Partial sums of A000084.

Original entry on oeis.org

0, 1, 3, 7, 17, 41, 107, 287, 809, 2341, 6965, 21101, 65031, 202939, 640441, 2039509, 6546861, 21158437, 68791923, 224839127, 738316629, 2434622357, 8058616301, 26765349429, 89173526191, 297942766331, 998072479961, 3351459203873
Offset: 0

Views

Author

N. J. A. Sloane, Dec 16 2000

Keywords

References

  • Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.

Links

Crossrefs

Cf. A000084, A006349, A006350, A058352, A058353, A000669.

A058352 a(n) Sum_{d|n, 1<=dA000084(d).

Original entry on oeis.org

0, 1, 1, 5, 1, 17, 1, 45, 13, 125, 1, 453, 1, 1265, 133, 4221, 1, 14201, 1, 46405, 1273, 155501, 1, 531789, 121, 1792809, 13801, 6126333, 1, 21032793, 1, 72121853, 155509, 248396797, 1381, 857944149, 1, 2964896881, 1792817, 10269600621, 1, 35628546989, 1
Offset: 0

Views

Author

N. J. A. Sloane, Dec 16 2000

Keywords

References

  • Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.

Crossrefs

Cf. A000084, A006349, A006350, A058351, A058353, A000669.

Programs

  • Maple
    A058352 := proc(n) local d,t1; t1 := 0; for d from 1 to n-1 do if n mod d = 0 then t1 := t1+d*A000084(d); fi; od; t1; end;

Extensions

More terms from Sean A. Irvine, Aug 04 2022

A058353 n*A000084(n).

Original entry on oeis.org

1, 4, 12, 40, 120, 396, 1260, 4176, 13788, 46240, 155496, 527160, 1792804, 6125028, 20986020, 72117632, 248396792, 857402748, 2964896876, 10269550040, 35622420288, 123727866768, 430254861944, 1497796242288, 5219231003500
Offset: 1

Views

Author

N. J. A. Sloane, Dec 16 2000

Keywords

References

  • Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.

Crossrefs

Cf. A000084, A006349, A006350, A058351, A058352, A000669.

A001573 Another approximation to A000084(n).

Original entry on oeis.org

1, 2, 4, 9, 23, 63, 177, 514, 1527, 4625, 14230, 44357, 139779, 444558, 1425151, 4600339, 14939849, 48778197, 160019885, 527200711
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Appears to have been calculated with very low precision. A093156 is a better version. - Pab Ter (pabrlos(AT)yahoo.com), May 11 2004

References

  • J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000084, A058585.

A058585 An approximation to A000084(n).

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 63, 177, 514, 1526, 4624, 14230, 44357, 139779, 444557, 1425151, 4600338, 14939849, 48778197, 160019884, 527200711, 1743607825, 5786756469, 19266336882, 64331275266, 215377539119, 722840378041, 2431459847363
Offset: 0

Views

Author

N. J. A. Sloane, Dec 26 2000

Keywords

References

  • J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.

Crossrefs

Cf. A000084, A001573, A093156.

Formula

a(n) = floor(u(n)) where u(n) has g.f. (1/2)*(5-3*x-2*x^2-sqrt(9-30*x-11*x^2+12*x^3+4*x^4)).

A093156 Another approximation to A000084(n).

Original entry on oeis.org

1, 2, 4, 9, 24, 63, 177, 514, 1527, 4625, 14230, 44358, 139779, 444558, 1425151, 4600339, 14939849, 48778197, 160019885, 527200711, 1743607826, 5786756470, 19266336882, 64331275266, 215377539119, 722840378042, 2431459847364
Offset: 1

Views

Author

Pab Ter (pabrlos(AT)yahoo.com), May 11 2004

Keywords

Comments

A more accurate version of A001573.

References

  • J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.

Crossrefs

Cf. A000084, A058585.

Formula

a(n) is the nearest integer to u(n) (with round(0.5)=0), where u(n) has o.g.f.: U(x) = (5 - 3*x - 2*x^2 - sqrt(9 - 30*x - 11*x^2 + 12*x^3 + 4*x^4))/2. [Riordan & Shannon]

A144962 Eigentriangle, row sums = A000084.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 2, 4, 5, 3, 2, 4, 10, 17, 5, 6, 4, 10, 24, 41, 17, 10, 12, 10, 24, 66, 127, 41, 34, 20, 30, 24, 66, 180, 365, 127, 82, 68, 50, 72, 66, 180, 522, 1119, 365, 254, 164, 170, 120, 198, 180, 522, 1532
Offset: 1

Views

Author

Gary W. Adamson, Sep 27 2008

Keywords

Comments

Row sums = A000084: (1, 2, 4, 10, 24, 66,...).
Right border = A000084 shifted: (1, 1, 2, 4, 10, 24,...)
Left border = A001572: (1, 1, 1, 3, 5, 17, 41,...).
A000084 = the INVERT transform of A001572.
Sum of n-th row terms = rightmost term of next row.

Examples

			First few rows of the triangle =
1;
1, 1;
1, 1, 2;
3, 1, 2, 4;
5, 3, 2, 4, 10;
17, 5, 6, 4, 10, 24;
41, 17, 10, 12, 10, 24, 66;
127, 41, 34, 20, 30, 24, 66, 180;
365, 127, 82, 68, 50, 72, 66, 180, 522;
1119, 365, 254, 164, 170, 120, 198, 180, 522, 1532;
...
Example: row 5 = (5, 3, 2, 4, 10) = termwise products of (5, 3, 1, 1, 1) and (1, 1, 2, 4, 10).

Crossrefs

A000084, Cf. A001572

Formula

Triangle read by rows, T(n,k) = A001572(n-k+1) * (A000084 * 0^(n-k)), 1<=k<=n.
Given an A001572 "decrescendo" triangle: (1; 1,1; 1,1,1; 3,1,1,1; 5,3,1,1,1;...), where A001572 begins: (1, 1, 1, 3, 5, 17, 41, 127,...); apply termwise products of the decrescendo triangle row terms to A000084 terms: (1, 2, 4, 10, 24, 66, 180, 522,...).

A058756 G.f. is (1-S)/(1+S), where S = g.f. for A000084.

Original entry on oeis.org

1, -2, -2, -2, -6, -10, -34, -82, -254, -730, -2238, -6826, -21370, -67122, -213654, -684258, -2208694, -7169298, -23402738, -76748130, -252790518, -835816658, -2773236614, -9230776706, -30814377058, -103139338858, -346067984622, -1163810570178
Offset: 0

Views

Author

N. J. A. Sloane, Jan 01 2001

Keywords

Crossrefs

Cf. A036655, A000137.

A058757 a(n) = n*coefficient of x^n in log(1+S), S = g.f. for A000084.

Original entry on oeis.org

1, 3, 7, 23, 61, 207, 631, 2111, 6901, 23183, 77749, 263807, 896403, 3063147, 10493077, 36060927, 124198397, 428708475, 1482448439, 5134798223, 17811210781, 61864011135, 215127430973, 748898387039, 2609615501811, 9101687173595
Offset: 1

Views

Author

N. J. A. Sloane, Jan 01 2001

Keywords

Comments

Useful for computing A000669 and A000084.

Crossrefs

Cf. A000084, A000669.

Programs

  • Maple
    For Maple code see A000669, method 2.

A057734 Erroneous version of A000084.

Original entry on oeis.org

1, 2, 4, 10, 24, 66, 180, 522, 1532, 4984
Offset: 1

Views

Author

Keywords

References

  • P. A. MacMahon, The combination of resistances, The Electrician, 28 (1892), 601-602; reprinted in Coll. Papers I, pp. 617-619.
Showing 1-10 of 44 results. Next

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

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